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An operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function, usually called the symbol of the pseudo-differential operator, that satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.
 
An operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function, usually called the symbol of the pseudo-differential operator, that satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756601.png" /> be an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756602.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756603.png" /> be the space of infinitely-differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756604.png" /> with compact support belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756605.png" />. The simplest pseudo-differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756606.png" /> is the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756607.png" /> given by
+
Let $  \Omega $
 +
be an open set in $  \mathbf R  ^ {n} $,  
 +
and let $  C _ {0}  ^  \infty  ( \Omega ) $
 +
be the space of infinitely-differentiable functions on $  \Omega $
 +
with compact support belonging to $  \Omega $.  
 +
The simplest pseudo-differential operator on $  \Omega $
 +
is the operator  $  P :  C _ {0}  ^  \infty  ( \Omega ) \rightarrow C  ^  \infty  ( \Omega ) $
 +
given by
 +
 
 +
$$ \tag{1 }
 +
P u ( x)  = \
 +
 
 +
\frac{1}{( 2 \pi )  ^ {n} }
 +
 
 +
\int\limits e ^ {i {x \cdot \xi } } p ( x , \xi ) \widehat{u}  ( \xi )  d \xi .
 +
$$
 +
 
 +
Here,  $  u \in C _ {0}  ^  \infty  ( \Omega ) $,
 +
$  \xi \in \mathbf R  ^ {n} $,
 +
$  d \xi $
 +
is Lebesgue measure on  $  \mathbf R  ^ {n} $,
 +
$  x \cdot \xi $
 +
is the usual inner product of the vectors  $  x $
 +
and  $  \xi $,
 +
$  \widehat{u}  ( \xi ) $
 +
is the [[Fourier transform|Fourier transform]] of the function  $  u $,
 +
i.e.
 +
 
 +
$$
 +
\widehat{u}  ( \xi )  =  \int\limits e ^ {- i x \cdot \xi } u ( x )  d x
 +
$$
 +
 
 +
(the integral, like the one in (1), is over all of  $  \mathbf R  ^ {n} $),
 +
and  $  p ( x , \xi ) $
 +
is a smooth function on  $  \Omega \times \mathbf R  ^ {n} $
 +
satisfying certain conditions and is called the symbol of the pseudo-differential operator $  P $(
 +
cf. also [[Symbol of an operator|Symbol of an operator]]). An operator  $  P $
 +
of the form (1) is denoted by  $  p ( x , D ) $
 +
or  $  p ( x , D _ {x} ) $.
 +
If
 +
 
 +
$$
 +
p ( x , \xi )  = \
 +
\sum _ {| \alpha | \leq  m }
 +
p _  \alpha  ( x) \xi  ^  \alpha
 +
$$
 +
 
 +
is a polynomial in  $  \xi $
 +
with coefficients  $  p _  \alpha  \in C _ {0}  ^  \infty  ( \Omega ) $(
 +
here  $  \alpha $
 +
is a multi-index, i.e.  $  \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $,
 +
$  \alpha _ {j} \geq  0 $,
 +
$  \alpha _ {j} $
 +
are integers,  $  | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $,
 +
$  \xi  ^  \alpha  = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $),
 +
then  $  p ( x , D ) $
 +
coincides with the [[Differential operator|differential operator]] obtained when  $  D = \partial  / i \partial  x $
 +
is substituted for  $  \xi $
 +
in the expression for  $  p ( x , \xi ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
One often uses the class of symbols  $  p ( x , \xi ) \in C  ^  \infty  ( \Omega \times \mathbf R  ^ {n} ) $
 +
satisfying the conditions
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p0756609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566011.png" /> is Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566013.png" /> is the usual inner product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566016.png" /> is the [[Fourier transform|Fourier transform]] of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566017.png" />, i.e.
+
$$ \tag{2 }
 +
| \partial  _  \xi  ^  \alpha  \partial  _ {x}  ^  \beta  p ( x , \xi ) |
 +
\leq  C _ {\alpha , \beta , {\mathcal K} }  \
 +
( 1 + | \xi | ) ^ {m - \rho | \alpha | + \delta | \beta | } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566018.png" /></td> </tr></table>
+
$$
 +
x  \in  {\mathcal K} ,\  \xi  \in  \mathbf R  ^ {n} .
 +
$$
  
(the integral, like the one in (1), is over all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566019.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566020.png" /> is a smooth function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566021.png" /> satisfying certain conditions and is called the symbol of the pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566022.png" /> (cf. also [[Symbol of an operator|Symbol of an operator]]). An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566023.png" /> of the form (1) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566025.png" />. If
+
Here  $  \alpha , \beta $
 +
are multi-indices, $  \partial  _ {x} = \partial  / \partial  x $,
 +
$  \partial  _  \xi  = \partial  / \partial  \xi $,  
 +
and $  {\mathcal K} $
 +
is a compact set in  $  \Omega $.  
 +
This class is denoted by  $  S _ {\rho , \delta }  ^ {m} $(
 +
or by  $  S _ {\rho , \delta }  ^ {m} ( \Omega \times \mathbf R  ^ {n} ) $).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566026.png" /></td> </tr></table>
+
It is usually assumed that  $  0 \leq  \rho , \delta \leq  1 $.
 +
By  $  L _ {\rho , \delta }  ^ {m} $(
 +
or  $  L _ {\rho , \delta }  ^ {m} ( \Omega ) $)
 +
one denotes the class of operators of the form  $  p ( x , D ) + K $,
 +
where  $  p \in S _ {\rho , \delta }  ^ {m} $
 +
and  $  K $
 +
is an integral operator with a  $  C  ^  \infty  $-
 +
kernel, i.e. an operator of the form
  
is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566027.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566028.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566029.png" /> is a multi-index, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566032.png" /> are integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566034.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566035.png" /> coincides with the [[Differential operator|differential operator]] obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566036.png" /> is substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566037.png" /> in the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566038.png" />.
+
$$
 +
K u ( x)  = \int\limits K ( x , y ) u ( y) d y ,
 +
$$
  
One often uses the class of symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566039.png" /> satisfying the conditions
+
where  $  K ( x , y ) \in C  ^  \infty  ( \Omega \times \Omega ) $.
 +
(Such operators  $  p ( x , D ) + K $
 +
are also called pseudo-differential operators in  $  \Omega $.)
 +
The function  $  p ( x , \xi ) $
 +
is called, like before, the symbol of $  p ( x , D ) + K $.
 +
However, in this case it is not uniquely defined, but only up to a symbol from  $  S ^ {- \infty } = \cap _ {m \in \mathbf R }  S _ {1,0}  ^ {m} $.
 +
An operator  $  A \in L _ {\rho , \delta }  ^ {m} $
 +
is called a pseudo-differential operator of order not exceeding  $  m $
 +
and type  $  \rho , \delta $.  
 +
The differential operator described above belongs to the class  $  L _ {1,0}  ^ {m} $.  
 +
The smallest possible value of  $  m $
 +
is called the order of the pseudo-differential operator. The classes  $  S _ {\rho , \delta }  ^ {m} $
 +
and  $  L _ {\rho , \delta }  ^ {m} $
 +
are often called the Hörmander classes.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
One may specify pseudo-differential operators in  $  \Omega $
 +
by double symbols or amplitudes, i.e. write them in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566041.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
P u  =
 +
\frac{1}{( 2 \pi )  ^ {n} }
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566042.png" /> are multi-indices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566044.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566045.png" /> is a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566046.png" />. This class is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566047.png" /> (or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566048.png" />).
+
\int\limits \int\limits e ^ {i ( x - y ) \cdot \xi }
 +
a ( x , y , \xi ) u ( y) d y  d \xi .
 +
$$
  
It is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566049.png" />. By <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566050.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566051.png" />) one denotes the class of operators of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566054.png" /> is an integral operator with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566055.png" />-kernel, i.e. an operator of the form
+
For  $  a ( x , y , \xi ) = p ( x , \xi ) $
 +
this formula turns into (1). It is usually assumed that $  a ( x , y , \xi ) \in S _ {\rho , \delta }  ^ {m} ( \Omega \times \Omega \times \mathbf R  ^ {n} ) $,  
 +
i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566056.png" /></td> </tr></table>
+
$$ \tag{4 }
 +
| \partial  _  \xi  ^  \alpha  \partial  _ {x} ^ {\beta  ^  \prime  }
 +
\partial  _ {y} ^ {\beta  ^ {\prime\prime} } a ( x , y , \xi ) | \leq
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566057.png" />. (Such operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566058.png" /> are also called pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566059.png" />.) The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566060.png" /> is called, like before, the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566061.png" />. However, in this case it is not uniquely defined, but only up to a symbol from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566062.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566063.png" /> is called a pseudo-differential operator of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566066.png" /> and type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566067.png" />. The differential operator described above belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566068.png" />. The smallest possible value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566069.png" /> is called the order of the pseudo-differential operator. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566071.png" /> are often called the Hörmander classes.
+
$$
 +
\leq  \
 +
C _ {\alpha , \beta  ^  \prime  , \beta  ^ {\prime\prime} , {\mathcal K} }  ( 1 +
 +
| \xi | ) ^ {m - \rho | \alpha | + \delta | \beta
 +
^  \prime  + \beta  ^ {\prime\prime} | } ,\  x , y \in {\mathcal K} ;
 +
$$
  
