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Microlocal analysis considers (generalized, hyper-) functions, operators, etc. in the  "microlocal"  range. Here,  "microlocal"  means seeing the matter more locally than usual by introducing the (cotangential) direction at every point. In Fourier analysis it corresponds to viewing things locally in both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637602.png" />. In view of the uncertainty principle, this is possible only by considering the objects modulo regular parts. This idea was first used in the study of pseudo-differential operators by P.D. Lax, S. Mizohata, L. Hörmander, etc. V.P. Maslov has enriched the theory by the introduction of a canonical structure. M. Sato has constructed the sheaf of micro-functions on the cotangent sphere bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637603.png" /> of the base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637604.png" /> as the basic object of microlocal analysis.
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Microlocal analysis considers (generalized, hyper-) functions, operators, etc. in the  "microlocal"  range. Here,  "microlocal"  means seeing the matter more locally than usual by introducing the (cotangential) direction at every point. In Fourier analysis it corresponds to viewing things locally in both $  x $
 +
and $  \xi $.  
 +
In view of the uncertainty principle, this is possible only by considering the objects modulo regular parts. This idea was first used in the study of pseudo-differential operators by P.D. Lax, S. Mizohata, L. Hörmander, etc. V.P. Maslov has enriched the theory by the introduction of a canonical structure. M. Sato has constructed the sheaf of micro-functions on the cotangent sphere bundle $  S  ^ {*} M $
 +
of the base space $  M $
 +
as the basic object of microlocal analysis.
  
 
==Micro-analyticity of hyperfunctions.==
 
==Micro-analyticity of hyperfunctions.==
A [[Hyperfunction|hyperfunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637605.png" /> is said to be micro-analytic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637606.png" /> if on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637607.png" /> it admits [[Analytic continuation|analytic continuation]] into the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637608.png" />, in the sense that it admits a boundary-value representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m0637609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376010.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376011.png" />. This is equivalent to saying that near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376014.png" /> is the germ of a real-analytic function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376015.png" /> is a Fourier hyperfunction (cf. [[Hyperfunction|Hyperfunction]]) exponentially decreasing in a conic neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376016.png" />. The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376017.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376018.png" /> is not micro-analytic is called the singular spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376019.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376020.png" />. By definition,
+
A [[Hyperfunction|hyperfunction]] $  f \in {\mathcal B} ( M) $
 +
is said to be micro-analytic at $  ( x _ {0} , \xi _ {0} ) \in S  ^ {*} M $
 +
if on a neighbourhood of $  x _ {0} $
 +
it admits [[Analytic continuation|analytic continuation]] into the half-space $  \langle  \mathop{\rm Im}  z , \xi _ {0} \rangle < 0 $,  
 +
in the sense that it admits a boundary-value representation $  \sum _ {j=} 1  ^ {N} F _ {j} ( x + i \Gamma _ {j} 0 ) $
 +
such that $  \Gamma _ {j} \cap \{ \langle  \mathop{\rm Im}  z , \xi _ {0} \rangle < 0 \} \neq \emptyset $
 +
for every $  j $.  
 +
This is equivalent to saying that near $  x _ {0} $,  
 +
$  f = {\mathcal F}  ^ {-} 1 ( g) + h $,  
 +
where $  h $
 +
is the germ of a real-analytic function and $  g $
 +
is a Fourier hyperfunction (cf. [[Hyperfunction|Hyperfunction]]) exponentially decreasing in a conic neighbourhood of $  \xi _ {0} $.  
 +
The set of points $  ( x _ {0} , \xi _ {0} ) \in \Omega \times S  ^ {n-} 1 $
 +
at which $  f $
 +
is not micro-analytic is called the singular spectrum of $  f $,  
 +
and is denoted by $  \mathop{\rm S}.S. f $.  
 +
By definition,
  
<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/m/m063/m063760/m06376021.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S. F ( x + i \Gamma 0 )  \subset  \Omega \times
 +
( \Gamma  ^  \circ  \cap S  ^ {n-} 1 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376022.png" /> is the dual cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376023.png" />. Conversely, a hyperfunction satisfying this estimate can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376024.png" />.
+
where $  \Gamma  ^  \circ  = \{ {\xi \in \mathbf R  ^ {n} } : {\langle  y , \xi \rangle \geq  0 \textrm{ for  all  }  y \in \Gamma } \} $
 +
is the dual cone of $  \Gamma $.  
 +
Conversely, a hyperfunction satisfying this estimate can be written in the form $  F ( x + i \Gamma 0 ) $.
  
