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 be an open set in
, and let
be the space of infinitely-differentiable functions on
with compact support belonging to
. The simplest pseudo-differential operator on
is the operator
given by
![]() | (1) |
Here, ,
,
is Lebesgue measure on
,
is the usual inner product of the vectors
and
,
is the Fourier transform of the function
, i.e.
![]() |
(the integral, like the one in (1), is over all of ), and
is a smooth function on
satisfying certain conditions and is called the symbol of the pseudo-differential operator
(cf. also Symbol of an operator). An operator
of the form (1) is denoted by
or
. If
![]() |
is a polynomial in with coefficients
(here
is a multi-index, i.e.
,
,
are integers,
,
), then
coincides with the differential operator obtained when
is substituted for
in the expression for
.
One often uses the class of symbols satisfying the conditions
![]() | (2) |
![]() |
Here are multi-indices,
,
, and
is a compact set in
. This class is denoted by
(or by
).
It is usually assumed that . By
(or
) one denotes the class of operators of the form
, where
and
is an integral operator with a
-kernel, i.e. an operator of the form
![]() |
where . (Such operators
are also called pseudo-differential operators in
.) The function
is called, like before, the symbol of
. However, in this case it is not uniquely defined, but only up to a symbol from
. An operator
is called a pseudo-differential operator of order not exceeding
and type
. The differential operator described above belongs to the class
. The smallest possible value of
is called the order of the pseudo-differential operator. The classes
and
are often called the Hörmander classes.
One may specify pseudo-differential operators in by double symbols or amplitudes, i.e. write them in the form
![]() | (3) |
For this formula turns into (1). It is usually assumed that
, i.e.
![]() | (4) |
![]() |
here is a compact set in
. If
, then the class of operators (3) (for all possible functions
) coincides with
. In this case the symbol
(determined up to a symbol from
) has the following asymptotic expansion:
![]() |
where and the summation extends over all multi-indices. This formula means that the difference between
and the partial sum over all
for which
is a symbol in
, i.e. is a symbol of order at most equal to the largest of the orders of the rest terms.
A pseudo-differential operator can be extended, by continuity or duality, to an operator
. Here
and
are the space of generalized functions and the space of generalized functions with compact support in
, respectively (cf. Generalized functions, space of). If
, then the pseudo-differential operator has the following pseudo-locality property: If
, where
, then
. Another formulation of this property is: The kernel
(in the sense of L. Schwartz) of
is infinitely differentiable in
for
.
A classical pseudo-differential operator of order in
is an operator
whose symbol
has the asymptotic expansion
![]() |
where ,
for
,
for
, and where
is positively homogeneous in
of order
:
![]() |
A differential operator (with smooth coefficients) serves as an example of a classical pseudo-differential operator. The function is called the principal symbol of a classical pseudo-differential operator of order
.
A pseudo-differential operator in
is called properly supported if the projections of
onto each factor when restricted to the support of the kernel of
are proper mappings (cf. also Proper morphism). A properly supported pseudo-differential operator maps
into
and can be extended, by continuity, to mappings
,
and
. It can be written in the form (1) with symbol
, where the exponent is understood as a function of
with
as parameter.
Suppose that are pseudo-differential operators in
one of which is properly supported. Then their product (composition)
makes sense. The composition theorem plays an important role in the theory of pseudo-differential operators: If
,
,
, then
. If, moreover,
and
,
and
are the symbols of
,
and
, then
![]() |
In particular, if are classical pseudo-differential operators of orders
and
, then
is a classical pseudo-differential operator of order
with principal symbol
, where
and
are the principal symbols of
and
.
If ,
, then there exists a, moreover unique, adjoint pseudo-differential operator
for which
,
, where
is the inner product of
and
in
. If, moreover,
,
is the symbol of
and
is the symbol of
, then
![]() |
Thus, the properly supported pseudo-differential operators for 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 -norm, in its most precise form, asserts the following (cf. [8]): Let
and let
be an operator of the form (3) with double symbol
satisfying (4), in which the numbers
satisfy the conditions
![]() | (5) |
then can be extended to a bounded operator
. In particular, under the conditions (5) pseudo-differential operators of the form (1) with symbols satisfying conditions (2) uniformly in
(i.e. such that the constants
do not depend on
) are bounded in
. This implies, e.g., the boundedness in
of operators
if
and if the kernel of
has compact support (when the bounds on the symbol are, again, uniform in
). For
or for
, 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 -norms, as well as in Hölder and in Gevrey norms (cf. [8]).