One may specify pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566072.png" /> by double symbols or amplitudes, i.e. write them in the form
+
here  $  {\mathcal K} $
 +
is a compact set in  $  \Omega $.
 +
If  $  0 \leq  \delta < \rho \leq  1 $,
 +
then the class of operators (3) (for all possible functions  $  a \in S _ {\rho , \delta }  ^ {m} $)
 +
coincides with  $  L _ {\rho , \delta }  ^ {m} ( \Omega ) $.  
 +
In this case the symbol  $  p ( x , \xi ) $(
 +
determined up to a symbol from  $  S ^ {- \infty } $)
 +
has the following asymptotic expansion:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
\left .
 +
p ( x , \xi ) \sim \
 +
\sum _  \alpha
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566074.png" /> this formula turns into (1). It is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566075.png" />, i.e.
+
\frac{1}{\alpha ! }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\partial  _  \xi  ^  \alpha  D _ {y}  ^  \alpha  a ( x , y , \xi ) \
 +
\right | _ {y = x }  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566077.png" /></td> </tr></table>
+
where  $  \alpha ! = \alpha _ {1} ! \dots \alpha _ {n} ! $
 +
and the summation extends over all multi-indices. This formula means that the difference between  $  p ( x , \xi ) $
 +
and the partial sum over all  $  \alpha $
 +
for which  $  | \alpha | \leq  N $
 +
is a symbol in  $  S _ {\rho , \delta }  ^ {m - ( \rho - \delta ) N } $,
 +
i.e. is a symbol of order at most equal to the largest of the orders of the rest terms.
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566078.png" /> is a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566079.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566080.png" />, then the class of operators (3) (for all possible functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566081.png" />) coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566082.png" />. In this case the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566083.png" /> (determined up to a symbol from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566084.png" />) has the following asymptotic expansion:
+
A pseudo-differential operator  $  P $
 +
can be extended, by continuity or duality, to an operator  $  P : {\mathcal E} ^ { \prime } ( \Omega ) \rightarrow D  ^  \prime  ( \Omega ) $.  
 +
Here  $  D  ^  \prime  ( \Omega ) $
 +
and  $  {\mathcal E} ^ { \prime } ( \Omega ) $
 +
are the space of generalized functions and the space of generalized functions with compact support in $  \Omega $,
 +
respectively (cf. [[Generalized functions, space of|Generalized functions, space of]]). If $  \delta < 1 $,  
 +
then the pseudo-differential operator has the following pseudo-locality property: If  $  u \in {\mathcal E} ^ { \prime } ( \Omega ) \cap C  ^  \infty  ( \Omega  ^  \prime  ) $,
 +
where  $  \Omega  ^  \prime  \subset  \Omega $,
 +
then  $  P u \in C  ^  \infty  ( \Omega  ^  \prime  ) $.  
 +
Another formulation of this property is: The kernel  $  K ( x , y ) $(
 +
in the sense of L. Schwartz) of  $  P $
 +
is infinitely differentiable in  $  x , y $
 +
for  $  x \neq y $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566085.png" /></td> </tr></table>
+
A classical pseudo-differential operator of order  $  m $
 +
in  $  \Omega $
 +
is an operator  $  P \in L _ {1,0}  ^ {m} $
 +
whose symbol  $  p ( x , \xi ) $
 +
has the asymptotic expansion
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566086.png" /> and the summation extends over all multi-indices. This formula means that the difference between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566087.png" /> and the partial sum over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566088.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566089.png" /> is a symbol in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566090.png" />, i.e. is a symbol of order at most equal to the largest of the orders of the rest terms.
+
$$
 +
p ( x , \xi )  \sim \
 +
\sum _ { j= } 0 ^  \infty  \chi ( \xi ) p _ {m-} j ( x , \xi ) ,
 +
$$
  
A pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566091.png" /> can be extended, by continuity or duality, to an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566092.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566094.png" /> are the space of generalized functions and the space of generalized functions with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566095.png" />, respectively (cf. [[Generalized functions, space of|Generalized functions, space of]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566096.png" />, then the pseudo-differential operator has the following pseudo-locality property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566098.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p07566099.png" />. Another formulation of this property is: The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660100.png" /> (in the sense of L. Schwartz) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660101.png" /> is infinitely differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660102.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660103.png" />.
+
where  $  \chi ( \xi ) \in C  ^  \infty  ( \mathbf R  ^ {n} ) $,  
 +
$  \chi ( \xi ) = 1 $
 +
for  $  | \xi | \geq  1 $,  
 +
$  \chi ( \xi ) = 0 $
 +
for  $  | \xi | \leq  1/2 $,  
 +
and where p _ {m- j} ( x , \xi ) \in C  ^  \infty  ( \Omega \times ( \mathbf R  ^ {n} \setminus  0 ) ) $
 +
is positively homogeneous in $  \xi $
 +
of order  $  m - j $:
  
A classical pseudo-differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660105.png" /> is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660106.png" /> whose symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660107.png" /> has the asymptotic expansion
+
$$
 +
p _ {m- j} ( x , t \xi )  = t  ^ {m- j} p _ {m- j} ( x , \xi ) ,\ \
 +
x \in \Omega ,\ \
 +
\xi \in \mathbf R  ^ {n} \setminus  0 ,\  t > 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660108.png" /></td> </tr></table>
+
A differential operator (with smooth coefficients) serves as an example of a classical pseudo-differential operator. The function  $  p _ {m} ( x , \xi ) $
 +
is called the principal symbol of a classical pseudo-differential operator of order  $  m $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660110.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660112.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660113.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660114.png" /> is positively homogeneous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660115.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660116.png" />:
+
A pseudo-differential operator  $  P $
 +
in  $  \Omega $
 +
is called properly supported if the projections of  $  \Omega \times \Omega $
 +
onto each factor when restricted to the support of the kernel of  $  P $
 +
are proper mappings (cf. also [[Proper morphism|Proper morphism]]). A properly supported pseudo-differential operator maps  $  C _ {0}  ^  \infty  ( \Omega ) $
 +
into  $  C _ {0}  ^  \infty  ( \Omega ) $
 +
and can be extended, by continuity, to mappings  $  C  ^  \infty  ( \Omega ) \rightarrow C  ^  \infty  ( \Omega ) $,
 +
$  {\mathcal E} ^ { \prime } ( \Omega ) \rightarrow {\mathcal E} ^ { \prime } ( \Omega ) $
 +
and  $  D  ^  \prime  ( \Omega ) \rightarrow D  ^  \prime  ( \Omega ) $.  
 +
It can be written in the form (1) with symbol  $  p ( x , \xi ) = e ^ {- i x \cdot \xi } P ( e ^ {i x \cdot \xi } ) $,  
 +
where the exponent is understood as a function of $  x $
 +
with  $  \xi $
 +
as parameter.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660117.png" /></td> </tr></table>
+
Suppose that  $  A , B $
 +
are pseudo-differential operators in  $  \Omega $
 +
one of which is properly supported. Then their product (composition)  $  C = A B $
 +
makes sense. The composition theorem plays an important role in the theory of pseudo-differential operators: If  $  A \in L _ {\rho , \delta }  ^ {m _ {1} } $,
 +
$  B \in L _ {\rho , \delta }  ^ {m _ {2} } $,
 +
0 \leq  \delta \leq  \rho \leq  1 $,
 +
then  $  C \in L _ {\rho , \delta }  ^ {m _ {1} + m _ {2} } $.  
 +
If, moreover,  $  \delta < \rho $
 +
and  $  c ( x , \xi ) $,
 +
$  a ( x , \xi ) $
 +
and  $  b ( x , \xi ) $
 +
are the symbols of  $  C $,
 +
$  A $
 +
and  $  B $,
 +
then
  
A differential operator (with smooth coefficients) serves as an example of a classical pseudo-differential operator. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660118.png" /> is called the principal symbol of a classical pseudo-differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660119.png" />.
+
$$
 +
c ( x , \xi ) \sim \
 +
\sum _  \alpha
  
A pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660121.png" /> is called properly supported if the projections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660122.png" /> onto each factor when restricted to the support of the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660123.png" /> are proper mappings (cf. also [[Proper morphism|Proper morphism]]). A properly supported pseudo-differential operator maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660124.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660125.png" /> and can be extended, by continuity, to mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660128.png" />. It can be written in the form (1) with symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660129.png" />, where the exponent is understood as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660130.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660131.png" /> as parameter.
+
\frac{1}{\alpha ! }
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660132.png" /> are pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660133.png" /> one of which is properly supported. Then their product (composition) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660134.png" /> makes sense. The composition theorem plays an important role in the theory of pseudo-differential operators: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660137.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660138.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660139.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660142.png" /> are the symbols of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660144.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660145.png" />, then
+
[ \partial  _  \xi  ^  \alpha  a ( x , \xi ) ]
 +
[ D _ {x}  ^  \alpha  b ( x , \xi ) ] .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660146.png" /></td> </tr></table>
+
In particular, if  $  A , B $
 +
are classical pseudo-differential operators of orders  $  m _ {1} $
 +
and  $  m _ {2} $,
 +
then  $  C $
 +
is a classical pseudo-differential operator of order  $  m _ {1} + m _ {2} $
 +
with principal symbol  $  c _ {m _ {1}  + m _ {2} } ( x , \xi ) = a _ {m _ {1}  } ( x , \xi ) b _ {m _ {2}  } ( x , \xi ) $,
 +
where  $  a _ {m _ {1}  } ( x , \xi ) $
 +
and  $  b _ {m _ {2}  } ( x , \xi ) $
 +
are the principal symbols of  $  A $
 +
and  $  B $.
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660147.png" /> are classical pseudo-differential operators of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660149.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660150.png" /> is a classical pseudo-differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660151.png" /> with principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660152.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660153.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660154.png" /> are the principal symbols of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660156.png" />.
+
If  $  P \in L _ {\rho , \delta }  ^ {m} $,
 +
0 \leq  \delta \leq  \rho \leq  1 $,  
 +
then there exists a, moreover unique, adjoint pseudo-differential operator $  P  ^ {*} \in L _ {\rho , \delta }  ^ {m} $
 +
for which  $  ( P u , v ) = ( u , P  ^ {*} v ) $,
 +
$  u , v \in C _ {0}  ^  \infty  ( \Omega ) $,  
 +
where $  ( u , v ) = \int u ( x) v ( x)  d x $
 +
is the inner product of  $  u $
 +
and  $  v $
 +
in  $  L _ {2} ( \Omega ) $.  
 +
If, moreover,  $  \delta < \rho $,
 +
p ^ {*} ( x , \xi ) $
 +
is the symbol of $  P  ^ {*} $
 +
and p ( x , \xi ) $
 +
is the symbol of  $  P $,
 +
then
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660158.png" />, then there exists a, moreover unique, adjoint pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660159.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660161.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660162.png" /> is the inner product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660164.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660165.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660167.png" /> is the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660169.png" /> is the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660170.png" />, then
+
$$
 +
p ^ {*} ( x , \xi )  \sim \
 +
\sum _  \alpha
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660171.png" /></td> </tr></table>
+
\frac{1}{\alpha ! }
  
