# Pseudo-differential operator

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

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The phrase "pseudo-differential operator" is often abbreviated to $\Psi DO$, just like $PDO$ for "partial differential operator" .
For algebras of $\Psi DOS$ on manifolds with singularities, in particular $\Psi DOS$ with discontinuous symbols, see [a2].