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) |