Thus, the properly supported pseudo-differential operators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660172.png" /> form an algebra with involution given by transition to the adjoint operator. The arbitrary pseudo-differential operators form a module over this algebra.
+
\partial  _  \xi  ^  \alpha  D _ {x}  ^  \alpha  \overline{ {p ( x , \xi ) }}\; .
 +
$$
  
The theorem on the boundedness of pseudo-differential operators from the Hörmander classes in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660173.png" />-norm, in its most precise form, asserts the following (cf. [[#References|[8]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660174.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660175.png" /> be an operator of the form (3) with double symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660176.png" /> satisfying (4), in which the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660177.png" /> satisfy the conditions
+
Thus, the properly supported pseudo-differential operators for  $  \delta \leq  \rho $
 +
form an algebra with involution given by transition to the adjoint operator. The arbitrary pseudo-differential operators form a module over this algebra.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660178.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
The theorem on the boundedness of pseudo-differential operators from the Hörmander classes in the  $  L _ {2} $-norm, in its most precise form, asserts the following (cf. [[#References|[8]]]): Let  $  \Omega = \mathbf R  ^ {n} $
 +
and let  $  P $
 +
be an operator of the form (3) with double symbol  $  a ( x , y , \xi ) $
 +
satisfying (4), in which the numbers  $  m , \rho , \delta $
 +
satisfy the conditions
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660179.png" /> can be extended to a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660180.png" />. In particular, under the conditions (5) pseudo-differential operators of the form (1) with symbols satisfying conditions (2) uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660181.png" /> (i.e. such that the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660182.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660183.png" />) are bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660184.png" />. This implies, e.g., the boundedness in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660185.png" /> of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660186.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660187.png" /> and if the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660188.png" /> has compact support (when the bounds on the symbol are, again, uniform in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660189.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660190.png" /> or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660191.png" />, operators of such a form need not be bounded [[#References|[19a]]]. Analogously, in general, if one of the two latter conditions of (5) are not fulfilled, then one already obtains a class of pseudo-differential operators that contains unbounded ones.
+
$$ \tag{5 }
 +
0 \leq  \rho  \leq  1 ,\ \
 +
0 \leq  \delta  < 1 ,\ \
 +
m  \leq  0 ,\ \
 +
\rho - \delta -
 +
\frac{m}{n}
 +
  \geq  0 ,
 +
$$
  
In terms of bounds on symbols one can give conditions for the boundedness of pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660192.png" />-norms, as well as in Hölder and in Gevrey norms (cf. [[#References|[8]]]).
+
then  $  P $
 +
can be extended to a bounded operator  $  P :  L _ {2} ( \mathbf R  ^ {n} ) \rightarrow L _ {2} ( \mathbf R  ^ {n} ) $.
 +
In particular, under the conditions (5) pseudo-differential operators of the form (1) with symbols satisfying conditions (2) uniformly in $  x $(
 +
i.e. such that the constants  $  C _ {\alpha , \beta , {\mathcal K} }  = C _ {\alpha , \beta }  $
 +
do not depend on  $  {\mathcal K} $)
 +
are bounded in  $  L _ {2} ( \mathbf R  ^ {n} ) $.  
 +
This implies, e.g., the boundedness in  $  L _ {2} ( \mathbf R  ^ {n} ) $
 +
of operators  $  P \in L _ {p , \delta }  ^ {0} $
 +
if  $  0 \leq  \delta \leq  \rho < 1 $
 +
and if the kernel of  $  P $
 +
has compact support (when the bounds on the symbol are, again, uniform in $  x $).  
 +
For  $  \rho < \delta $
 +
or for  $  \delta = 1 $,
 +
operators of such a form need not be bounded [[#References|[19a]]]. Analogously, in general, if one of the two latter conditions of (5) are not fulfilled, then one already obtains a class of pseudo-differential operators that contains unbounded ones.
  
If an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660193.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660194.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660195.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660196.png" /> and where (2) holds uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660197.png" />, then this operator can be extended to a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660198.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660199.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660200.png" /> denotes the usual Sobolev space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660201.png" /> (which is sometimes denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660202.png" />).
+
In terms of bounds on symbols one can give conditions for the boundedness of pseudo-differential operators in  $  L _ {p} $-norms, as well as in Hölder and in Gevrey norms (cf. [[#References|[8]]]).
  
The class of pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660203.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660204.png" /> is naturally invariant under diffeomorphisms. Its subclass of classical pseudo-differential operators has the same property. This makes it possible to define the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660205.png" /> and classical pseudo-differential operators on an arbitrary smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660206.png" />. The formula for change of variables in the symbol under a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660208.png" /> are domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660209.png" />, has the form
+
If an operator  $  P = p ( x , D ) $
 +
is given on  $  \mathbf R  ^ {n} $,
 +
where  $  P \in S _ {\rho , \delta }  ^ {m} $,
 +
0 \leq  \delta \leq  \rho \leq  1 $
 +
and where (2) holds uniformly in  $  x \in \mathbf R  ^ {n} $,
 +
then this operator can be extended to a bounded operator  $  P : H  ^ {s} ( \mathbf R  ^ {n} ) \rightarrow H  ^ {s-} m ( \mathbf R  ^ {n} ) $,  
 +
$  s \in \mathbf R $,  
 +
where  $  H  ^ {t} ( \mathbf R  ^ {n} ) $
 +
denotes the usual Sobolev space over  $  \mathbf R  ^ {n} $(
 +
which is sometimes denoted also by  $  W _ {2}  ^ {t} ( \mathbf R  ^ {n} ) $).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660210.png" /></td> </tr></table>
+
The class of pseudo-differential operators in  $  L _ {\rho , \delta }  ^ {m} $
 +
for  $  1 - \rho \leq  \delta < \rho \leq  1 $
 +
is naturally invariant under diffeomorphisms. Its subclass of classical pseudo-differential operators has the same property. This makes it possible to define the class  $  L _ {\rho , \delta }  ^ {m} ( x) $
 +
and classical pseudo-differential operators on an arbitrary smooth manifold  $  X $.  
 +
The formula for change of variables in the symbol under a diffeomorphism  $  \kappa :  \Omega \rightarrow \Omega _ {1} $,
 +
where  $  \Omega , \Omega _ {1} $
 +
are domains in  $  X $,
 +
has the form
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660211.png" /> is the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660212.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660213.png" /> is the symbol of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660214.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660215.png" />, i.e. that obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660216.png" /> by a change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660217.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660218.png" /> denotes the Jacobian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660219.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660220.png" /> is the transposed matrix; and
+
$$
 +
a _ {1} ( y , \eta ) \mid  _ {y = \kappa ( x) }  \sim \
 +
\left . \sum _  \alpha
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660221.png" /></td> </tr></table>
+
\frac{1}{\alpha ! }
  
In particular, this implies that the principal symbol of a classical pseudo-differential operator on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660222.png" /> is a well-defined function on the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660223.png" />.
+
a ^ {( \alpha ) } ( x , {}  ^ {t} \kappa  ^  \prime  ( x) \eta )
 +
D _ {z}  ^  \alpha  e ^ {i \kappa _ {x}  ^ {\prime\prime} ( z) \cdot \eta }
 +
\right | _ {z = x }  .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660224.png" /> is a compact manifold (without boundary), then the pseudo-differential operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660225.png" /> form an algebra with involution, if the involution is introduced by means of an inner product, given by a smooth positive density. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660226.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660227.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660228.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660229.png" />, then it is compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660230.png" />. For classical pseudo-differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660231.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660232.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660233.png" />,
+
Here  $  a ( x , \xi ) $
 +
is the symbol of  $  A \in L _ {\rho , \delta }  ^ {m} ( \Omega ) $;
 +
$  a _ {1} ( x , \xi ) $
 +
is the symbol of the operator  $  A _ {1} \in L _ {\rho , \delta }  ^ {m} ( \Omega _ {1} ) $
 +
given by $  A _ {1} u = [ A ( u \circ \kappa ) ] \circ \kappa  ^ {- 1} $,
 +
i.e. that obtained from  $  A $
 +
by a change of variables  $  \kappa $;
 +
$  \kappa  ^  \prime  ( x) $
 +
denotes the Jacobian of  $  \kappa $;
 +
$  {}  ^ {t} \kappa  ^  \prime  ( x) $
 +
is the transposed matrix; and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660234.png" /></td> </tr></table>
+
$$
 +
a ^ {( \alpha ) } ( x , \xi )  = \
 +
\partial  _  \xi  ^  \alpha  a ( x , \xi ) ,\ \
 +
\kappa _ {x}  ^ {\prime\prime} ( z)  = \kappa ( z) - \kappa ( x) - \kappa  ^  \prime
 +
( x) ( z - x ) .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660235.png" /> is the principal symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660236.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660237.png" /> runs over the set of compact operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660238.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660239.png" /> can by continuity be extended to a bounded linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660240.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660241.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660242.png" />.
+
In particular, this implies that the principal symbol of a classical pseudo-differential operator on a manifold  $  X $
 +
is a well-defined function on the cotangent bundle  $  T  ^ {*} X $.
  
