Microlocal analysis

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 $ x $ and $ \xi $. 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 $ S ^ {*} M $ of the base space $ M $ as the basic object of microlocal analysis.

Micro-analyticity of hyperfunctions.

A hyperfunction $ f \in {\mathcal B} ( M) $ is said to be micro-analytic at $ ( x _ {0} , \xi _ {0} ) \in S ^ {*} M $ if on a neighbourhood of $ x _ {0} $ it admits analytic continuation into the half-space $ \langle \mathop{\rm Im} z , \xi _ {0} \rangle < 0 $, in the sense that it admits a boundary-value representation $ \sum_{j=1}^ {N} F _ {j} ( x + i \Gamma _ {j} 0 ) $ such that $ \Gamma _ {j} \cap \{ \langle \mathop{\rm Im} z , \xi _ {0} \rangle < 0 \} \neq \emptyset $ for every $ j $. This is equivalent to saying that near $ x _ {0} $, $ f = {\mathcal F} ^ {-1} ( g) + h $, where $ h $ is the germ of a real-analytic function and $ g $ is a Fourier hyperfunction (cf. Hyperfunction) exponentially decreasing in a conic neighbourhood of $ \xi _ {0} $. The set of points $ ( x _ {0} , \xi _ {0} ) \in \Omega \times S ^ {n-1} $ at which $ f $ is not micro-analytic is called the singular spectrum of $ f $, and is denoted by $ \mathop{\rm S}.S. f $. By definition,

$$ \mathop{\rm S}.S. F ( x + i \Gamma 0 ) \subset \Omega \times ( \Gamma ^ \circ \cap S ^ {n-1} ) , $$

where $ \Gamma ^ \circ = \{ {\xi \in \mathbf R ^ {n} } : {\langle y , \xi \rangle \geq 0 \textrm{ for all } y \in \Gamma } \} $ is the dual cone of $ \Gamma $. Conversely, a hyperfunction satisfying this estimate can be written in the form $ F ( x + i \Gamma 0 ) $.

Operations and the singular spectrum.

The following inclusions hold:

$$ \mathop{\rm S}.S. ( f g ) \subset $$

$$ \subset \ \{ {( x , \xi + \eta ) } : {( x , \xi ) \in \mathop{\rm S}.S. f , ( x , \eta ) \in \mathop{\rm S}.S. g } \} \cup $$

$$ \cup \{ {( x , \xi ) } : {x \in \supp f , ( x , \xi ) \in \mathop{\rm S}.S. g , \textrm{ or } x \in \supp g , ( x , \xi ) \in \mathop{\rm S}.S. f } \} ; $$

$$ \mathop{\rm S}.S. f ( \Phi ( \widetilde{x} ) ) \subset ( \Phi ^ {-1} \times {} ^ {t} d \Phi ) ( \mathop{\rm S}.S. f ) . $$

Here, the operations are legitimate if the vector $ 0 $ does not appear in the $ \xi $- component of the right-hand side. In particular, under coordinate transformation the singular spectrum behaves like a subset of $ S ^ {*} \mathbf R ^ {n} $. Restriction $ f ( x , 0 ) = f ( x , t ) \mid_{t=0}$ is possible if $ \mathop{\rm S}.S. f \cap S _ {\{ t = 0 \} } ^ {*} M = \emptyset $, in which case $ f $ is said to contain $ t $ as real-analytic parameter at $ t = 0 $, and then

$$ \mathop{\rm S}.S. f ( x , 0 ) \subset \rho _ {*} ( \mathop{\rm S}.S. f ) \mid_{t=0} , $$

where $ \rho $ denotes projection on the $ d x $ components. The dual assertion is:

$$ \mathop{\rm S}.S. \int\limits f ( x , t ) d x \subset $$

$$ \subset \ \{ ( t , \theta ) : ( x , t , 0 , \theta ) \in \mathop{\rm S}.S. \ f \textrm{ for some } x \} . $$

The combination of these assertions gives a convolution, dual to the product:

$$ \mathop{\rm S}.S. ( f \star g ) \subset $$

$$ \subset \ \{ ( x + y , \xi ) : \ ( x , \xi ) \in fnem S.S. f , ( y , \xi ) \in fnnem S.S. g \} . $$

