Wave front
wave front set, of a generalized function (distribution) or hyperfunction
A conical set in the cotangent bundle to the manifold on which the generalized function or hyperfunction in question is given, which characterizes its singularities. A hyperfunction is a sum of formal boundary values of holomorphic functions. Two such sums are identified if they are equivalent in the sense of equivalence given by an analogue of Bogolyubov's "edge-of-the-wedge" theorem (cf. Bogolyubov theorem), in which, however, one in no sense assumes that the holomorphic functions in question have limits.
The wave front set of a hyperfunction is also often called the analytic wave front set or the singular support (the last term is more often used in a completely-different sense, when it denotes the complement to the set of some sort of regularity of the generalized function on the manifold itself, and not in the cotangent bundle). The concept of the wave front set lies behind micro-local analysis, which is a complex of ideas and methods using wave front sets and other related concepts and techniques (in particular, pseudo-differential operators and Fourier integral operators) for studying partial differential equations (mainly linear equations).
Let be a domain in
and let
, that is,
is a generalized function on
. Then the wave front set
of
is the closed conical subset of
defined as follows: If
, then
means that there is a function
, equal to
in a neighbourhood of
, and a conical neighbourhood
of
in
, such that for every
,
![]() |
where
![]() |
that is, is the Fourier transform of
.
If is a manifold and
is a generalized function on
(or, more generally, a generalized section of a smooth vector bundle), then
is defined in the same way as above (after transition to local coordinates). In this case
turns out to be a well-defined conical subset of
(the cotangent bundle without the zero section).
One introduces the canonical projection . Then
![]() | (1) |
where is the complement of the largest open subset of
on which
coincides with an infinitely-differentiable function. This relationship shows that
is actually a finer characteristic of the singularities of
than
.
Let be a pseudo-differential operator of order
on
with principal symbol
, and let
be the set of its characteristic directions, that is,
![]() |
Then
![]() | (2) |
Here the first inclusion characterizes the pseudo-locality of , and the second is a far-reaching generalization of the theorem on the smoothness of solutions of elliptic equations with smooth coefficients.
If the principal symbol of
is real-valued, then the following theorem on the propagation of singularities holds: If one is given a connected piece
of a bicharacteristic (that is, a trajectory of the Hamiltonian vector field on
with Hamiltonian
) that does not intersect
, then either
or
.
This theorem shows that the singularities of the solutions (that is, their wave front sets) of an equation with a smooth right-hand side
propagate along the bicharacteristics of the principal symbol
of
(see [3], [4], [8], [11], [12], [16]).
The analytic wave front set for a generalized function
can be defined in one of the following three equivalent (see [13]) ways (here, for simplicity,
is a domain in
):
1) if there are a neighbourhood
of
, open proper convex cones
in
and functions
, holomorphic in
, such that
,
, and
, where
is the cone dual to
and
is the boundary value of the holomorphic function
for
,
, understood in the sense of weak convergence of generalized functions. This definition is also applicable to hyperfunctions if the boundary value is interpreted differently.
2) Let
![]() |
(a generalized Fourier transform); then if and only if for any function
that is analytic in a neighbourhood of
there are a conical neighbourhood
of
and positive constants
such that
![]() |
3) if and only if there are a neighbourhood
of
in
, a bounded sequence of generalized functions
,
with compact support, and a constant
, such that
in
and
![]() |
There is an analogue of the property (1) for the analytic wave front:
![]() |
where is the complement of the largest set on which
is real-analytic. There is an analogue of the property (2), where one can take for
a differential operator with real-analytic coefficients or an analytic pseudo-differential operator (see [6], [9], [11], [15], [16]). For such an operator
with a real principal symbol, a theorem on the propagation of the analytic wave front set holds, analogous to the theorem stated above for the ordinary wave front set (see [11]).
References
[1] | M. Sato, "Hyperfunctions and partial differential equations" , Proc. 2nd Conf. Functional Anal. Related Topics , Tokyo Univ. Press (1969) pp. 91–94 MR0650826 Zbl 0208.35801 |
[2] | L. Hörmander, "Fourier integral operators I" Acta Math. , 127 (1971) pp. 79–183 MR0388463 Zbl 0212.46601 |
[3] | J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" Acta Math. , 128 (1972) pp. 183–269 MR0388464 Zbl 0232.47055 |
[4] | J.J. Duistermaat, "Fourier integral operators" , Courant Inst. Math. (1973) MR0451313 Zbl 0272.47028 |
[5] | M.A. Shubin, "Pseudo-differential operators and spectral theory" , Springer (1983) (Translated from Russian) |
[6] | F. Trèves, "Introduction to pseudo-differential and Fourier integral operators" , 1–2 , Plenum (1980) |
[7] | M.E. Taylor, "Pseudo-differential operators" , Princeton Univ. Press (1981) MR1567325 Zbl 0289.35001 Zbl 0207.45402 |
[8] | L. Nirenberg, "Lectures on linear partial differential equations" , Amer. Math. Soc. (1972) MR0450756 MR0450755 Zbl 0267.35001 |
[9] | M. Kashiwara, "Microfunctions and pseudo-differential equations" H. Komatsu (ed.) , Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971 , Lect. notes in math. , 287 , Springer (1973) pp. 265–529 MR0420735 Zbl 0277.46039 |
[10] | P. Schapira, "Théorie des hyperfonctions" , Lect. notes in math. , 126 , Springer (1970) MR0631543 MR0270151 Zbl 0201.44805 Zbl 0192.47305 |
[11] | J. Sjöstrand, "Singularités analytiques microlocales" , Univ. Paris-Sud (1982) ((Prepublication.)) MR0699623 Zbl 0524.35007 |
[12] | R. Lascar, "Propagation des singularités des solutions d'Aeequations pseudo-differentielles à caractéristiques de multiplicités variables" , Springer (1981) |
[13] | J. Bony, "Equivalence des diverses notions de spectre singulier analytique" Sém. Goulaouic–Schwartz , III (1976–1977) MR0650834 Zbl 0367.46036 |
[14a] | J. Bros, D. Iagolnitzer, "Tuboides et structure analytique des distributions I. Tuboides et généralisation d'un théorème de Grauert" Sém. Goulaouic–Lions–Schwartz , 16 (1974) MR0399493 |
[14b] | J. Bros, D. Iagolnitzer, "Tuboides et structure analytique des distributions II. Support essential et structure analytique des distributions" Sém. Goulaouic–Lions–Schwartz , 18 (1975) MR0399494 |
[15] | L. Hörmander, "On the singularities of solutions of partial differential equations" Comm. Pure Appl. Math. , 23 (1970) pp. 329–358 MR0262646 Zbl 0193.06603 Zbl 0191.10901 Zbl 0188.40901 |
[16] | 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 |
Comments
References
[a1] | V. Guillemin, S. Sternberg, "Geometric asymptotics" , Amer. Math. Soc. (1977) MR0516965 Zbl 0364.53011 |
[a2] | V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) Zbl 0734.53001 |
Wave front. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wave_front&oldid=49176