 
==Operations and the singular spectrum.==
 
==Operations and the singular spectrum.==
 
The following inclusions hold:
 
The following inclusions hold:
  
<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/m/m063/m063760/m06376025.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S. ( f g ) \subset
 +
$$
  
<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/m/m063/m063760/m06376026.png" /></td> </tr></table>
+
$$
 +
\subset  \
 +
\{ {( x , \xi + \eta ) } : {( x , \xi )
 +
\in  \mathop{\rm S}.S. f , ( x , \eta ) \in  \mathop{\rm S}.S. g } \} \cup
 +
$$
  
<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/m/m063/m063760/m06376027.png" /></td> </tr></table>
+
$$
 +
\cup
 +
\{ {( x , \xi ) } : {x \in \supp  f
 +
, ( x , \xi ) \in  \mathop{\rm S}.S. g ,
 +
\textrm{ or }  x \in \supp  g , ( x , \xi ) \in  \mathop{\rm S}.S. f } \} ;
 +
$$
  
<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/m/m063/m063760/m06376028.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S.  f ( \Phi ( \widetilde{x}  ) )  \subset  ( \Phi  ^ {-} 1 \times {}  ^ {t} d \Phi ) (  \mathop{\rm S}.S. f  ) .
 +
$$
  
Here, the operations are legitimate if the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376029.png" /> does not appear in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376030.png" />-component of the right-hand side. In particular, under coordinate transformation the singular spectrum behaves like a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376031.png" />. Restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376032.png" /> is possible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376033.png" />, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376034.png" /> is said to contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376035.png" /> as real-analytic parameter at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376037.png" />, and then
+
Here, the operations are legitimate if the vector 0 $
 +
does not appear in the $  \xi $-
 +
component of the right-hand side. In particular, under coordinate transformation the singular spectrum behaves like a subset of $  S  ^ {*} \mathbf R  ^ {n} $.  
 +
Restriction $  f ( x , 0 ) = f ( x , t ) \mid  _ {t=} 0 $
 +
is possible if $  \mathop{\rm S}.S.  f \cap S _ {\{ t = 0 \} }  ^ {*} M = \emptyset $,  
 +
in which case $  f $
 +
is said to contain $  t $
 +
as real-analytic parameter at $  t = 0 $,  
 +
and then
  
<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/m/m063/m063760/m06376038.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S.  f ( x , 0 )  \subset  \rho _ {*} (  \mathop{\rm S}.S. f  ) \mid  _ {t=} 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376039.png" /> denotes projection on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376040.png" /> components. The dual assertion is:
+
where $  \rho $
 +
denotes projection on the $  d x $
 +
components. The dual assertion is:
  
<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/m/m063/m063760/m06376041.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S. \int\limits f ( x , t )  d x \subset
 +
$$
  
<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/m/m063/m063760/m06376042.png" /></td> </tr></table>
+
$$
 +
\subset  \
 +
\{ ( t , \theta ) : ( x , t , 0 , \theta ) \in  \mathop{\rm S}.S. \
 +
f  \textrm{ for  some  }  x \} .
 +
$$
  
 
The combination of these assertions gives a convolution, dual to the product:
 
The combination of these assertions gives a convolution, dual to the product:
  
<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/m/m063/m063760/m06376043.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S. ( f \star g ) \subset
 +
$$
  
<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/m/m063/m063760/m06376044.png" /></td> </tr></table>
+
$$
 +
\subset  \
 +
\{ ( x + y , \xi ) : \
 +
( x , \xi ) \in fnem S.S.  f , ( y , \xi ) \in fnnem S.S. g \} .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376045.png" /> be a [[Linear differential operator|linear differential operator]] with real-analytic coefficients and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376046.png" /> be its [[Characteristic manifold|characteristic manifold]]. Then
+
Let $  P ( x , \partial  ) $
 +
be a [[Linear differential operator|linear differential operator]] with real-analytic coefficients and let $  \mathop{\rm Char}  P = \{ {( x , \xi ) } : {P _ {m} ( x , \xi ) = 0 } \} $
 +
be its [[Characteristic manifold|characteristic manifold]]. Then
  
<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/m/m063/m063760/m06376047.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm S}.S. P ( x , \partial  ) u ( x)  \subset  \
 +
\mathop{\rm S}.S.  u ( x) \subset
 +
$$
  
<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/m/m063/m063760/m06376048.png" /></td> </tr></table>
+
$$
 +
\subset  \
 +
\mathop{\rm S}.S. P ( x , \partial  ) u ( x) \cup  \mathop{\rm Char}  P
 +
$$
  
(Sato's fundamental theorem). Hence Cauchy data for solutions can be specified on a non-characteristic manifold. Holmgren's uniqueness theorem holds with these data. More generally, for any hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376050.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376051.png" /> (the Kashiwara–Kawai Holmgren-type theorem); further, the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376053.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376054.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376056.png" /> denotes projection to the equator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376057.png" /> (the watermelon theorem).
+
(Sato's fundamental theorem). Hence Cauchy data for solutions can be specified on a non-characteristic manifold. Holmgren's uniqueness theorem holds with these data. More generally, for any hyperfunction $  u $,
 +
0 \in \supp  u \subset  \{ x _ {1} \geq  0 \} $
 +
implies $  ( 0 , \pm  d x _ {1} ) \in  \mathop{\rm S}.S. u $(
 +
the Kashiwara–Kawai Holmgren-type theorem); further, the fibre $  E $
 +
of $  \mathop{\rm S}.S. u $
 +
at 0 $
 +
has the form $  \rho  ^ {-} 1 \rho ( E \setminus  \{ \pm  d x _ {1} \} ) \cup \{ \pm  d x _ {1} \} $,  
 +
where $  \rho : S  ^ {n-} 1 \setminus  \{ \pm  d x _ {1} \} \rightarrow S  ^ {n-} 2 $
 +
denotes projection to the equator $  \xi _ {1} = 0 $(
 +
the watermelon theorem).
  
