Hyperfunction

A hyperfunction is a kind of generalized function, and the class of hyperfunctions is characterized as the largest class of localizable generalized functions. According to the three general recipes for constructing generalized functions, hyperfunctions are obtained 1) as ideal limits or boundary values on the real axis of holomorphic functions; 2) as locally finite sums of continuous linear functionals on real-analytic functions; or 3) as formal derivatives of continuous functions by infinite-order differential operators of local type. The theory of hyperfunctions has a prehistory starting with G. Köthe, A. Grothendieck, etc., and also in some physical literature. But the localization together with the accurate setting in the case of several variables was given by M. Sato, who employed cohomology theory.

A hyperfunction of one variable $ f ( x) $ on an open set $ \Omega \subset \mathbf R $ is a formal expression of the form $ F _ {+} ( x + i0) - F _ {-} ( x - i0) $, where $ F _ \pm ( z) $ is a function holomorphic on the upper, respectively lower, half-neighbourhood $ U _ \pm = U \cap \{ {z } : {\pm \mathop{\rm Im} z > 0 } \} $, for a complex neighbourhood $ U \supset \Omega $ satisfying $ U \cap \mathbf R = \Omega $. The expression $ f ( x) $ is identified with 0 if and only if $ F _ \pm ( z) $ agrees on $ \Omega $ as a holomorphic function (or, equivalently, as a continuous function, by the Painlevé theorem). If the limits exist in distribution sense, the formula gives the natural imbedding of the space of distributions into that of hyperfunctions. The precise definition of the space $ {\mathcal B} ( \Omega ) $ of hyperfunctions on $ \Omega $ is: $ {\mathcal B} ( \Omega ) = \lim\limits _ {\rightarrow U \supset \Omega } {\mathcal O} ( U \setminus \Omega )/ {\mathcal O} ( \Omega ) $. (Actually the limit is superfluous.) For $ F ( z) \in {\mathcal O} ( U \setminus \Omega ) $, let $ [ F] $ denote the hyperfunction $ f $ on $ \Omega $ defined by $ F $. In turn, $ F $ is called a defining function of $ f $. Examples of typical generalized functions interpreted as hyperfunctions are:

$$ \delta ( x) = \ \left [ - { \frac{1}{2 \pi i } } { \frac{1}{z} } \right ] = - { \frac{1}{2 \pi i } } \left ( { \frac{1}{x + i0 } } - { \frac{1}{x - i0 } } \right ) $$

(Dirac's delta-function);

$$ Y ( x) = \ \left [ - { \frac{1}{2 \pi i } } \ \mathop{\rm log} (- z) \right ] = $$

$$ = \ - { \frac{1}{2 \pi i } } ( \mathop{\rm log} (- x - i0) - \mathop{\rm log} (- x + i0)) $$

(Heaviside's function);

$$ \textrm{ p }.v. { \frac{1}{x} } = { \frac{1}{2} } \left ( { \frac{1}{x + i0 } } + { \frac{1}{x - i0 } } \right ) $$

(Cauchy's principal value);

$$ \textrm{ p }.f. \ x ^ {-} m = { \frac{1}{2} } \left ( { \frac{1}{( x + i0) ^ {m} } } + { \frac{1}{( x - i0) ^ {m} } } \right ) $$

(Hadamard's finite part);

$$ x _ \pm ^ \lambda = \ \left [ \frac{\mps ( \mps z) ^ \lambda }{2i \sin \pi \lambda } \right ] = \pm \frac{( \mps x - i0) ^ \lambda - ( \mps x + i0) ^ \lambda }{2i \sin \pi \lambda } \ \ \textrm{ for } \lambda \notin \mathbf Z , $$

$$ x _ \pm ^ {m} = \left [ \mps { \frac{1}{2 \pi i } } ( \pm z) ^ {m} \mathop{\rm ln} ( \mps z) \right ] \ \textrm{ for } \lambda = m \in \mathbf Z . $$