A parametrix of a pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660243.png" /> is a pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660244.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660245.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660246.png" /> are pseudo-differential operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660247.png" />, i.e. are integral operators with a smooth kernel. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660248.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660249.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660250.png" /> is the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660251.png" />. A sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660252.png" /> to have a parametrix is that the conditions
+
If  $  X $
 +
is a compact manifold (without boundary), then the pseudo-differential operators on  $  X $
 +
form an algebra with involution, if the involution is introduced by means of an inner product, given by a smooth positive density. An operator  $  A \in L _ {\rho , \delta }  ^ {0} ( X) $
 +
is bounded in  $  L _ {2} ( X) $,  
 +
and if  $  A \in L _ {\rho , \delta }  ^ {m} ( X) $
 +
for  $  m < 0 $,
 +
then it is compact in  $  L _ {2} ( X) $.  
 +
For classical pseudo-differential operators  $  A $
 +
of order  $  0 $
 +
on  $  X $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660253.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$
 +
\inf  \| A + K \|  = \
 +
\sup _ {( x , \xi ) \in T  ^ {*} X } \
 +
| a _ {0} ( x , \xi ) | ,
 +
$$
 +
 
 +
where  $  a _ {0} ( x , \xi ) $
 +
is the principal symbol of  $  A $
 +
and  $  K $
 +
runs over the set of compact operators in  $  L _ {2} ( X) $.
 +
An operator  $  A \in L _ {\rho , \delta }  ^ {m} ( X) $
 +
can by continuity be extended to a bounded linear operator from  $  H  ^ {s} ( X) $
 +
into  $  H  ^ {s- m} ( X) $
 +
for any  $  s \in \mathbf R $.
 +
 
 +
A parametrix of a pseudo-differential operator  $  A $
 +
is a pseudo-differential operator  $  B $
 +
such that  $  I - A B $
 +
and  $  I - B A $
 +
are pseudo-differential operators of order  $  - \infty $,
 +
i.e. are integral operators with a smooth kernel. Suppose that  $  A \in L _ {\rho , \delta }  ^ {m} ( \Omega ) $,
 +
$  0 \leq  \delta < p \leq  1 $,
 +
and that  $  a ( x , \xi ) $
 +
is the symbol of  $  A $.  
 +
A sufficient condition for  $  A $
 +
to have a parametrix is that the conditions
 +
 
 +
$$ \tag{6 }
 +
\left . \begin{array}{c}
 +
 
 +
| a ( x , \xi ) |  \geq  \epsilon  | \xi | ^ {m _ {0} } ,\ \
 +
| \xi | \geq  R ,\ \
 +
\epsilon > 0 ,\ \
 +
m _ {0} \in \mathbf R ;
 +
\\
 +
 
 +
| a  ^ {- 1} ( x , \xi ) \partial  _  \xi  ^  \alpha  \partial  _ {x}  ^  \beta
 +
a ( x , \xi ) |  \leq
 +
\\
 +
 
 +
\leq  c _ {\alpha , \beta , K }
 +
| \xi | ^ {- \rho | \alpha | + \delta | \beta | } ,\ \
 +
| \xi | \geq  R ,\  x \in K ,
 +
 +
\end{array}
 +
\right \}
 +
$$
  
 
are fulfilled.
 
are fulfilled.
  
In this case a parametrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660254.png" /> exists. The simplest implication from the existence of a parametrix is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660255.png" /> is a hypo-elliptic operator: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660256.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660257.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660258.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660259.png" /> (cf. [[Support of a generalized function|Support of a generalized function]]). The following exact result (the regularity theorem) is also valid: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660260.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660261.png" />. A micro-local regularity theorem is also valid: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660262.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660263.png" /> denotes the [[Wave front|wave front]] of the generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660264.png" />.
+
In this case a parametrix $  B \in L _ {\rho , \delta }  ^ {- m _ {0} } ( \Omega ) $
 +
exists. The simplest implication from the existence of a parametrix is that $  A $
 +
is a hypo-elliptic operator: If $  A u \in C  ^  \infty  ( \Omega  ^  \prime  ) $,  
 +
where $  \Omega  ^  \prime  \subset  \Omega $,  
 +
then $  u \in C  ^  \infty  ( \Omega  ^  \prime  ) $.  
 +
In other words, $  \textrm{ sing  supp  }  A u = \textrm{ sing  supp  }  u $(
 +
cf. [[Support of a generalized function|Support of a generalized function]]). The following exact result (the regularity theorem) is also valid: If $  A u \in H _ { \mathop{\rm loc}  }  ^ {s} ( \Omega  ^  \prime  ) $,  
 +
then $  u \in H _ { \mathop{\rm loc}  } ^ {s + m _ {0} } ( \Omega  ^  \prime  ) $.  
 +
A micro-local regularity theorem is also valid: $  \mathop{\rm WF} ( A u ) = \mathop{\rm WF} ( u) $,  
 +
where $  \mathop{\rm WF} ( u) $
 +
denotes the [[Wave front|wave front]] of the generalized function $  u $.
  
Condition (6) is invariant under diffeomorphisms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660265.png" />. Therefore the corresponding class of pseudo-differential operators on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660266.png" /> has a meaning. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660267.png" /> is compact, then such an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660268.png" /> is Fredholm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660269.png" /> (cf. [[Fredholm-operator(2)|Fredholm operator]]), i.e. has finite-dimensional kernel and co-kernel in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660270.png" />, and has a closed image.
+
Condition (6) is invariant under diffeomorphisms for $  1 - \rho \leq  \delta < \rho \leq  1 $.  
 +
Therefore the corresponding class of pseudo-differential operators on a manifold $  X $
 +
has a meaning. If $  X $
 +
is compact, then such an operator $  A $
 +
is Fredholm in $  C  ^  \infty  ( X) $(
 +
cf. [[Fredholm-operator(2)|Fredholm operator]]), i.e. has finite-dimensional kernel and co-kernel in $  C  ^  \infty  ( X) $,  
 +
and has a closed image.
  
A classical pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660271.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660272.png" /> with smooth symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660273.png" /> is called elliptic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660274.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660275.png" />. For such an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660276.png" /> condition (6) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660277.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660278.png" /> has a parametrix that is also a classical pseudo-differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660279.png" />. On a compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660280.png" /> such an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660281.png" /> gives rise to a Fredholm operator
+
A classical pseudo-differential operator $  A $
 +
of order $  m $
 +
with smooth symbol $  a _ {m} ( x , \xi ) $
 +
is called elliptic if $  a _ {m} ( x , \xi ) \neq 0 $
 +
for $  \xi \neq 0 $.  
 +
For such an operator $  A $
 +
condition (6) holds with $  m _ {0} = m $,  
 +
and $  A $
 +
has a parametrix that is also a classical pseudo-differential operator of order $  m $.  
 +
On a compact manifold $  X $
 +
such an operator $  A $
 +
gives rise to a Fredholm operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660282.png" /></td> </tr></table>
+
$$
 +
A : H  ^ {s} ( X)  \rightarrow  H  ^ {s- m} ( X),\ \
 +
s \in \mathbf R .
 +
$$
  
All these definitions and statements can be transferred to pseudo-differential operators acting on vector functions, or, more generally, on sections of vector bundles. For an elliptic operator on a compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660283.png" /> the index of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660284.png" /> determined by it on the Sobolev classes of sections does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660285.png" /> and can be explicitly computed (cf. [[Index formulas|Index formulas]]).
+
All these definitions and statements can be transferred to pseudo-differential operators acting on vector functions, or, more generally, on sections of vector bundles. For an elliptic operator on a compact manifold $  X $
 +
the index of the mapping $  A : H  ^ {s} ( X) \rightarrow H  ^ {s- m} ( X) $
 +
determined by it on the Sobolev classes of sections does not depend on $  s \in \mathbf R $
 +
and can be explicitly computed (cf. [[Index formulas|Index formulas]]).
  
 
The role of pseudo-differential operators lies in the fact that there is a number of operations leading outside the class of differential operators but preserving the class of pseudo-differential operators. E.g., the resolvent and complex powers of an elliptic differential operator on a compact manifold are classical pseudo-differential operators; they arise when reducing an elliptic boundary value problem to the boundary (cf., e.g., [[#References|[7]]], [[#References|[8]]], and [[#References|[1e]]]).
 
The role of pseudo-differential operators lies in the fact that there is a number of operations leading outside the class of differential operators but preserving the class of pseudo-differential operators. E.g., the resolvent and complex powers of an elliptic differential operator on a compact manifold are classical pseudo-differential operators; they arise when reducing an elliptic boundary value problem to the boundary (cf., e.g., [[#References|[7]]], [[#References|[8]]], and [[#References|[1e]]]).
  
There are several versions of the theory of pseudo-differential operators, adapted to the solution of various problems in analysis and mathematical physics. Often, pseudo-differential operators with a parameter arise; they are necessary, e.g., in the study of resolvent and asymptotic expansions for eigen values. An important role is played by different versions of the theory of pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660286.png" />, taking into account effects related to the description of the behaviour of functions at infinity, and often inspired by mathematical problems in quantum mechanics arising in the study of quantization of classical systems (cf. [[#References|[5]]], [[#References|[11]]]). In the theory of local solvability of partial differential equations and in spectral theory it is expedient to use pseudo-differential operators whose behaviour can be described by weight functions replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660287.png" /> in estimates of the type (2) (cf. [[#References|[8]]], [[#References|[14]]]). One has constructed an algebra of pseudo-differential operators on manifolds with boundary, containing, in particular, the parametrix of elliptic boundary value problems (cf. [[#References|[3]]], [[#References|[13]]]).
+
There are several versions of the theory of pseudo-differential operators, adapted to the solution of various problems in analysis and mathematical physics. Often, pseudo-differential operators with a parameter arise; they are necessary, e.g., in the study of resolvent and asymptotic expansions for eigen values. An important role is played by different versions of the theory of pseudo-differential operators in $  \mathbf R  ^ {n} $,  
 +
taking into account effects related to the description of the behaviour of functions at infinity, and often inspired by mathematical problems in quantum mechanics arising in the study of quantization of classical systems (cf. [[#References|[5]]], [[#References|[11]]]). In the theory of local solvability of partial differential equations and in spectral theory it is expedient to use pseudo-differential operators whose behaviour can be described by weight functions replacing $  | \xi | $
 +
in estimates of the type (2) (cf. [[#References|[8]]], [[#References|[14]]]). One has constructed an algebra of pseudo-differential operators on manifolds with boundary, containing, in particular, the parametrix of elliptic boundary value problems (cf. [[#References|[3]]], [[#References|[13]]]).
  
 
A particular case of pseudo-differential operators are the multi-dimensional singular integral and integro-differential operators, whose study prepared the emergence of the theory of pseudo-differential operators (cf. [[#References|[12]]] and also [[Singular integral|Singular integral]]).
 