 
==Decomposition of singular spectra.==
 
==Decomposition of singular spectra.==
 
One has
 
One has
  
<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/m/m063/m063760/m06376058.png" /></td> </tr></table>
+
$$
 +
\delta ( x)  = \int\limits _ {S  ^ {n-} 1 } W ( x , \omega )  d \omega ,
 +
$$
 +
 
 +
$$
 +
W ( x , \omega )  =
 +
\frac{( n - 1 ) ! }{( - 2 \pi i
 +
)  ^ {n} }
 +
 +
\frac{ \mathop{\rm det} (  \mathop{\rm grad} _  \omega  \psi ( x , \omega )
 +
) }{( \phi ( x , \omega ) + i 0 )  ^ {n} }
 +
,
 +
$$
  
<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/m/m063/m063760/m06376059.png" /></td> </tr></table>
+
where the twisted phase  $  \phi ( x , \omega ) $
 +
is a real-analytic function of  $  ( x , \omega ) $
 +
which is of positive type in  $  x $(
 +
that is,  $  \mathop{\rm Re}  \phi ( x , \omega ) = 0 $
 +
implies  $  \mathop{\rm Im}  \phi ( x , \omega ) \geq  0 $),
 +
is homogeneous of degree 1 in  $  \omega $,
 +
and  $  \phi ( 0 , \omega ) = 0 $,
 +
$  \mathop{\rm grad} _ {x}  \phi ( 0 , \omega ) = \omega $;  
 +
and the vector  $  \psi ( x , \omega ) $
 +
is such that  $  \langle  \psi ( x , \omega ) , x \rangle = \phi ( x , \omega ) $.  
 +
This is a generalization of the classical Radon decomposition, in which
  
where the twisted phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376060.png" /> is a real-analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376061.png" /> which is of positive type in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376062.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376063.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376064.png" />), is homogeneous of degree 1 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376065.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376067.png" />; and the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376068.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376069.png" />. This is a generalization of the classical Radon decomposition, in which
+
$$
 +
W ( x , \omega )  =
 +
\frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {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/m/m063/m063760/m06376070.png" /></td> </tr></table>
+
\frac{1}{( x \omega + i 0 )  ^ {n} }
 +
.
 +
$$
  
The component, regarded as a hyperfunction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376071.png" />, has a singular spectrum with only one direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376072.png" />. Via convolution it gives a similar decomposition of general hyperfunctions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376073.png" /> also satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376074.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376075.png" />, then the singular spectrum of the component as a hyperfunction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376076.png" /> is precisely one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376077.png" />; this is useful in applications. Typical examples are:
+
The component, regarded as a hyperfunction of $  x $,  
 +
has a singular spectrum with only one direction $  \omega $.  
 +
Via convolution it gives a similar decomposition of general hyperfunctions. If $  \phi ( x , \omega ) $
 +
also satisfies $  \phi ( x , \omega ) \neq 0 $
 +
for $  x \neq 0 $,  
 +
then the singular spectrum of the component as a hyperfunction of $  x $
 +
is precisely one point $  ( 0 , \omega ) $;  
 +
this is useful in applications. Typical examples are:
  
<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/m/m063/m063760/m06376078.png" /></td> </tr></table>
+
$$
 +
W ( x , \omega )  =
 +
\frac{( n - 1 ) ! }{( - 2 \pi i )  ^ {n} }
 +
 
 +
\frac{( 1 - i x \omega )  ^ {n-} 1 - ( 1 - i x \omega )  ^ {n-} 2
 +
( x  ^ {2} - ( x \omega )  ^ {2} ) }{( x \omega + i ( x  ^ {2} - ( x \omega )  ^ {2} ) + i 0 )  ^ {n} }
 +
 
 +
$$
  
 
(Kashiwara example);
 
(Kashiwara example);
  
<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/m/m063/m063760/m06376079.png" /></td> </tr></table>
+
$$
 +
W ( x , \omega )  =
 +
\frac{( n - 1 ) ! }{( - 2 \pi i )  ^ {n} }
 +
 
 +
\frac{1 + i \alpha x \omega }{( x \omega + i \alpha x  ^ {2} + i 0 ) ^ {n } }
 +
,\  \alpha > 0,
 +
$$
 +
 
 +
(Bony example). For such a decomposition  $  f $
 +
is micro-analytic at  $  ( x _ {0} , \xi _ {0} ) $
 +
if and only if  $  f \star _ {x} W ( x , \omega ) $
 +
is real analytic in  $  ( x , \omega ) $
 +
at  $  ( x _ {0} , \omega _ {0} ) $.
  