Hyperfunctions of several variables. In the sequel a cone $ \Gamma \subset \mathbf R ^ {n} $ means an open convex cone with vertex at the origin. For two cones $ \Delta , \Gamma $, the notation $ \Delta \subset \subset \Gamma $ means that $ \Delta \cap S ^ {n - 1 } $ is relatively compact in $ \Gamma \cap S ^ {n - 1 } $. A wedge means an open subset of $ \mathbf C ^ {n} $ of the form $ \Omega + i \Gamma $, where $ \Omega \subset \mathbf R ^ {n} $ is a real open set and $ \Gamma \subset \mathbf R ^ {n} $ is a cone. $ \Omega $ is called the edge of the wedge and $ \Gamma $ the opening. An infinitesimal wedge ( $ 0 $- wedge) or a tuboid of opening $ \Gamma $ and edge $ \Omega $( or simply of type $ \Omega + i \Gamma $) is a complex open set $ U $ such that $ U \subset \Omega + i \Gamma $ and such that for any $ \Delta \subset \subset \Gamma $ the set $ U $ contains the part of $ \Omega + i \Delta $ contained in some complex neighbourhood of the edge $ \Omega $. The symbol $ \Omega + i \Gamma 0 $ denotes any one such open set, and $ {\mathcal O} ( \Omega + i \Gamma 0) $ denotes the totality of functions holomorphic on some of them (i.e. $ {\mathcal O} ( \Omega + i \Gamma 0) = \lim\limits _ {\rightarrow U } {\mathcal O} ( U) $, the limit being taken with respect to all the $ 0 $- wedges $ U $ of type $ \Omega + i \Gamma $).

A hyperfunction $ f ( x) $ on an open set $ \Omega \subset \mathbf R ^ {n} $ is an equivalence class (in the obvious sense) of formal expressions of the form

$$ \sum _ {j = 1 } ^ { N } F _ {j} ( x + i \Gamma _ {j} 0), $$

where $ F _ {j} ( z) \in {\mathcal O} ( \Omega + i \Gamma _ {j} 0) $. $ \{ F _ {j} ( z) \} $ is called a set of defining functions of $ f ( x) $. The totality of hyperfunctions on $ \Omega $ is denoted by $ {\mathcal B} ( \Omega ) $. It becomes a $ \mathbf C $- linear space by the $ \mathbf C $- linear structure of the defining functions. Linear differential operators with real-analytic coefficients act in the same way via the defining functions. The above boundary value representation corresponds to the case of one variable when $ \Gamma _ {1} = \mathbf R _ {+} $, $ \Gamma _ {2} = \mathbf R _ {-} $.

Hyperfunctions are localizable, i.e. the correspondence $ \Omega \mapsto {\mathcal B} ( \Omega ) $ satisfies the axioms for a sheaf. It is a flabby sheaf, i.e. any section extends to the whole space. The sheaf of hyperfunctions is defined via the derived sheaf as $ {\mathcal B} = {\mathcal H} _ {\mathbf R ^ {n} } ^ {n} ( {\mathcal O} ) \otimes \omega $, where $ \omega $ is the orientation sheaf of $ \mathbf R ^ {n} $. Actually $ {\mathcal B} ( \Omega ) \cong H _ \Omega ^ {n} ( \mathbf C ^ {n} , {\mathcal O} ) $ holds, the cohomologies in the other degrees being trivial. By choosing a Stein neighbourhood $ U \supset \Omega $, one can represent the relative cohomology by a covering: If $ U \cap \mathbf R ^ {n} = \Omega $, $ U _ {j} = \{ {z \in U } : { \mathop{\rm Im} z _ {j} \neq 0 } \} $, $ j = 1 \dots n $, then the pair $ {\mathcal U} = \{ U, U _ {1} \dots U _ {n} \} $, $ {\mathcal U} ^ \prime = \{ U _ {1} \dots U _ {n} \} $ constitutes a relative Stein covering of the pair $ ( U, U \setminus \Omega ) $, hence by Leray's theorem,

