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
and
. 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
of the base space
as the basic object of microlocal analysis.
Micro-analyticity of hyperfunctions.
A hyperfunction
is said to be micro-analytic at
if on a neighbourhood of
it admits analytic continuation into the half-space
, in the sense that it admits a boundary-value representation
such that
for every
. This is equivalent to saying that near
,
, where
is the germ of a real-analytic function and
is a Fourier hyperfunction (cf. Hyperfunction) exponentially decreasing in a conic neighbourhood of
. The set of points
at which
is not micro-analytic is called the singular spectrum of
, and is denoted by
. By definition,
where
is the dual cone of
. Conversely, a hyperfunction satisfying this estimate can be written in the form
.
Operations and the singular spectrum.
The following inclusions hold:
Here, the operations are legitimate if the vector
does not appear in the
-component of the right-hand side. In particular, under coordinate transformation the singular spectrum behaves like a subset of
. Restriction
is possible if
, in which case
is said to contain
as real-analytic parameter at
, and then
where
denotes projection on the
components. The dual assertion is:
The combination of these assertions gives a convolution, dual to the product:
Let
be a linear differential operator with real-analytic coefficients and let
be its characteristic manifold. Then
(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
,
implies
(the Kashiwara–Kawai Holmgren-type theorem); further, the fibre
of
at
has the form
, where
denotes projection to the equator
(the watermelon theorem).
Decomposition of singular spectra.
One has
where the twisted phase
is a real-analytic function of
which is of positive type in
(that is,
implies
), is homogeneous of degree 1 in
, and
,
; and the vector
is such that
. This is a generalization of the classical Radon decomposition, in which
The component, regarded as a hyperfunction of
, has a singular spectrum with only one direction
. Via convolution it gives a similar decomposition of general hyperfunctions. If
also satisfies
for
, then the singular spectrum of the component as a hyperfunction of
is precisely one point
; this is useful in applications. Typical examples are:
(Kashiwara example);
(Bony example). For such a decomposition
is micro-analytic at
if and only if
is real analytic in
at
.
The Fourier–Bros–Iagolnitzer transform of a hyperfunction
(the FBI-transform of
) is
where
is a real-analytic function satisfying: 1) for
one has
and
; and 2)
for some
. A typical example of such a
is
A hyperfunction
is micro-analytic at
if and only if for some (equivalently, any) modification of
with compact support its FBI-transform is exponentially decreasing with respect to
uniformly in
in a neighbourhood of
. Integration of the inversion formula over the radial variable gives the formula
This supplies a partition of unity of the sheaf
. All these arguments are compatible with the corresponding theory of (analytic) wave front sets for distributions.
The sheaf
of micro-functions on
is the sheaf associated with the pre-sheaf
The sequence of sheaves on
:
is exact. Here
denotes projection. By definition,
for a hyperfunction
. The sheaf
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
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
is constructed from the sheaf
by Sato micro-localization: Let
be a real-analytic manifold and
a complex neighbourhood of it. Let
be the real blowing-up,
,
the canonical inclusions, and let
be the subset of the fibre product of
and
over
defined by
; let
be the canonical projections to the factors. Let
be the orientation sheaf of
. Then
(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) |