(Bony example). For such a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376080.png" /> is micro-analytic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376081.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376082.png" /> is real analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376083.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376084.png" />.
+
The Fourier–Bros–Iagolnitzer transform of a hyperfunction  $  f $(
 +
the FBI-transform of  $  f  $)  
 +
is
  
The Fourier–Bros–Iagolnitzer transform of a hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376085.png" /> (the FBI-transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376086.png" />) is
+
$$
 +
\int\limits _ {\mathbf R  ^ {n} } e ^ {i \lambda \phi ( x, y, \xi , \alpha ) }
 +
f( y)  dy ,
 +
$$
  
<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/m/m063/m063760/m06376087.png" /></td> </tr></table>
+
where  $  \phi ( x , y , \xi , \alpha ) $
 +
is a real-analytic function satisfying: 1) for  $  x = y = \alpha $
 +
one has  $  \phi = 0 $
 +
and  $  \phi _ {x} = - \phi _ {y} = \xi $;  
 +
and 2)  $  \mathop{\rm Im}  \phi \geq  C ( | x - \alpha |  ^ {2} + | y - \alpha |  ^ {2} ) $
 +
for some  $  C > 0 $.  
 +
A typical example of such a  $  \phi $
 +
is
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376088.png" /> is a real-analytic function satisfying: 1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376089.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376091.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376092.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376093.png" />. A typical example of such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376094.png" /> is
+
$$
 +
\phi ( x , y , \xi , \alpha )  = ( x - y ) \xi + i
 +
( ( x - \alpha )  ^ {2} + ( y - \alpha ) ^ {2} ) .
 +
$$
  
<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/m/m063/m063760/m06376095.png" /></td> </tr></table>
+
A hyperfunction  $  f $
 +
is micro-analytic at  $  ( x _ {0} , \xi _ {0} ) $
 +
if and only if for some (equivalently, any) modification of  $  f $
 +
with compact support its FBI-transform is exponentially decreasing with respect to  $  \lambda $
 +
uniformly in  $  ( x , \xi , \alpha ) $
 +
in a neighbourhood of  $  ( x _ {0} , \xi _ {0} , x _ {0} ) $.  
 +
Integration of the inversion formula over the radial variable gives the formula
  
A hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376096.png" /> is micro-analytic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376097.png" /> if and only if for some (equivalently, any) modification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376098.png" /> with compact support its FBI-transform is exponentially decreasing with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m06376099.png" /> uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760100.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760101.png" />. Integration of the inversion formula over the radial variable gives the formula
+
$$
 +
\delta ( x - y ) =
 +
$$
  
<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/m/m063/m063760/m063760102.png" /></td> </tr></table>
+
$$
 +
= \
  
<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/m/m063/m063760/m063760103.png" /></td> </tr></table>
+
\frac{2  ^ {n} \Gamma ( 3 n / 2 ) }{( - 2 \pi i )  ^ {3n/2} }
 +
\times
 +
$$
  
<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/m/m063/m063760/m063760104.png" /></td> </tr></table>
+
$$
 +
\times
 +
\int\limits _ {\mathbf R  ^ {n} \times S  ^ {n-} 1 }
 +
\frac{1 + ( i / 2 ) ( x - y ) \omega }{( ( x - y ) \omega + i ( ( x - \alpha )  ^ {2} + (
 +
y - \alpha )  ^ {2} ) + i 0 )  ^ {3n/2} }
 +
  d \alpha  d \omega .
 +
$$
  
This supplies a partition of unity of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760105.png" />. All these arguments are compatible with the corresponding theory of (analytic) [[Wave front|wave front]] sets for distributions.
+
This supplies a partition of unity of the sheaf $  {\mathcal B} / {\mathcal A} $.  
 +
All these arguments are compatible with the corresponding theory of (analytic) [[Wave front|wave front]] sets for distributions.
  