$$ H _ \Omega ^ {n} ( \mathbf C ^ {n} , {\mathcal O} ) = $$

$$ = \ {C ^ {n} ( {\mathcal U} \mathop{\rm mod} {\mathcal U} ^ \prime , {\mathcal O} ) } / {\delta C ^ {n - 1 } ( {\mathcal U} \mathop{\rm mod} {\mathcal U} ^ \prime , {\mathcal O} ) } \cong $$

$$ \cong { {\mathcal O} ( U \srp \Omega ) } / {\sum _ {j = 1 } ^ { n } } {\mathcal O} ( U \srp _ {j} \Omega ), $$

where $ U \srp \Omega = \{ {z \in U } : { \mathop{\rm Im} z _ {k} \neq 0\spg \textrm{ for all } k } \} $, $ U \srp _ {j} \Omega = \{ {z \in U } : { \mathop{\rm Im} z _ {k} \neq 0 \textrm{ for all } k \neq j } \} $. In case $ n = 1 $ this reduces to $ H _ \Omega ^ {1} ( \mathbf C ^ {n} , {\mathcal O} ) = {\mathcal O} ( U \setminus \Omega )/ {\mathcal O} ( U) $. Another representation is given by choosing a set of vectors $ \eta ^ {0} , \eta ^ {1} \dots \eta ^ {n} \in \mathbf R ^ {n} $ such that the half-spaces $ E ^ {\eta ^ {j} } = \{ {y \in \mathbf R ^ {n} } : {\langle y, \eta ^ {j} > \rangle 0 } \} $ defined by them satisfy $ \cup _ {j = 0 } ^ {n} E ^ {\eta ^ {j} } = \mathbf R ^ {n} \setminus \{ 0 \} $, and putting $ U _ {j} = ( \Omega + iE ^ {\eta ^ {j} } ) \cap U $, $ j = 0 \dots n $. The $ n $- cocycles are of the form

$$ \sum _ {j = 0 } ^ { n } F _ {j} ( z) U \wedge U _ {0} \wedge \dots \wedge \widehat{U} _ {j} \wedge \dots \wedge U _ {n} , $$

where $ F _ {j} ( z) \in {\mathcal O} ( U _ {0} \cap \dots \cap \widehat{U} _ {j} \cap \dots \cap U _ {n} ) $, and $ H _ \Omega ^ {n} ( \mathbf C ^ {n} , {\mathcal O} ) $ is given as the quotient space of these by the $ n $- coboundary space. (The symbol $ \widehat{ {}} $ implies that the factor below it is omitted.)

The set $ {\mathcal B} ( \Omega ) $ is identified with the cohomology group $ H _ \Omega ^ {n} ( \mathbf C ^ {n} , {\mathcal O} ) $ by extending the following correspondences:

$$ {\mathcal O} ( U \srp \Omega ) \ni \ F ( z) \mapsto \ [ F ( z)] = \ \sum _ \sigma ( \mathop{\rm sgn} \sigma ) F ( x + i \Gamma _ \sigma 0) \in \ {\mathcal B} ( \Omega ), $$

where $ \Gamma _ \sigma $ denotes the $ \sigma $- th orthant $ \{ {y \in \mathbf R ^ {n} } : {\sigma _ {j} y _ {j} > 0, j = 1 \dots n } \} $ and $ \mathop{\rm sgn} \sigma = \sigma _ {1} \dots \sigma _ {n} $; or by

$$ \sum _ {j = 0 } ^ { n } F _ {j} ( z) U \wedge U _ {0} \wedge \dots \wedge \widehat{U} _ {j} \wedge \dots \wedge U _ {n\ } \mapsto $$

$$ \mapsto \ \sum _ {j = 0 } ^ { n } (- 1) ^ {j} F _ {j} ( x + i \Gamma _ {j} 0), $$

where $ \Gamma _ {j} = E ^ {\eta ^ {0} } \cap \dots \cap \widehat{E} {} ^ {\eta ^ {j} } \cap \dots \cap E ^ {\eta ^ {n} } $.