A particular case of pseudo-differential operators are the multi-dimensional singular integral and integro-differential operators, whose study prepared the emergence of the theory of pseudo-differential operators (cf. [[#References|[12]]] and also [[Singular integral|Singular integral]]).
Line 122: Line 498:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> J.J. Kohn,   L. Nirenberg,   "An algebra of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' : 1–2 (1965) pp. 269–305</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> L. Hörmander,   "Pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' : 3 (1965) pp. 501–517</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> J.J. Kohn,   L. Nirenberg,   "Non-coercive boundary value problems" ''Commun. Pure Appl. Math.'' , '''18''' : 3 (1965) pp. 443–492</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top"> L. Hörmander,   "Pseudo-differential operators and non-elliptic boundary problems" ''Ann. of Math.'' , '''83''' : 1 (1966) pp. 129–209</TD></TR><TR><TD valign="top">[1e]</TD> <TD valign="top"> L. Hörmander,   "Pseudo-differential operators and hypoelliptic equations" A.P. Calderòn (ed.) , ''Singular Integrals'' , ''Proc. Symp. Pure Math.'' , '''10''' , Amer. Math. Soc. (1966) pp. 138–183</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.S. Agranovich,   M.I. Vishik,   "Pseudo-differential operators" , Moscow (1988) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.I. Eskin,   "Boundary value problems for elliptic pseudodifferential equations" , Amer. Math. Soc. (1981) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.V. Grushin,   "Pseudodifferential operators" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.A. Shubin,   "Pseudo-differential operators and spectral theory" , Springer (1987) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> K.O. Friedrichs,   "Pseudo-differential operators" , Courant Inst. Math. (1970)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> F. Trèves,   "Introduction to pseudo-differential and Fourier integral operators" , '''1–2''' , Plenum (1980)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.E. Taylor,   "Pseudo-differential operators" , Springer (1974)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Kumanogo,   "Pseudo-differential operators" , M.I.T. (1981)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> J.J. Duistermaat,   "Fourier integral operators" , Courant Inst. Math. (1973)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.P. Maslov,   M.V. Fedoryuk,   "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.S. Agranovich,   "Elliptic singular integro-differential operators" ''Russian Math. Surveys'' , '''20''' : 5 (1965) pp. 1–121 ''Uspekhi Mat. Nauk'' , '''20''' : 5 (1965) pp. 3–120</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> L. Boutet de Monvel,   "Boundary value problems for pseudodifferential operators" ''Acta Math.'' , '''126''' (1971) pp. 11–51</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> L. Hörmander,   "The Weyl calculus of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''32''' : 3 (1979) pp. 359–443</TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top"> H.O. Cordes,   "Elliptic pseudo-differential operators - an abstract theory" , ''Lect. notes in math.'' , '''756''' , Springer (1979)</TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top"> H.O. Cordes,   "Spectral theory of linear differential operators and comparison algebras" , Cambridge Univ. Press (1986)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> Yu.V. Egorov,   "Linear differential equations of principal type" , Consultants Bureau (1986) (Translated from Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> G. Grubb,   "Functional calculus of pseudo-differential boundary problems" , Birkhäuser (1986)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> B. Helffer,   "Théorie spectrale pour des opérateurs globalement elliptiques" ''Astérisque'' , '''112''' (1984)</TD></TR><TR><TD valign="top">[19a]</TD> <TD valign="top"> L. Hörmander,   "Pseudo-differential operators of type 1,1" ''Comm. Partial Diff. Eq.'' , '''13''' : 9 (1988) pp. 1085–1111</TD></TR><TR><TD valign="top">[19b]</TD> <TD valign="top"> L. Hörmander,   "Continuity of pseudo-differential operators of type 1,1" ''Comm. Partial Diff. Eq.'' , '''14''' : 2 (1989) pp. 231–243</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> V. Ivrii,   "Precise spectral asymptotics for elliptic operators" , ''Lect. notes in math.'' , '''1100''' , Springer (1984)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> S. Rempel,   B.-W. Schulze,   "Index theory of elliptic boundary problems" , Akademie Verlag (1982)</TD></TR></table>
+
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> J.J. Kohn, L. Nirenberg, "An algebra of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' : 1–2 (1965) pp. 269–305 {{MR|0176362}} {{ZBL|0171.35101}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' : 3 (1965) pp. 501–517 {{MR|0180740}} {{ZBL|0125.33401}} </TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> J.J. Kohn, L. Nirenberg, "Non-coercive boundary value problems" ''Commun. Pure Appl. Math.'' , '''18''' : 3 (1965) pp. 443–492 {{MR|0181815}} {{ZBL|0125.33302}} </TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators and non-elliptic boundary problems" ''Ann. of Math.'' , '''83''' : 1 (1966) pp. 129–209 {{MR|0233064}} {{ZBL|0132.07402}} </TD></TR><TR><TD valign="top">[1e]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators and hypoelliptic equations" A.P. Calderòn (ed.) , ''Singular Integrals'' , ''Proc. Symp. Pure Math.'' , '''10''' , Amer. Math. Soc. (1966) pp. 138–183 {{MR|0383152}} {{ZBL|0167.09603}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.S. Agranovich, M.I. Vishik, "Pseudo-differential operators" , Moscow (1988) (In Russian) {{MR|1023117}} {{ZBL|0696.35123}} {{ZBL|0167.09801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.I. Eskin, "Boundary value problems for elliptic pseudodifferential equations" , Amer. Math. Soc. (1981) (Translated from Russian) {{MR|0623608}} {{ZBL|0458.35002}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.V. Grushin, "Pseudodifferential operators" , Moscow (1975) (In Russian) {{MR|}} {{ZBL|0498.35090}} {{ZBL|0255.35022}} {{ZBL|0238.47038}} {{ZBL|0223.35084}} {{ZBL|0238.35078}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.A. Shubin, "Pseudo-differential operators and spectral theory" , Springer (1987) (Translated from Russian) {{MR|883081}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> K.O. Friedrichs, "Pseudo-differential operators" , Courant Inst. Math. (1970) {{MR|}} {{ZBL|0226.47028}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> F. Trèves, "Introduction to pseudo-differential and Fourier integral operators" , '''1–2''' , Plenum (1980)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.E. Taylor, "Pseudo-differential operators" , Springer (1974) {{MR|0442523}} {{ZBL|0289.35001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Kumanogo, "Pseudo-differential operators" , M.I.T. (1981) {{MR|1414739}} {{MR|1414619}} {{MR|1406605}} {{MR|0518297}} {{MR|0412904}} {{MR|0361937}} {{MR|0355693}} {{MR|0328392}} {{MR|0315521}} {{MR|0303360}} {{MR|0438189}} {{MR|0291896}} {{MR|0254677}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> J.J. Duistermaat, "Fourier integral operators" , Courant Inst. Math. (1973) {{MR|0451313}} {{ZBL|0272.47028}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.P. Maslov, M.V. Fedoryuk, "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.S. Agranovich, "Elliptic singular integro-differential operators" ''Russian Math. Surveys'' , '''20''' : 5 (1965) pp. 1–121 ''Uspekhi Mat. Nauk'' , '''20''' : 5 (1965) pp. 3–120 {{MR|}} {{ZBL|0149.36101}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> L. Boutet de Monvel, "Boundary value problems for pseudodifferential operators" ''Acta Math.'' , '''126''' (1971) pp. 11–51</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> L. Hörmander, "The Weyl calculus of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''32''' : 3 (1979) pp. 359–443 {{MR|517939}} {{ZBL|0388.47032}} </TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top"> H.O. Cordes, "Elliptic pseudo-differential operators - an abstract theory" , ''Lect. notes in math.'' , '''756''' , Springer (1979) {{MR|551619}} {{ZBL|0417.35004}} </TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top"> H.O. Cordes, "Spectral theory of linear differential operators and comparison algebras" , Cambridge Univ. Press (1986) {{MR|0890743}} {{ZBL|0727.35092}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> Yu.V. Egorov, "Linear differential equations of principal type" , Consultants Bureau (1986) (Translated from Russian) {{MR|0872855}} {{ZBL|0669.35001}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> G. Grubb, "Functional calculus of pseudo-differential boundary problems" , Birkhäuser (1986) {{MR|885088}} {{ZBL|0622.35001}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> B. Helffer, "Théorie spectrale pour des opérateurs globalement elliptiques" ''Astérisque'' , '''112''' (1984) {{MR|0743094}} {{ZBL|0541.35002}} </TD></TR><TR><TD valign="top">[19a]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators of type 1,1" ''Comm. Partial Diff. Eq.'' , '''13''' : 9 (1988) pp. 1085–1111 {{MR|0946283}} {{ZBL|0667.35078}} </TD></TR><TR><TD valign="top">[19b]</TD> <TD valign="top"> L. Hörmander, "Continuity of pseudo-differential operators of type 1,1" ''Comm. Partial Diff. Eq.'' , '''14''' : 2 (1989) pp. 231–243 {{MR|0976972}} {{ZBL|0688.35107}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> V. Ivrii, "Precise spectral asymptotics for elliptic operators" , ''Lect. notes in math.'' , '''1100''' , Springer (1984) {{MR|0771297}} {{ZBL|}} </TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> S. Rempel, B.-W. Schulze, "Index theory of elliptic boundary problems" , Akademie Verlag (1982) {{MR|0690065}} {{ZBL|0504.35002}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The phrase "pseudo-differential operator" is often abbreviated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660288.png" />, just like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660289.png" /> for "partial differential operator" .
+
The phrase "pseudo-differential operator" is often abbreviated to $\rm{\Psi DO}$,  
 +
just like $\rm{PDO}$ for "partial differential operator".
  
For algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660290.png" /> on manifolds with singularities, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075660/p075660291.png" /> with discontinuous symbols, see [[#References|[a2]]].
+
For algebras of $\rm{\Psi DO}$s on manifolds with singularities, in particular $\rm{\Psi DO}$s
 +
with discontinuous symbols, see [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.A. Plamenevskii,   "Algebras of pseudodifferential operators" , Kluwer (1989) (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M.E. Taylor,   "Pseudo-differential operators" , Princeton Univ. Press (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Chazarain,   A. Piriou,   "Introduction to the theory of linear partial differential equations" , North-Holland (1982) (Translated from French)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985) {{MR|2512677}} {{MR|2304165}} {{MR|2108588}} {{MR|1996773}} {{MR|1481433}} {{MR|1313500}} {{MR|1065993}} {{MR|1065136}} {{MR|0961959}} {{MR|0925821}} {{MR|0881605}} {{MR|0862624}} {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{MR|0717035}} {{MR|0705278}} {{ZBL|1178.35003}} {{ZBL|1115.35005}} {{ZBL|1062.35004}} {{ZBL|1028.35001}} {{ZBL|0712.35001}} {{ZBL|0687.35002}} {{ZBL|0619.35002}} {{ZBL|0619.35001}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.A. Plamenevskii, "Algebras of pseudodifferential operators" , Kluwer (1989) (Translated from Russian) {{MR|1105811}} {{MR|1045105}} {{MR|1026642}} {{MR|0992982}} {{ZBL|0677.35090}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M.E. Taylor, "Pseudo-differential operators" , Princeton Univ. Press (1981) {{MR|1567325}} {{ZBL|0289.35001}} {{ZBL|0207.45402}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Chazarain, A. Piriou, "Introduction to the theory of linear partial differential equations" , North-Holland (1982) (Translated from French) {{MR|0678605}} {{ZBL|0487.35002}} </TD></TR></table>

Latest revision as of 04:02, 4 January 2022


An operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function, usually called the symbol of the pseudo-differential operator, that satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

Let $ \Omega $ be an open set in $ \mathbf R ^ {n} $, and let $ C _ {0} ^ \infty ( \Omega ) $ be the space of infinitely-differentiable functions on $ \Omega $ with compact support belonging to $ \Omega $. The simplest pseudo-differential operator on $ \Omega $ is the operator $ P : C _ {0} ^ \infty ( \Omega ) \rightarrow C ^ \infty ( \Omega ) $ given by

$$ \tag{1 } P u ( x) = \ \frac{1}{( 2 \pi ) ^ {n} } \int\limits e ^ {i {x \cdot \xi } } p ( x , \xi ) \widehat{u} ( \xi ) d \xi . $$

Here, $ u \in C _ {0} ^ \infty ( \Omega ) $, $ \xi \in \mathbf R ^ {n} $, $ d \xi $ is Lebesgue measure on $ \mathbf R ^ {n} $, $ x \cdot \xi $ is the usual inner product of the vectors $ x $ and $ \xi $, $ \widehat{u} ( \xi ) $ is the Fourier transform of the function $ u $, i.e.

$$ \widehat{u} ( \xi ) = \int\limits e ^ {- i x \cdot \xi } u ( x ) d x $$

(the integral, like the one in (1), is over all of $ \mathbf R ^ {n} $), and $ p ( x , \xi ) $ is a smooth function on $ \Omega \times \mathbf R ^ {n} $ satisfying certain conditions and is called the symbol of the pseudo-differential operator $ P $( cf. also Symbol of an operator). An operator $ P $ of the form (1) is denoted by $ p ( x , D ) $ or $ p ( x , D _ {x} ) $. If

$$ p ( x , \xi ) = \ \sum _ {| \alpha | \leq m } p _ \alpha ( x) \xi ^ \alpha $$

is a polynomial in $ \xi $ with coefficients $ p _ \alpha \in C _ {0} ^ \infty ( \Omega ) $( here $ \alpha $ is a multi-index, i.e. $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $, $ \alpha _ {j} \geq 0 $, $ \alpha _ {j} $ are integers, $ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $, $ \xi ^ \alpha = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $), then $ p ( x , D ) $ coincides with the differential operator obtained when $ D = \partial / i \partial x $ is substituted for $ \xi $ in the expression for $ p ( x , \xi ) $.

One often uses the class of symbols $ p ( x , \xi ) \in C ^ \infty ( \Omega \times \mathbf R ^ {n} ) $ satisfying the conditions

$$ \tag{2 } | \partial _ \xi ^ \alpha \partial _ {x} ^ \beta p ( x , \xi ) | \leq C _ {\alpha , \beta , {\mathcal K} } \ ( 1 + | \xi | ) ^ {m - \rho | \alpha | + \delta | \beta | } , $$

$$ x \in {\mathcal K} ,\ \xi \in \mathbf R ^ {n} . $$

Here $ \alpha , \beta $ are multi-indices, $ \partial _ {x} = \partial / \partial x $, $ \partial _ \xi = \partial / \partial \xi $, and $ {\mathcal K} $ is a compact set in $ \Omega $. This class is denoted by $ S _ {\rho , \delta } ^ {m} $( or by $ S _ {\rho , \delta } ^ {m} ( \Omega \times \mathbf R ^ {n} ) $).

It is usually assumed that $ 0 \leq \rho , \delta \leq 1 $. By $ L _ {\rho , \delta } ^ {m} $( or $ L _ {\rho , \delta } ^ {m} ( \Omega ) $) one denotes the class of operators of the form $ p ( x , D ) + K $, where $ p \in S _ {\rho , \delta } ^ {m} $ and $ K $ is an integral operator with a $ C ^ \infty $- kernel, i.e. an operator of the form

$$ K u ( x) = \int\limits K ( x , y ) u ( y) d y , $$

where $ K ( x , y ) \in C ^ \infty ( \Omega \times \Omega ) $. (Such operators $ p ( x , D ) + K $ are also called pseudo-differential operators in $ \Omega $.) The function $ p ( x , \xi ) $ is called, like before, the symbol of $ p ( x , D ) + K $. However, in this case it is not uniquely defined, but only up to a symbol from $ S ^ {- \infty } = \cap _ {m \in \mathbf R } S _ {1,0} ^ {m} $. An operator $ A \in L _ {\rho , \delta } ^ {m} $ is called a pseudo-differential operator of order not exceeding $ m $ and type $ \rho , \delta $. The differential operator described above belongs to the class $ L _ {1,0} ^ {m} $. The smallest possible value of $ m $ is called the order of the pseudo-differential operator. The classes $ S _ {\rho , \delta } ^ {m} $ and $ L _ {\rho , \delta } ^ {m} $ are often called the Hörmander classes.

One may specify pseudo-differential operators in $ \Omega $ by double symbols or amplitudes, i.e. write them in the form

$$ \tag{3 } P u = \frac{1}{( 2 \pi ) ^ {n} } \int\limits \int\limits e ^ {i ( x - y ) \cdot \xi } a ( x , y , \xi ) u ( y) d y d \xi . $$

For $ a ( x , y , \xi ) = p ( x , \xi ) $ this formula turns into (1). It is usually assumed that $ a ( x , y , \xi ) \in S _ {\rho , \delta } ^ {m} ( \Omega \times \Omega \times \mathbf R ^ {n} ) $, i.e.

$$ \tag{4 } | \partial _ \xi ^ \alpha \partial _ {x} ^ {\beta ^ \prime } \partial _ {y} ^ {\beta ^ {\prime\prime} } a ( x , y , \xi ) | \leq $$

$$ \leq \ C _ {\alpha , \beta ^ \prime , \beta ^ {\prime\prime} , {\mathcal K} } ( 1 + | \xi | ) ^ {m - \rho | \alpha | + \delta | \beta ^ \prime + \beta ^ {\prime\prime} | } ,\ x , y \in {\mathcal K} ; $$

here $ {\mathcal K} $ is a compact set in $ \Omega $. If $ 0 \leq \delta < \rho \leq 1 $, then the class of operators (3) (for all possible functions $ a \in S _ {\rho , \delta } ^ {m} $) coincides with $ L _ {\rho , \delta } ^ {m} ( \Omega ) $. In this case the symbol $ p ( x , \xi ) $( determined up to a symbol from $ S ^ {- \infty } $) has the following asymptotic expansion:

$$ \left . p ( x , \xi ) \sim \ \sum _ \alpha \frac{1}{\alpha ! } \partial _ \xi ^ \alpha D _ {y} ^ \alpha a ( x , y , \xi ) \ \right | _ {y = x } , $$

where $ \alpha ! = \alpha _ {1} ! \dots \alpha _ {n} ! $ and the summation extends over all multi-indices. This formula means that the difference between $ p ( x , \xi ) $ and the partial sum over all $ \alpha $ for which $ | \alpha | \leq N $ is a symbol in $ S _ {\rho , \delta } ^ {m - ( \rho - \delta ) N } $, i.e. is a symbol of order at most equal to the largest of the orders of the rest terms.

A pseudo-differential operator $ P $ can be extended, by continuity or duality, to an operator $ P : {\mathcal E} ^ { \prime } ( \Omega ) \rightarrow D ^ \prime ( \Omega ) $. Here $ D ^ \prime ( \Omega ) $ and $ {\mathcal E} ^ { \prime } ( \Omega ) $ are the space of generalized functions and the space of generalized functions with compact support in $ \Omega $, respectively (cf. Generalized functions, space of). If $ \delta < 1 $, then the pseudo-differential operator has the following pseudo-locality property: If $ u \in {\mathcal E} ^ { \prime } ( \Omega ) \cap C ^ \infty ( \Omega ^ \prime ) $, where $ \Omega ^ \prime \subset \Omega $, then $ P u \in C ^ \infty ( \Omega ^ \prime ) $. Another formulation of this property is: The kernel $ K ( x , y ) $( in the sense of L. Schwartz) of $ P $ is infinitely differentiable in $ x , y $ for $ x \neq y $.

A classical pseudo-differential operator of order $ m $ in $ \Omega $ is an operator $ P \in L _ {1,0} ^ {m} $ whose symbol $ p ( x , \xi ) $ has the asymptotic expansion

$$ p ( x , \xi ) \sim \ \sum _ { j= } 0 ^ \infty \chi ( \xi ) p _ {m-} j ( x , \xi ) , $$

where $ \chi ( \xi ) \in C ^ \infty ( \mathbf R ^ {n} ) $, $ \chi ( \xi ) = 1 $ for $ | \xi | \geq 1 $, $ \chi ( \xi ) = 0 $ for $ | \xi | \leq 1/2 $, and where $ p _ {m- j} ( x , \xi ) \in C ^ \infty ( \Omega \times ( \mathbf R ^ {n} \setminus 0 ) ) $ is positively homogeneous in $ \xi $ of order $ m - j $:

$$ p _ {m- j} ( x , t \xi ) = t ^ {m- j} p _ {m- j} ( x , \xi ) ,\ \ x \in \Omega ,\ \ \xi \in \mathbf R ^ {n} \setminus 0 ,\ t > 0 . $$

A differential operator (with smooth coefficients) serves as an example of a classical pseudo-differential operator. The function $ p _ {m} ( x , \xi ) $ is called the principal symbol of a classical pseudo-differential operator of order $ m $.