The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760106.png" /> of micro-functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760107.png" /> is the sheaf associated with the pre-sheaf
+
The sheaf $  {\mathcal C} $
 +
of micro-functions on $  S  ^ {*} M $
 +
is the sheaf associated with the pre-sheaf
  
<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/m/m063/m063760/m063760108.png" /></td> </tr></table>
+
$$
 +
S  ^ {*} M  \supset  \Omega \times \Delta  \mapsto  {\mathcal C}
 +
( \Omega \times \Delta ) =
 +
$$
  
<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/m/m063/m063760/m063760109.png" /></td> </tr></table>
+
$$
 +
= \
 +
{ {\mathcal B} ( \Omega ) } / {\{ f \in {\mathcal B} ( \Omega ) :
 +
  \mathop{\rm S}.S. f \cap \Omega \times \Delta = \emptyset \} } .
 +
$$
  
The sequence of sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760110.png" />:
+
The sequence of sheaves on $  M $:
  
<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/m/m063/m063760/m063760111.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  {\mathcal A}  \rightarrow  {\mathcal B}  \mathop \rightarrow \limits ^ { { \mathop{\rm sp}}  }  \pi _ {*} {\mathcal C}  \rightarrow  0
 +
$$
  
is exact. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760112.png" /> denotes projection. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760113.png" /> for a hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760114.png" />. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760115.png" /> is flabby, which implies the possibility of arbitrary modification of hyperfunctions preserving the singular spectrum. Analytic pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]) and micro-differential operators naturally act on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760116.png" /> as sheaf homomorphisms. They act isomorphically at a non-characteristic point. Canonical transforms induce ring isomorphisms of the sheaf of pseudo-differential operators. Using these, a simple characteristic system of pseudo-differential equations can be locally reduced to the direct sum of copies of de Rham, Cauchy–Riemann and Lewy–Mizohata equations (the fundamental structure theorem).
+
is exact. Here $  \pi : S  ^ {*} M \rightarrow M $
 +
denotes projection. By definition, $  \mathop{\rm S}.S. f = \supp  \mathop{\rm sp}  [ f  ] $
 +
for a hyperfunction $  f $.  
 +
The sheaf $  {\mathcal C} $
 +
is flabby, which implies the possibility of arbitrary modification of hyperfunctions preserving the singular spectrum. Analytic pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]) and micro-differential operators naturally act on $  {\mathcal C} $
 +
as sheaf homomorphisms. They act isomorphically at a non-characteristic point. Canonical transforms induce ring isomorphisms of the sheaf of pseudo-differential operators. Using these, a simple characteristic system of pseudo-differential equations can be locally reduced to the direct sum of copies of de Rham, Cauchy–Riemann and Lewy–Mizohata equations (the fundamental structure theorem).
  
The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760117.png" /> is constructed from the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760118.png" /> by Sato micro-localization: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760119.png" /> be a real-analytic manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760120.png" /> a complex neighbourhood of it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760121.png" /> be the real blowing-up, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760123.png" /> the canonical inclusions, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760124.png" /> be the subset of the fibre product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760126.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760127.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760128.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760129.png" /> be the canonical projections to the factors. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760130.png" /> be the [[Orientation|orientation]] sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063760/m063760131.png" />. Then
+
The sheaf $  {\mathcal C} $
 +
is constructed from the sheaf $  {\mathcal O} $
 +
by Sato micro-localization: Let $  M $
 +
be a real-analytic manifold and $  X $
 +
a complex neighbourhood of it. Let $  {}  ^ {M} {\widetilde{X}  } = ( X \setminus  M ) \amalg S _ {M} X $
 +
be the real blowing-up, $  \iota : X \setminus  M \rightarrow X $,  
 +
$  j : X \setminus  M \rightarrow {}  ^ {M} {\widetilde{X}  } $
 +
the canonical inclusions, and let $  D M $
 +
be the subset of the fibre product of $  S _ {M} X $
 +
and $  S _ {M}  ^ {*} X $
 +
over $  M $
 +
defined by $  \langle  \xi , \eta \rangle \leq  0 $;  
 +
let $  \phi , \tau $
 +
be the canonical projections to the factors. Let $  \omega $
 +
be the [[Orientation|orientation]] sheaf of $  M $.  
 +
Then
  
<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/m/m063/m063760/m063760132.png" /></td> </tr></table>
+
$$
 +
{\mathcal C}  = \mathbf R \tau _ {*} \pi  ^ {-} 1 \mathbf R \Gamma _ {S _ {M}  X } ( j _ {*} \iota  ^ {-} 1 {\mathcal O} ) \otimes \omega [ n] =
 +
$$
  
<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/m/m063/m063760/m063760133.png" /></td> </tr></table>
+
$$
 +
= \
 +
\mathbf R  ^ {n-} 1 \tau _ {*} \pi  ^ {-} 1 {\mathcal H} _ {S _ {M}  X }  ^ {1} ( j _ {*} \iota  ^ {-} 1 {\mathcal O} ) \otimes \omega
 +
$$
  
 
(the fundamental vanishing theorem). This argument can be generalized to the second micro-localization of micro-functions with respect to a holomorphic parameter or to micro-localization of any sheaf.
 
(the fundamental vanishing theorem). This argument can be generalized to the second micro-localization of micro-functions with respect to a holomorphic parameter or to micro-localization of any sheaf.