Let $ P ( x , \partial ) $ be a linear differential operator with real-analytic coefficients and let $ \mathop{\rm Char} P = \{ {( x , \xi ) } : {P _ {m} ( x , \xi ) = 0 } \} $ be its characteristic manifold. Then

$$ \mathop{\rm S}.S. P ( x , \partial ) u ( x) \subset \ \mathop{\rm S}.S. u ( x) \subset $$

$$ \subset \ \mathop{\rm S}.S. P ( x , \partial ) u ( x) \cup \mathop{\rm Char} P $$

(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 $ u $, $ 0 \in \supp u \subset \{ x _ {1} \geq 0 \} $ implies $ ( 0 , \pm d x _ {1} ) \in \mathop{\rm S}.S. u $( the Kashiwara–Kawai Holmgren-type theorem); further, the fibre $ E $ of $ \mathop{\rm S}.S. u $ at $ 0 $ has the form $ \rho ^ {-1} \rho ( E \setminus \{ \pm d x _ {1} \} ) \cup \{ \pm d x _ {1} \} $, where $ \rho : S ^ {n-1} \setminus \{ \pm d x _ {1} \} \rightarrow S ^ {n-2} $ denotes projection to the equator $ \xi _ {1} = 0 $( the watermelon theorem).

Decomposition of singular spectra.

One has

$$ \delta ( x) = \int\limits _ {S ^ {n-1} } W ( x , \omega ) d \omega , $$

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{ \mathop{\rm det} ( \mathop{\rm grad} _ \omega \psi ( x , \omega ) ) }{( \phi ( x , \omega ) + i 0 ) ^ {n} } , $$

where the twisted phase $ \phi ( x , \omega ) $ is a real-analytic function of $ ( x , \omega ) $ which is of positive type in $ x $( that is, $ \mathop{\rm Re} \phi ( x , \omega ) = 0 $ implies $ \mathop{\rm Im} \phi ( x , \omega ) \geq 0 $), is homogeneous of degree 1 in $ \omega $, and $ \phi ( 0 , \omega ) = 0 $, $ \mathop{\rm grad} _ {x} \phi ( 0 , \omega ) = \omega $; and the vector $ \psi ( x , \omega ) $ is such that $ \langle \psi ( x , \omega ) , x \rangle = \phi ( x , \omega ) $. This is a generalization of the classical Radon decomposition, in which

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{1}{( x \omega + i 0 ) ^ {n} } . $$

The component, regarded as a hyperfunction of $ x $, has a singular spectrum with only one direction $ \omega $. Via convolution it gives a similar decomposition of general hyperfunctions. If $ \phi ( x , \omega ) $ also satisfies $ \phi ( x , \omega ) \neq 0 $ for $ x \neq 0 $, then the singular spectrum of the component as a hyperfunction of $ x $ is precisely one point $ ( 0 , \omega ) $; this is useful in applications. Typical examples are:

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{( 1 - i x \omega ) ^ {n-1} - ( 1 - i x \omega ) ^ {n-2} ( x ^ {2} - ( x \omega ) ^ {2} ) }{( x \omega + i ( x ^ {2} - ( x \omega ) ^ {2} ) + i 0 ) ^ {n} } $$

(Kashiwara example);

$$ W ( x , \omega ) = \frac{( n - 1 ) ! }{( - 2 \pi i ) ^ {n} } \frac{1 + i \alpha x \omega }{( x \omega + i \alpha x ^ {2} + i 0 ) ^ {n } } ,\ \alpha > 0, $$

(Bony example). For such a decomposition $ f $ is micro-analytic at $ ( x _ {0} , \xi _ {0} ) $ if and only if $ f \star _ {x} W ( x , \omega ) $ is real analytic in $ ( x , \omega ) $ at $ ( x _ {0} , \omega _ {0} ) $.

