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
[2]	L. Hörmander, "Fourier integral operators I" Acta Math. , 127 (1971) pp. 79–183
[3]	J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" Acta Math. , 128 (1972) pp. 183–269
[4]	J.J. Duistermaat, "Fourier integral operators" , Courant Inst. Math. (1973)
[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)
[8]	L. Nirenberg, "Lectures on linear partial differential equations" , Amer. Math. Soc. (1972)
[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
[10]	P. Schapira, "Théorie des hyperfonctions" , Lect. notes in math. , 126 , Springer (1970)
[11]	J. Sjöstrand, "Singularités analytiques microlocales" , Univ. Paris-Sud (1982) ((Prepublication.))
[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)
[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)
[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)
[15]	L. Hörmander, "On the singularities of solutions of partial differential equations" Comm. Pure Appl. Math. , 23 (1970) pp. 329–358
[16]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985)

References

[a1]	V. Guillemin, S. Sternberg, "Geometric asymptotics" , Amer. Math. Soc. (1977)
[a2]	V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990)