Latest revision as of 08:00, 6 June 2020


Microlocal analysis considers (generalized, hyper-) functions, operators, etc. in the "microlocal" range. Here, "microlocal" means seeing the matter more locally than usual by introducing the (cotangential) direction at every point. In Fourier analysis it corresponds to viewing things locally in both $ x $ and $ \xi $. In view of the uncertainty principle, this is possible only by considering the objects modulo regular parts. This idea was first used in the study of pseudo-differential operators by P.D. Lax, S. Mizohata, L. Hörmander, etc. V.P. Maslov has enriched the theory by the introduction of a canonical structure. M. Sato has constructed the sheaf of micro-functions on the cotangent sphere bundle $ S ^ {*} M $ of the base space $ M $ as the basic object of microlocal analysis.

Micro-analyticity of hyperfunctions.

A hyperfunction $ f \in {\mathcal B} ( M) $ is said to be micro-analytic at $ ( x _ {0} , \xi _ {0} ) \in S ^ {*} M $ if on a neighbourhood of $ x _ {0} $ it admits analytic continuation into the half-space $ \langle \mathop{\rm Im} z , \xi _ {0} \rangle < 0 $, in the sense that it admits a boundary-value representation $ \sum _ {j=} 1 ^ {N} F _ {j} ( x + i \Gamma _ {j} 0 ) $ such that $ \Gamma _ {j} \cap \{ \langle \mathop{\rm Im} z , \xi _ {0} \rangle < 0 \} \neq \emptyset $ for every $ j $. This is equivalent to saying that near $ x _ {0} $, $ f = {\mathcal F} ^ {-} 1 ( g) + h $, where $ h $ is the germ of a real-analytic function and $ g $ is a Fourier hyperfunction (cf. Hyperfunction) exponentially decreasing in a conic neighbourhood of $ \xi _ {0} $. The set of points $ ( x _ {0} , \xi _ {0} ) \in \Omega \times S ^ {n-} 1 $ at which $ f $ is not micro-analytic is called the singular spectrum of $ f $, and is denoted by $ \mathop{\rm S}.S. f $. By definition,

$$ \mathop{\rm S}.S. F ( x + i \Gamma 0 ) \subset \Omega \times ( \Gamma ^ \circ \cap S ^ {n-} 1 ) , $$

where $ \Gamma ^ \circ = \{ {\xi \in \mathbf R ^ {n} } : {\langle y , \xi \rangle \geq 0 \textrm{ for all } y \in \Gamma } \} $ is the dual cone of $ \Gamma $. Conversely, a hyperfunction satisfying this estimate can be written in the form $ F ( x + i \Gamma 0 ) $.

Operations and the singular spectrum.

The following inclusions hold:

$$ \mathop{\rm S}.S. ( f g ) \subset $$

$$ \subset \ \{ {( x , \xi + \eta ) } : {( x , \xi ) \in \mathop{\rm S}.S. f , ( x , \eta ) \in \mathop{\rm S}.S. g } \} \cup $$

$$ \cup \{ {( x , \xi ) } : {x \in \supp f , ( x , \xi ) \in \mathop{\rm S}.S. g , \textrm{ or } x \in \supp g , ( x , \xi ) \in \mathop{\rm S}.S. f } \} ; $$

$$ \mathop{\rm S}.S. f ( \Phi ( \widetilde{x} ) ) \subset ( \Phi ^ {-} 1 \times {} ^ {t} d \Phi ) ( \mathop{\rm S}.S. f ) . $$

Here, the operations are legitimate if the vector $ 0 $ does not appear in the $ \xi $- component of the right-hand side. In particular, under coordinate transformation the singular spectrum behaves like a subset of $ S ^ {*} \mathbf R ^ {n} $. Restriction $ f ( x , 0 ) = f ( x , t ) \mid _ {t=} 0 $ is possible if $ \mathop{\rm S}.S. f \cap S _ {\{ t = 0 \} } ^ {*} M = \emptyset $, in which case $ f $ is said to contain $ t $ as real-analytic parameter at $ t = 0 $, and then

$$ \mathop{\rm S}.S. f ( x , 0 ) \subset \rho _ {*} ( \mathop{\rm S}.S. f ) \mid _ {t=} 0 , $$

where $ \rho $ denotes projection on the $ d x $ components. The dual assertion is:

$$ \mathop{\rm S}.S. \int\limits f ( x , t ) d x \subset $$

$$ \subset \ \{ ( t , \theta ) : ( x , t , 0 , \theta ) \in \mathop{\rm S}.S. \ f \textrm{ for some } x \} . $$

The combination of these assertions gives a convolution, dual to the product:

$$ \mathop{\rm S}.S. ( f \star g ) \subset $$

$$ \subset \ \{ ( x + y , \xi ) : \ ( x , \xi ) \in fnem S.S. f , ( y , \xi ) \in fnnem S.S. g \} . $$