The Fourier–Bros–Iagolnitzer transform of a hyperfunction $ f $( the FBI-transform of $ f $) is

$$ \int\limits _ {\mathbf R ^ {n} } e ^ {i \lambda \phi ( x, y, \xi , \alpha ) } f( y) dy , $$

where $ \phi ( x , y , \xi , \alpha ) $ is a real-analytic function satisfying: 1) for $ x = y = \alpha $ one has $ \phi = 0 $ and $ \phi _ {x} = - \phi _ {y} = \xi $; and 2) $ \mathop{\rm Im} \phi \geq C ( | x - \alpha | ^ {2} + | y - \alpha | ^ {2} ) $ for some $ C > 0 $. A typical example of such a $ \phi $ is

$$ \phi ( x , y , \xi , \alpha ) = ( x - y ) \xi + i ( ( x - \alpha ) ^ {2} + ( y - \alpha ) ^ {2} ) . $$

A hyperfunction $ f $ is micro-analytic at $ ( x _ {0} , \xi _ {0} ) $ if and only if for some (equivalently, any) modification of $ f $ with compact support its FBI-transform is exponentially decreasing with respect to $ \lambda $ uniformly in $ ( x , \xi , \alpha ) $ in a neighbourhood of $ ( x _ {0} , \xi _ {0} , x _ {0} ) $. Integration of the inversion formula over the radial variable gives the formula

$$ \delta ( x - y ) = $$

$$ = \ \frac{2 ^ {n} \Gamma ( 3 n / 2 ) }{( - 2 \pi i ) ^ {3n/2} } \times $$

$$ \times \int\limits _ {\mathbf R ^ {n} \times S ^ {n-1} } \frac{1 + ( i / 2 ) ( x - y ) \omega }{( ( x - y ) \omega + i ( ( x - \alpha ) ^ {2} + ( y - \alpha ) ^ {2} ) + i 0 ) ^ {3n/2} } d \alpha d \omega . $$

This supplies a partition of unity of the sheaf $ {\mathcal B} / {\mathcal A} $. All these arguments are compatible with the corresponding theory of (analytic) wave front sets for distributions.

The sheaf $ {\mathcal C} $ of micro-functions on $ S ^ {*} M $ is the sheaf associated with the pre-sheaf

$$ S ^ {*} M \supset \Omega \times \Delta \mapsto {\mathcal C} ( \Omega \times \Delta ) = $$

$$ = \ { {\mathcal B} ( \Omega ) } / {\{ f \in {\mathcal B} ( \Omega ) : \mathop{\rm S}.S. f \cap \Omega \times \Delta = \emptyset \} } . $$

The sequence of sheaves on $ M $:

$$ 0 \rightarrow {\mathcal A} \rightarrow {\mathcal B} \mathop \rightarrow \limits ^ { { \mathop{\rm sp}} } \pi _ {*} {\mathcal C} \rightarrow 0 $$

is exact. Here $ \pi : S ^ {*} M \rightarrow M $ denotes projection. By definition, $ \mathop{\rm S}.S. f = \supp \mathop{\rm sp} [ f ] $ for a hyperfunction $ f $. The sheaf $ {\mathcal C} $ 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 $ {\mathcal C} $ 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 $ {\mathcal C} $ is constructed from the sheaf $ {\mathcal O} $ by Sato micro-localization: Let $ M $ be a real-analytic manifold and $ X $ a complex neighbourhood of it. Let $ {} ^ {M} {\widetilde{X} } = ( X \setminus M ) \amalg S _ {M} X $ be the real blowing-up, $ \iota : X \setminus M \rightarrow X $, $ j : X \setminus M \rightarrow {} ^ {M} {\widetilde{X} } $ the canonical inclusions, and let $ D M $ be the subset of the fibre product of $ S _ {M} X $ and $ S _ {M} ^ {*} X $ over $ M $ defined by $ \langle \xi , \eta \rangle \leq 0 $; let $ \phi , \tau $ be the canonical projections to the factors. Let $ \omega $ be the orientation sheaf of $ M $. Then

$$ {\mathcal C} = \mathbf R \tau _ {*} \pi ^ {-1} \mathbf R \Gamma _ {S _ {M} X } ( j _ {*} \iota ^ {-1} {\mathcal O} ) \otimes \omega [ n] = $$

$$ = \ \mathbf R ^ {n-1} \tau _ {*} \pi ^ {-1} {\mathcal H} _ {S _ {M} X } ^ {1} ( j _ {*} \iota ^ {-1} {\mathcal O} ) \otimes \omega $$

(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)