If an operator is given on
, where
,
and where (2) holds uniformly in
, then this operator can be extended to a bounded operator
,
, where
denotes the usual Sobolev space over
(which is sometimes denoted also by
).
The class of pseudo-differential operators in for
is naturally invariant under diffeomorphisms. Its subclass of classical pseudo-differential operators has the same property. This makes it possible to define the class
and classical pseudo-differential operators on an arbitrary smooth manifold
. The formula for change of variables in the symbol under a diffeomorphism
, where
are domains in
, has the form
![]() |
Here is the symbol of
;
is the symbol of the operator
given by
, i.e. that obtained from
by a change of variables
;
denotes the Jacobian of
;
is the transposed matrix; and
![]() |
In particular, this implies that the principal symbol of a classical pseudo-differential operator on a manifold is a well-defined function on the cotangent bundle
.
If is a compact manifold (without boundary), then the pseudo-differential operators on
form an algebra with involution, if the involution is introduced by means of an inner product, given by a smooth positive density. An operator
is bounded in
, and if
for
, then it is compact in
. For classical pseudo-differential operators
of order
on
,
![]() |
where is the principal symbol of
and
runs over the set of compact operators in
. An operator
can by continuity be extended to a bounded linear operator from
into
for any
.
A parametrix of a pseudo-differential operator is a pseudo-differential operator
such that
and
are pseudo-differential operators of order
, i.e. are integral operators with a smooth kernel. Suppose that
,
, and that
is the symbol of
. A sufficient condition for
to have a parametrix is that the conditions
![]() | (6) |
are fulfilled.
In this case a parametrix exists. The simplest implication from the existence of a parametrix is that
is a hypo-elliptic operator: If
, where
, then
. In other words,
(cf. Support of a generalized function). The following exact result (the regularity theorem) is also valid: If
, then
. A micro-local regularity theorem is also valid:
, where
denotes the wave front of the generalized function
.
Condition (6) is invariant under diffeomorphisms for . Therefore the corresponding class of pseudo-differential operators on a manifold
has a meaning. If
is compact, then such an operator
is Fredholm in
(cf. Fredholm operator), i.e. has finite-dimensional kernel and co-kernel in
, and has a closed image.
A classical pseudo-differential operator of order
with smooth symbol
is called elliptic if
for
. For such an operator
condition (6) holds with
, and
has a parametrix that is also a classical pseudo-differential operator of order
. On a compact manifold
such an operator
gives rise to a Fredholm operator
![]() |
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 the index of the mapping
determined by it on the Sobolev classes of sections does not depend on
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 , 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
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 |
[2] | M.S. Agranovich, M.I. Vishik, "Pseudo-differential operators" , Moscow (1988) (In Russian) MR1023117 Zbl 0696.35123 Zbl 0167.09801 |
[3] | G.I. Eskin, "Boundary value problems for elliptic pseudodifferential equations" , Amer. Math. Soc. (1981) (Translated from Russian) MR0623608 Zbl 0458.35002 |
[4] | V.V. Grushin, "Pseudodifferential operators" , Moscow (1975) (In Russian) Zbl 0498.35090 Zbl 0255.35022 Zbl 0238.47038 Zbl 0223.35084 Zbl 0238.35078 |
[5] | M.A. Shubin, "Pseudo-differential operators and spectral theory" , Springer (1987) (Translated from Russian) MR883081 |
[6] | K.O. Friedrichs, "Pseudo-differential operators" , Courant Inst. Math. (1970) Zbl 0226.47028 |
[7] | F. Trèves, "Introduction to pseudo-differential and Fourier integral operators" , 1–2 , Plenum (1980) |
[8] | M.E. Taylor, "Pseudo-differential operators" , Springer (1974) MR0442523 Zbl 0289.35001 |
[9] | H. Kumanogo, "Pseudo-differential operators" , M.I.T. (1981) MR1414739 MR1414619 MR1406605 MR0518297 MR0412904 MR0361937 MR0355693 MR0328392 MR0315521 MR0303360 MR0438189 MR0291896 MR0254677 |
[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 , just like
for "partial differential operator" .
For algebras of on manifolds with singularities, in particular
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 |
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