Let $ P ( x , \partial ) $ be a linear differential operator with real-analytic coefficients and let $ \mathop{\rm Char} P = \{ {( x , \xi ) } : {P _ {m} ( x , \xi ) = 0 } \} $ be its characteristic manifold. Then

$$ \mathop{\rm S}.S. P ( x , \partial ) u ( x) \subset \ \mathop{\rm S}.S. u ( x) \subset $$

$$ \subset \ \mathop{\rm S}.S. P ( x , \partial ) u ( x) \cup \mathop{\rm Char} P $$

(Sato's fundamental theorem). Hence Cauchy data for solutions can be specified on a non-characteristic manifold. Holmgren's uniqueness theorem holds with these data. More generally, for any hyperfunction $ u $, $ 0 \in \supp u \subset \{ x _ {1} \geq 0 \} $ implies $ ( 0 , \pm d x _ {1} ) \in \mathop{\rm S}.S. u $( the Kashiwara–Kawai Holmgren-type theorem); further, the fibre $ E $ of $ \mathop{\rm S}.S. u $ at $ 0 $ has the form $ \rho ^ {-} 1 \rho ( E \setminus \{ \pm d x _ {1} \} ) \cup \{ \pm d x _ {1} \} $, where $ \rho : S ^ {n-} 1 \setminus \{ \pm d x _ {1} \} \rightarrow S ^ {n-} 2 $ denotes projection to the equator $ \xi _ {1} = 0 $( the watermelon theorem).

Decomposition of singular spectra.

One has

$$ \delta ( x) = \int\limits _ {S ^ {n-} 1 } W ( x , \omega ) d \omega , $$

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{ \mathop{\rm det} ( \mathop{\rm grad} _ \omega \psi ( x , \omega ) ) }{( \phi ( x , \omega ) + i 0 ) ^ {n} } , $$

where the twisted phase $ \phi ( x , \omega ) $ is a real-analytic function of $ ( x , \omega ) $ which is of positive type in $ x $( that is, $ \mathop{\rm Re} \phi ( x , \omega ) = 0 $ implies $ \mathop{\rm Im} \phi ( x , \omega ) \geq 0 $), is homogeneous of degree 1 in $ \omega $, and $ \phi ( 0 , \omega ) = 0 $, $ \mathop{\rm grad} _ {x} \phi ( 0 , \omega ) = \omega $; and the vector $ \psi ( x , \omega ) $ is such that $ \langle \psi ( x , \omega ) , x \rangle = \phi ( x , \omega ) $. This is a generalization of the classical Radon decomposition, in which

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{1}{( x \omega + i 0 ) ^ {n} } . $$

The component, regarded as a hyperfunction of $ x $, has a singular spectrum with only one direction $ \omega $. Via convolution it gives a similar decomposition of general hyperfunctions. If $ \phi ( x , \omega ) $ also satisfies $ \phi ( x , \omega ) \neq 0 $ for $ x \neq 0 $, then the singular spectrum of the component as a hyperfunction of $ x $ is precisely one point $ ( 0 , \omega ) $; this is useful in applications. Typical examples are:

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{( 1 - i x \omega ) ^ {n-} 1 - ( 1 - i x \omega ) ^ {n-} 2 ( x ^ {2} - ( x \omega ) ^ {2} ) }{( x \omega + i ( x ^ {2} - ( x \omega ) ^ {2} ) + i 0 ) ^ {n} } $$

(Kashiwara example);

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{1 + i \alpha x \omega }{( x \omega + i \alpha x ^ {2} + i 0 ) ^ {n } } ,\ \alpha > 0, $$

(Bony example). For such a decomposition $ f $ is micro-analytic at $ ( x _ {0} , \xi _ {0} ) $ if and only if $ f \star _ {x} W ( x , \omega ) $ is real analytic in $ ( x , \omega ) $ at $ ( x _ {0} , \omega _ {0} ) $.

The Fourier–Bros–Iagolnitzer transform of a hyperfunction $ f $( the FBI-transform of $ f $) is

$$ \int\limits _ {\mathbf R ^ {n} } e ^ {i \lambda \phi ( x, y, \xi , \alpha ) } f( y) dy , $$

where $ \phi ( x , y , \xi , \alpha ) $ is a real-analytic function satisfying: 1) for $ x = y = \alpha $ one has $ \phi = 0 $ and $ \phi _ {x} = - \phi _ {y} = \xi $; and 2) $ \mathop{\rm Im} \phi \geq C ( | x - \alpha | ^ {2} + | y - \alpha | ^ {2} ) $ for some $ C > 0 $. A typical example of such a $ \phi $ is

$$ \phi ( x , y , \xi , \alpha ) = ( x - y ) \xi + i ( ( x - \alpha ) ^ {2} + ( y - \alpha ) ^ {2} ) . $$