A pseudo-differential operator $ P $ in $ \Omega $ is called properly supported if the projections of $ \Omega \times \Omega $ onto each factor when restricted to the support of the kernel of $ P $ are proper mappings (cf. also Proper morphism). A properly supported pseudo-differential operator maps $ C _ {0} ^ \infty ( \Omega ) $ into $ C _ {0} ^ \infty ( \Omega ) $ and can be extended, by continuity, to mappings $ C ^ \infty ( \Omega ) \rightarrow C ^ \infty ( \Omega ) $, $ {\mathcal E} ^ { \prime } ( \Omega ) \rightarrow {\mathcal E} ^ { \prime } ( \Omega ) $ and $ D ^ \prime ( \Omega ) \rightarrow D ^ \prime ( \Omega ) $. It can be written in the form (1) with symbol $ p ( x , \xi ) = e ^ {- i x \cdot \xi } P ( e ^ {i x \cdot \xi } ) $, where the exponent is understood as a function of $ x $ with $ \xi $ as parameter.

Suppose that $ A , B $ are pseudo-differential operators in $ \Omega $ one of which is properly supported. Then their product (composition) $ C = A B $ makes sense. The composition theorem plays an important role in the theory of pseudo-differential operators: If $ A \in L _ {\rho , \delta } ^ {m _ {1} } $, $ B \in L _ {\rho , \delta } ^ {m _ {2} } $, $ 0 \leq \delta \leq \rho \leq 1 $, then $ C \in L _ {\rho , \delta } ^ {m _ {1} + m _ {2} } $. If, moreover, $ \delta < \rho $ and $ c ( x , \xi ) $, $ a ( x , \xi ) $ and $ b ( x , \xi ) $ are the symbols of $ C $, $ A $ and $ B $, then

$$ c ( x , \xi ) \sim \ \sum _ \alpha \frac{1}{\alpha ! } [ \partial _ \xi ^ \alpha a ( x , \xi ) ] [ D _ {x} ^ \alpha b ( x , \xi ) ] . $$

In particular, if $ A , B $ are classical pseudo-differential operators of orders $ m _ {1} $ and $ m _ {2} $, then $ C $ is a classical pseudo-differential operator of order $ m _ {1} + m _ {2} $ with principal symbol $ c _ {m _ {1} + m _ {2} } ( x , \xi ) = a _ {m _ {1} } ( x , \xi ) b _ {m _ {2} } ( x , \xi ) $, where $ a _ {m _ {1} } ( x , \xi ) $ and $ b _ {m _ {2} } ( x , \xi ) $ are the principal symbols of $ A $ and $ B $.

If $ P \in L _ {\rho , \delta } ^ {m} $, $ 0 \leq \delta \leq \rho \leq 1 $, then there exists a, moreover unique, adjoint pseudo-differential operator $ P ^ {*} \in L _ {\rho , \delta } ^ {m} $ for which $ ( P u , v ) = ( u , P ^ {*} v ) $, $ u , v \in C _ {0} ^ \infty ( \Omega ) $, where $ ( u , v ) = \int u ( x) v ( x) d x $ is the inner product of $ u $ and $ v $ in $ L _ {2} ( \Omega ) $. If, moreover, $ \delta < \rho $, $ p ^ {*} ( x , \xi ) $ is the symbol of $ P ^ {*} $ and $ p ( x , \xi ) $ is the symbol of $ P $, then

$$ p ^ {*} ( x , \xi ) \sim \ \sum _ \alpha \frac{1}{\alpha ! } \partial _ \xi ^ \alpha D _ {x} ^ \alpha \overline{ {p ( x , \xi ) }}\; . $$

Thus, the properly supported pseudo-differential operators for $ \delta \leq \rho $ form an algebra with involution given by transition to the adjoint operator. The arbitrary pseudo-differential operators form a module over this algebra.

The theorem on the boundedness of pseudo-differential operators from the Hörmander classes in the $ L _ {2} $-norm, in its most precise form, asserts the following (cf. [8]): Let $ \Omega = \mathbf R ^ {n} $ and let $ P $ be an operator of the form (3) with double symbol $ a ( x , y , \xi ) $ satisfying (4), in which the numbers $ m , \rho , \delta $ satisfy the conditions

$$ \tag{5 } 0 \leq \rho \leq 1 ,\ \ 0 \leq \delta < 1 ,\ \ m \leq 0 ,\ \ \rho - \delta - \frac{m}{n} \geq 0 , $$

then $ P $ can be extended to a bounded operator $ P : L _ {2} ( \mathbf R ^ {n} ) \rightarrow L _ {2} ( \mathbf R ^ {n} ) $. In particular, under the conditions (5) pseudo-differential operators of the form (1) with symbols satisfying conditions (2) uniformly in $ x $( i.e. such that the constants $ C _ {\alpha , \beta , {\mathcal K} } = C _ {\alpha , \beta } $ do not depend on $ {\mathcal K} $) are bounded in $ L _ {2} ( \mathbf R ^ {n} ) $. This implies, e.g., the boundedness in $ L _ {2} ( \mathbf R ^ {n} ) $ of operators $ P \in L _ {p , \delta } ^ {0} $ if $ 0 \leq \delta \leq \rho < 1 $ and if the kernel of $ P $ has compact support (when the bounds on the symbol are, again, uniform in $ x $). For $ \rho < \delta $ or for $ \delta = 1 $, operators of such a form need not be bounded [19a]. Analogously, in general, if one of the two latter conditions of (5) are not fulfilled, then one already obtains a class of pseudo-differential operators that contains unbounded ones.

In terms of bounds on symbols one can give conditions for the boundedness of pseudo-differential operators in $ L _ {p} $-norms, as well as in Hölder and in Gevrey norms (cf. [8]).

If an operator $ P = p ( x , D ) $ is given on $ \mathbf R ^ {n} $, where $ P \in S _ {\rho , \delta } ^ {m} $, $ 0 \leq \delta \leq \rho \leq 1 $ and where (2) holds uniformly in $ x \in \mathbf R ^ {n} $, then this operator can be extended to a bounded operator $ P : H ^ {s} ( \mathbf R ^ {n} ) \rightarrow H ^ {s-} m ( \mathbf R ^ {n} ) $, $ s \in \mathbf R $, where $ H ^ {t} ( \mathbf R ^ {n} ) $ denotes the usual Sobolev space over $ \mathbf R ^ {n} $( which is sometimes denoted also by $ W _ {2} ^ {t} ( \mathbf R ^ {n} ) $).

The class of pseudo-differential operators in $ L _ {\rho , \delta } ^ {m} $ for $ 1 - \rho \leq \delta < \rho \leq 1 $ is naturally invariant under diffeomorphisms. Its subclass of classical pseudo-differential operators has the same property. This makes it possible to define the class $ L _ {\rho , \delta } ^ {m} ( x) $ and classical pseudo-differential operators on an arbitrary smooth manifold $ X $. The formula for change of variables in the symbol under a diffeomorphism $ \kappa : \Omega \rightarrow \Omega _ {1} $, where $ \Omega , \Omega _ {1} $ are domains in $ X $, has the form

$$ a _ {1} ( y , \eta ) \mid _ {y = \kappa ( x) } \sim \ \left . \sum _ \alpha \frac{1}{\alpha ! } a ^ {( \alpha ) } ( x , {} ^ {t} \kappa ^ \prime ( x) \eta ) D _ {z} ^ \alpha e ^ {i \kappa _ {x} ^ {\prime\prime} ( z) \cdot \eta } \right | _ {z = x } . $$

Here $ a ( x , \xi ) $ is the symbol of $ A \in L _ {\rho , \delta } ^ {m} ( \Omega ) $; $ a _ {1} ( x , \xi ) $ is the symbol of the operator $ A _ {1} \in L _ {\rho , \delta } ^ {m} ( \Omega _ {1} ) $ given by $ A _ {1} u = [ A ( u \circ \kappa ) ] \circ \kappa ^ {- 1} $, i.e. that obtained from $ A $ by a change of variables $ \kappa $; $ \kappa ^ \prime ( x) $ denotes the Jacobian of $ \kappa $; $ {} ^ {t} \kappa ^ \prime ( x) $ is the transposed matrix; and

$$ a ^ {( \alpha ) } ( x , \xi ) = \ \partial _ \xi ^ \alpha a ( x , \xi ) ,\ \ \kappa _ {x} ^ {\prime\prime} ( z) = \kappa ( z) - \kappa ( x) - \kappa ^ \prime ( x) ( z - x ) . $$

In particular, this implies that the principal symbol of a classical pseudo-differential operator on a manifold $ X $ is a well-defined function on the cotangent bundle $ T ^ {*} X $.

If $ X $ is a compact manifold (without boundary), then the pseudo-differential operators on $ X $ form an algebra with involution, if the involution is introduced by means of an inner product, given by a smooth positive density. An operator $ A \in L _ {\rho , \delta } ^ {0} ( X) $ is bounded in $ L _ {2} ( X) $, and if $ A \in L _ {\rho , \delta } ^ {m} ( X) $ for $ m < 0 $, then it is compact in $ L _ {2} ( X) $. For classical pseudo-differential operators $ A $ of order $ 0 $ on $ X $,

$$ \inf \| A + K \| = \ \sup _ {( x , \xi ) \in T ^ {*} X } \ | a _ {0} ( x , \xi ) | , $$

where $ a _ {0} ( x , \xi ) $ is the principal symbol of $ A $ and $ K $ runs over the set of compact operators in $ L _ {2} ( X) $. An operator $ A \in L _ {\rho , \delta } ^ {m} ( X) $ can by continuity be extended to a bounded linear operator from $ H ^ {s} ( X) $ into $ H ^ {s- m} ( X) $ for any $ s \in \mathbf R $.