Several practical criteria to determine whether a hyperfunction is zero are collectively known as the edge-of-the-wedge theorem: A hyperfunction $ F ( x + i \Gamma 0) $ with single term is zero if and only if $ F ( z) $ itself is zero (injectivity of the boundary value operation). $ F _ {1} ( x + i \Gamma _ {1} 0) = F _ {2} ( x + i \Gamma _ {2} 0) $ if and only if they stick together to a function in $ {\mathcal O} ( \Omega + i ( \Gamma _ {1} + \Gamma _ {2} ) 0) $( an Epstein-type theorem). In case $ \Gamma _ {2} = - \Gamma _ {1} $ the result becomes real analytic (a Bogolyubov-type theorem). $ \sum _ {j = 1 } ^ {N} F _ {j} ( x + i \Gamma _ {j} 0) = 0 $ if and only if there exist $ G _ {jk} ( z) \in {\mathcal O} ( \Omega + i ( \Gamma _ {j} + \Gamma _ {k} ) 0) $, $ j, k = 1 \dots N $, such that $ G _ {jk} ( z) = - G _ {kj} ( z) $ and $ F _ {j} ( z) = \sum _ {k = 1 } ^ {N} G _ {jk} ( z) $ in $ {\mathcal O} ( \Omega + i \Gamma _ {j} 0) $, $ j = 1 \dots N $( a Martineau-type theorem). These are manifestations of the fact that the cohomology classes vanish by covering representations.

A real-analytic function $ \phi \in {\mathcal A} ( \Omega ) $ on $ \Omega $ is naturally included in $ {\mathcal B} ( \Omega ) $ by identifying it with the expression $ \phi ( x + i \Gamma 0) $, for any $ \Gamma $. They form a subsheaf. Thus, for a hyperfunction $ f $ on $ \Omega $ the notion of the singular support of $ f $, denoted by $ \sing \supp f $, is well defined: It is the complement in $ \Omega $ of the largest open subset $ \Omega ^ \prime \subset \Omega $ in which $ f ( x) $ is real analytic. If $ \sing \supp f \subset K $ for a closed subset $ K \subset \Omega $, then one can choose a boundary value representation such that each $ F _ {j} ( z) $ can be continued analytically to $ \Omega \setminus K $. The edge-of-the-wedge theorem supplies also criteria for determining whether or not a hyperfunction is real analytic in an open (sub)set $ \Omega $.

The definite integral $ \int _ {D} f ( x, t) dx $ of an $ f \in {\mathcal B} ( \Omega \times T) $ over a bounded domain $ D $ is well-defined if $ \sing \supp f \cap \partial D \times T = \emptyset $. Indeed, let $ f ( x, t) - \sum F _ {j} (( x, t) + i \Gamma _ {j} 0) $ be a boundary value representation such that each $ F _ {j} ( z, \tau ) $ may be extended to a neighbourhood of $ \partial D \times T $, or, more generally, to a $ 0 $- wedge with edge $ \partial D \times T $ and with an opening whose projection to the $ x $- space is the whole space. Then

$$ \int\limits _ { D } f ( x, t) dx = \ \sum _ { j } \left [ \int\limits _ {D _ {j} } F _ {j} ( z, \tau ) dz \right ] _ {\tau \mapsto t + i \Delta _ {j} 0 } , $$

where $ D _ {j} $ is a suitable deformation of $ D $ fixing $ \partial D $ and $ \Delta _ {j} $ is the projection of $ \Gamma _ {j} $ on the $ t $- plane. This integration commutes with differentiation or integration with respect to the remaining variable $ t $. Cf. Microlocal analysis for further operations.