A hyperfunction $ f $ is micro-analytic at $ ( x _ {0} , \xi _ {0} ) $ if and only if for some (equivalently, any) modification of $ f $ with compact support its FBI-transform is exponentially decreasing with respect to $ \lambda $ uniformly in $ ( x , \xi , \alpha ) $ in a neighbourhood of $ ( x _ {0} , \xi _ {0} , x _ {0} ) $. Integration of the inversion formula over the radial variable gives the formula

$$ \delta ( x - y ) = $$

$$ = \ \frac{2 ^ {n} \Gamma ( 3 n / 2 ) }{( - 2 \pi i ) ^ {3n/2} } \times $$

$$ \times \int\limits _ {\mathbf R ^ {n} \times S ^ {n-} 1 } \frac{1 + ( i / 2 ) ( x - y ) \omega }{( ( x - y ) \omega + i ( ( x - \alpha ) ^ {2} + ( y - \alpha ) ^ {2} ) + i 0 ) ^ {3n/2} } d \alpha d \omega . $$

This supplies a partition of unity of the sheaf $ {\mathcal B} / {\mathcal A} $. All these arguments are compatible with the corresponding theory of (analytic) wave front sets for distributions.

The sheaf $ {\mathcal C} $ of micro-functions on $ S ^ {*} M $ is the sheaf associated with the pre-sheaf

$$ S ^ {*} M \supset \Omega \times \Delta \mapsto {\mathcal C} ( \Omega \times \Delta ) = $$

$$ = \ { {\mathcal B} ( \Omega ) } / {\{ f \in {\mathcal B} ( \Omega ) : \mathop{\rm S}.S. f \cap \Omega \times \Delta = \emptyset \} } . $$

The sequence of sheaves on $ M $:

$$ 0 \rightarrow {\mathcal A} \rightarrow {\mathcal B} \mathop \rightarrow \limits ^ { { \mathop{\rm sp}} } \pi _ {*} {\mathcal C} \rightarrow 0 $$

is exact. Here $ \pi : S ^ {*} M \rightarrow M $ denotes projection. By definition, $ \mathop{\rm S}.S. f = \supp \mathop{\rm sp} [ f ] $ for a hyperfunction $ f $. The sheaf $ {\mathcal C} $ is flabby, which implies the possibility of arbitrary modification of hyperfunctions preserving the singular spectrum. Analytic pseudo-differential operators (cf. Pseudo-differential operator) and micro-differential operators naturally act on $ {\mathcal C} $ as sheaf homomorphisms. They act isomorphically at a non-characteristic point. Canonical transforms induce ring isomorphisms of the sheaf of pseudo-differential operators. Using these, a simple characteristic system of pseudo-differential equations can be locally reduced to the direct sum of copies of de Rham, Cauchy–Riemann and Lewy–Mizohata equations (the fundamental structure theorem).

The sheaf $ {\mathcal C} $ is constructed from the sheaf $ {\mathcal O} $ by Sato micro-localization: Let $ M $ be a real-analytic manifold and $ X $ a complex neighbourhood of it. Let $ {} ^ {M} {\widetilde{X} } = ( X \setminus M ) \amalg S _ {M} X $ be the real blowing-up, $ \iota : X \setminus M \rightarrow X $, $ j : X \setminus M \rightarrow {} ^ {M} {\widetilde{X} } $ the canonical inclusions, and let $ D M $ be the subset of the fibre product of $ S _ {M} X $ and $ S _ {M} ^ {*} X $ over $ M $ defined by $ \langle \xi , \eta \rangle \leq 0 $; let $ \phi , \tau $ be the canonical projections to the factors. Let $ \omega $ be the orientation sheaf of $ M $. Then

$$ {\mathcal C} = \mathbf R \tau _ {*} \pi ^ {-} 1 \mathbf R \Gamma _ {S _ {M} X } ( j _ {*} \iota ^ {-} 1 {\mathcal O} ) \otimes \omega [ n] = $$

$$ = \ \mathbf R ^ {n-} 1 \tau _ {*} \pi ^ {-} 1 {\mathcal H} _ {S _ {M} X } ^ {1} ( j _ {*} \iota ^ {-} 1 {\mathcal O} ) \otimes \omega $$

(the fundamental vanishing theorem). This argument can be generalized to the second micro-localization of micro-functions with respect to a holomorphic parameter or to micro-localization of any sheaf.

For additional references see also Hyperfunction.

References

[a1] M. Kashiwara, "Microfunctions and pseudo-differential equations" H. Komatsu (ed.) , Hyperfunctions and Pseudo-differential Equations, Part II , Lect. notes in math. , 287 , Springer (1973) pp. 263–529
[a2] M. Kashiwara, "Systems of micro-differential equations" , Birkhäuser (1983) (Translated from French)
[a3] J. Sjöstrand, "Singularités analytiques microlocales" Astérisque , 95 (1982)
[a4] Y. Laurent, "Théorie de la deuxième microlocalisation dans le domaine complexe" , Birkhäuser (1985)
[a5] M. Kashiwara, P. Schapira, "Microlocal study of sheaves" Astérisque , 128 (1986)
How to Cite This Entry:
Microlocal analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Microlocal_analysis&oldid=15497
This article was adapted from an original article by A. Kaneko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article