A parametrix of a pseudo-differential operator $ A $ is a pseudo-differential operator $ B $ such that $ I - A B $ and $ I - B A $ are pseudo-differential operators of order $ - \infty $, i.e. are integral operators with a smooth kernel. Suppose that $ A \in L _ {\rho , \delta } ^ {m} ( \Omega ) $, $ 0 \leq \delta < p \leq 1 $, and that $ a ( x , \xi ) $ is the symbol of $ A $. A sufficient condition for $ A $ to have a parametrix is that the conditions

$$ \tag{6 } \left . \begin{array}{c} | a ( x , \xi ) | \geq \epsilon | \xi | ^ {m _ {0} } ,\ \ | \xi | \geq R ,\ \ \epsilon > 0 ,\ \ m _ {0} \in \mathbf R ; \\ | a ^ {- 1} ( x , \xi ) \partial _ \xi ^ \alpha \partial _ {x} ^ \beta a ( x , \xi ) | \leq \\ \leq c _ {\alpha , \beta , K } | \xi | ^ {- \rho | \alpha | + \delta | \beta | } ,\ \ | \xi | \geq R ,\ x \in K , \end{array} \right \} $$

are fulfilled.

In this case a parametrix $ B \in L _ {\rho , \delta } ^ {- m _ {0} } ( \Omega ) $ exists. The simplest implication from the existence of a parametrix is that $ A $ is a hypo-elliptic operator: If $ A u \in C ^ \infty ( \Omega ^ \prime ) $, where $ \Omega ^ \prime \subset \Omega $, then $ u \in C ^ \infty ( \Omega ^ \prime ) $. In other words, $ \textrm{ sing supp } A u = \textrm{ sing supp } u $( cf. Support of a generalized function). The following exact result (the regularity theorem) is also valid: If $ A u \in H _ { \mathop{\rm loc} } ^ {s} ( \Omega ^ \prime ) $, then $ u \in H _ { \mathop{\rm loc} } ^ {s + m _ {0} } ( \Omega ^ \prime ) $. A micro-local regularity theorem is also valid: $ \mathop{\rm WF} ( A u ) = \mathop{\rm WF} ( u) $, where $ \mathop{\rm WF} ( u) $ denotes the wave front of the generalized function $ u $.

Condition (6) is invariant under diffeomorphisms for $ 1 - \rho \leq \delta < \rho \leq 1 $. Therefore the corresponding class of pseudo-differential operators on a manifold $ X $ has a meaning. If $ X $ is compact, then such an operator $ A $ is Fredholm in $ C ^ \infty ( X) $( cf. Fredholm operator), i.e. has finite-dimensional kernel and co-kernel in $ C ^ \infty ( X) $, and has a closed image.

A classical pseudo-differential operator $ A $ of order $ m $ with smooth symbol $ a _ {m} ( x , \xi ) $ is called elliptic if $ a _ {m} ( x , \xi ) \neq 0 $ for $ \xi \neq 0 $. For such an operator $ A $ condition (6) holds with $ m _ {0} = m $, and $ A $ has a parametrix that is also a classical pseudo-differential operator of order $ m $. On a compact manifold $ X $ such an operator $ A $ gives rise to a Fredholm operator

$$ A : H ^ {s} ( X) \rightarrow H ^ {s- m} ( X),\ \ s \in \mathbf R . $$

All these definitions and statements can be transferred to pseudo-differential operators acting on vector functions, or, more generally, on sections of vector bundles. For an elliptic operator on a compact manifold $ X $ the index of the mapping $ A : H ^ {s} ( X) \rightarrow H ^ {s- m} ( X) $ determined by it on the Sobolev classes of sections does not depend on $ s \in \mathbf R $ and can be explicitly computed (cf. Index formulas).

The role of pseudo-differential operators lies in the fact that there is a number of operations leading outside the class of differential operators but preserving the class of pseudo-differential operators. E.g., the resolvent and complex powers of an elliptic differential operator on a compact manifold are classical pseudo-differential operators; they arise when reducing an elliptic boundary value problem to the boundary (cf., e.g., [7], [8], and [1e]).

There are several versions of the theory of pseudo-differential operators, adapted to the solution of various problems in analysis and mathematical physics. Often, pseudo-differential operators with a parameter arise; they are necessary, e.g., in the study of resolvent and asymptotic expansions for eigen values. An important role is played by different versions of the theory of pseudo-differential operators in $ \mathbf R ^ {n} $, taking into account effects related to the description of the behaviour of functions at infinity, and often inspired by mathematical problems in quantum mechanics arising in the study of quantization of classical systems (cf. [5], [11]). In the theory of local solvability of partial differential equations and in spectral theory it is expedient to use pseudo-differential operators whose behaviour can be described by weight functions replacing $ | \xi | $ in estimates of the type (2) (cf. [8], [14]). One has constructed an algebra of pseudo-differential operators on manifolds with boundary, containing, in particular, the parametrix of elliptic boundary value problems (cf. [3], [13]).

A particular case of pseudo-differential operators are the multi-dimensional singular integral and integro-differential operators, whose study prepared the emergence of the theory of pseudo-differential operators (cf. [12] and also Singular integral).

The theory of pseudo-differential operators serves as a basis for the study of Fourier integral operators (cf. Fourier integral operator; [7], [10]), which play the same role in the theory of hyperbolic equations as do pseudo-differential operators in the theory of elliptic equations.

References

[1a] J.J. Kohn, L. Nirenberg, "An algebra of pseudo-differential operators" Commun. Pure Appl. Math. , 18 : 1–2 (1965) pp. 269–305 MR0176362 Zbl 0171.35101
[1b] L. Hörmander, "Pseudo-differential operators" Commun. Pure Appl. Math. , 18 : 3 (1965) pp. 501–517 MR0180740 Zbl 0125.33401
[1c] J.J. Kohn, L. Nirenberg, "Non-coercive boundary value problems" Commun. Pure Appl. Math. , 18 : 3 (1965) pp. 443–492 MR0181815 Zbl 0125.33302
[1d] L. Hörmander, "Pseudo-differential operators and non-elliptic boundary problems" Ann. of Math. , 83 : 1 (1966) pp. 129–209 MR0233064 Zbl 0132.07402
[1e] L. Hörmander, "Pseudo-differential operators and hypoelliptic equations" A.P. Calderòn (ed.) , Singular Integrals , Proc. Symp. Pure Math. , 10 , Amer. Math. Soc. (1966) pp. 138–183 MR0383152 Zbl 0167.09603
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[10] J.J. Duistermaat, "Fourier integral operators" , Courant Inst. Math. (1973) MR0451313 Zbl 0272.47028
[11] V.P. Maslov, M.V. Fedoryuk, "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)
[12] M.S. Agranovich, "Elliptic singular integro-differential operators" Russian Math. Surveys , 20 : 5 (1965) pp. 1–121 Uspekhi Mat. Nauk , 20 : 5 (1965) pp. 3–120 Zbl 0149.36101
[13] L. Boutet de Monvel, "Boundary value problems for pseudodifferential operators" Acta Math. , 126 (1971) pp. 11–51
[14] L. Hörmander, "The Weyl calculus of pseudo-differential operators" Commun. Pure Appl. Math. , 32 : 3 (1979) pp. 359–443 MR517939 Zbl 0388.47032
[15a] H.O. Cordes, "Elliptic pseudo-differential operators - an abstract theory" , Lect. notes in math. , 756 , Springer (1979) MR551619 Zbl 0417.35004
[15b] H.O. Cordes, "Spectral theory of linear differential operators and comparison algebras" , Cambridge Univ. Press (1986) MR0890743 Zbl 0727.35092
[16] Yu.V. Egorov, "Linear differential equations of principal type" , Consultants Bureau (1986) (Translated from Russian) MR0872855 Zbl 0669.35001
[17] G. Grubb, "Functional calculus of pseudo-differential boundary problems" , Birkhäuser (1986) MR885088 Zbl 0622.35001
[18] B. Helffer, "Théorie spectrale pour des opérateurs globalement elliptiques" Astérisque , 112 (1984) MR0743094 Zbl 0541.35002
[19a] L. Hörmander, "Pseudo-differential operators of type 1,1" Comm. Partial Diff. Eq. , 13 : 9 (1988) pp. 1085–1111 MR0946283 Zbl 0667.35078
[19b] L. Hörmander, "Continuity of pseudo-differential operators of type 1,1" Comm. Partial Diff. Eq. , 14 : 2 (1989) pp. 231–243 MR0976972 Zbl 0688.35107
[20] V. Ivrii, "Precise spectral asymptotics for elliptic operators" , Lect. notes in math. , 1100 , Springer (1984) MR0771297
[21] S. Rempel, B.-W. Schulze, "Index theory of elliptic boundary problems" , Akademie Verlag (1982) MR0690065 Zbl 0504.35002

Comments

The phrase "pseudo-differential operator" is often abbreviated to $\rm{\Psi DO}$, just like $\rm{PDO}$ for "partial differential operator".

For algebras of $\rm{\Psi DO}$s on manifolds with singularities, in particular $\rm{\Psi DO}$s with discontinuous symbols, see [a2].

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985) MR2512677 MR2304165 MR2108588 MR1996773 MR1481433 MR1313500 MR1065993 MR1065136 MR0961959 MR0925821 MR0881605 MR0862624 MR1540773 MR0781537 MR0781536 MR0717035 MR0705278 Zbl 1178.35003 Zbl 1115.35005 Zbl 1062.35004 Zbl 1028.35001 Zbl 0712.35001 Zbl 0687.35002 Zbl 0619.35002 Zbl 0619.35001 Zbl 0612.35001 Zbl 0601.35001 Zbl 0521.35002 Zbl 0521.35001
[a2] B.A. Plamenevskii, "Algebras of pseudodifferential operators" , Kluwer (1989) (Translated from Russian) MR1105811 MR1045105 MR1026642 MR0992982 Zbl 0677.35090
[a3] M.E. Taylor, "Pseudo-differential operators" , Princeton Univ. Press (1981) MR1567325 Zbl 0289.35001 Zbl 0207.45402
[a4] J. Chazarain, A. Piriou, "Introduction to the theory of linear partial differential equations" , North-Holland (1982) (Translated from French) MR0678605 Zbl 0487.35002
How to Cite This Entry:
Pseudo-differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-differential_operator&oldid=12424
This article was adapted from an original article by M.A. Shubin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article