The totality $ {\mathcal B} [ K] $ of hyperfunctions with support in a fixed compact set $ K $ becomes a nuclear Fréchet space (cf. also Nuclear space). It is the strong dual of the nuclear (DF)-space $ {\mathcal A} ( K) $ of real-analytic functions defined on $ K $ and endowed with the inductive limit topology of $ {\mathcal O} ( U) $' s for $ U \supset K $. Namely, hyperfunctions with compact supports are analytic functionals with carrier in the real axis. The duality is given by the definite integral $ \langle f, \phi \rangle = \int _ {\mathbf R ^ {n} } f ( x) \phi ( x) dx $ for $ f \in {\mathcal B} [ K] $ and $ \phi \in {\mathcal A} ( K) $. Analytic functionals can be employed to reconstruct the space or the sheaf of hyperfunctions: a) $ {\mathcal B} ( \Omega ) $ is "the totality of locally finite sums of analytic functionals with carrier in W" , where rearrangement by decomposition of supports is admitted as an equivalence relation (Martineau's definition); or b) for a bounded $ \Omega $, one has $ {\mathcal B} ( \Omega ) = {\mathcal B} [ \overline \Omega \; ]/ {\mathcal B} [ \partial \Omega ] $( Schapira's definition).

The sheaf $ {\mathcal Z} $ of Fourier hyperfunctions is defined on the directional compactification $ \mathbf D ^ {n} = \mathbf R ^ {n} \cup S _ \infty ^ {n - 1 } $ of $ \mathbf R ^ {n} $, employing instead of $ {\mathcal O} $ the sheaf $ {\mathcal O} tilde $ of holomorphic functions of infra-exponential growth, i.e. of growth $ O ( e ^ {\epsilon | \mathop{\rm Re} z | } ) $ for all $ \epsilon > 0 $. Fourier hyperfunctions on $ \mathbf D ^ {n} $ admit a Fourier transform of Bochner–Carleman type, via decomposition by support or growth order into proper convex cones. They form a nuclear Fréchet space, dual to the nuclear (DF)-space $ {\mathcal P} _ {*} $ of functions holomorphic and exponentially decreasing on a strip neighbourhood of the real axis. The latter is invariant under the classical Fourier transform and the Parseval equality holds with this duality.

A real-analytic coordinate transformation of hyperfunctions is defined naturally via transformation of defining functions. Hence hyperfunctions can be defined on real-analytic manifolds. Fourier series are typical examples of hyperfunctions on a manifold: $ \sum _ {\alpha \in \mathbf Z ^ {n} } a _ \alpha e ^ {i \alpha \cdot x } $ converges as a hyperfunction if and only if $ a _ \alpha = O ( e ^ {\epsilon | \alpha | } ) $ for all $ \epsilon > 0 $. Resolution by differential forms with hyperfunction coefficients supplies a concrete flabby resolution of the sheaf $ \mathbf C $ or $ {\mathcal O} $, and serves to calculate relative cohomology groups.

References

[a1]	M. Sato, "Theory of hyperfunctions" SÛgaku , 10 (1958) pp. 1–27 ((in Japanese))
[a2]	M. Sato, "Theory of hyperfunctions I, II" J. Fac. Sci. Univ. Tokyo Sect. 1 , 8 (1959–1960) pp. 139–193; 387–437
[a3]	A. Martineau, "Les hyperfonctions de M. Sato" , Sem. Bourbaki (1960)
[a4]	R. Harvey, "Hyperfunctions and linear partial differential equations" , Stanford Univ. Press (1966) (Thesis)
[a5]	H. Komatsu, "Sato's hyperfunctions and linear partial differential equations with constant coefficients" Seminar Notes Univ. of Tokyo , 22 (1970) ((in Japanese))
[a6]	P. Schapira, "Théorie des hyperfonctions" , Lect. notes in math. , 126 , Springer (1970)
[a7]	M. Morimoto, "Introduction to Sato hyperfunctions" , Kyôritsu (1976) ((in Japanese))
[a8]	M. Kashiwara, T. Kawai, T. Kimura, "Foundation of algebraic analysis" , Princeton Univ. Press (1986) ((translated from the Japanese))
[a9]	A. Kaneko, "Introduction to hyperfunctions" , Kluwer (1988) ((translated from the Japanese))