岡田 靖則

A compactly supported distribution is called invertible in the sense of Ehrenpreis-Hörmander if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may be the contribution from the origin or from the boundary of the support of the function. For the proof, we propose a new method to calculate the asymptotic expansions of finite Hankel transforms of functions with singularities at a point other than the origin.

In this paper, we continue our study of integral operators in [12] and give a result on the analytic singular support of the kernel hyperfunction similar to the main result in [12]. A new feature with respect to [12] is that we use classes of ultradifferentiable functions associated with a weight function and a topological description of related spaces of quasi analytic functionals.

The aim of the paper is to study the relation between ultra-differentiable classes of functions defined in terms of estimates on derivatives on one hand and in terms of growth properties of Fourier transforms of suitably localized functions in the class on the other hand. We establish this relation for the ultra-differentiable classes introduced in , , and show that the classes of , , can be regarded as inhomogeneous Gevrey classes in the sense of . We also discuss a number of properties of the weight functions used to define the respective classes and of their Young conjugates.

The main result of this paper is that the integral operators between spaces of compactly supported hyperfunctions must have properly supported kernels. We also discuss the uniqueness and the regularity of the integral operators in hyperfunction theory. (C) 2012 Elsevier Inc. All rights reserved.

We introduce a sheaf of bounded hyperfunctions at infinity in one variable, and consider periodic linear functional equations. We show that a Massera type theorem holds in this framework. Moreover we consider its vector-valued variants and give applications to partial differential equations periodic with respect to one variable.

We study holomorphic solutions for convolution equations in tube domains. Let Qr be the sheaf of holomorphic functions in tube domains on the purely imaginary space root-1R(n) and Y the complex 0 --> O-tau mu*/--> O-tau --> 0 generated by the convolution operator mu* with hyperfunction kernel mu. In this paper, we give a new definition of "the characteristic set" Char(mu*) using terms of zeros of the total symbol of mu*, and show, under the abstract condition (S), the equivalence between two notions of characteristics outside of the zero section T*(root-1Rn).(root-1R(n)). Moreover we conclude that the micro-support SS(Y) of Y coincides with the characteristics Char(mu*).

Garding-Kotake-Leray studied a characteristic Cauchy problem Pu = v epsilon O with zero Cauchy data on a hypersurface S. They showed that u can be ramified of order q around another hypersurface K, which has a contact of order q - 1 with S along the characteristic points. Dunau generalized this result. In this paper, we further extend Dunau's result: we establish a one-to-one correspondence between u's as above and v's in a certain class of ramified functions.

We study the surjectivity of convolution operators on the spaces of hyperfunctions and Fourier hyperfunctions. On the space of hyperfunctions, we give a sufficient condition (the kernel is a nonzero ultradistribution), weaker than earlier conditions. On the space of Fourier hyperfunction, we give a new sufficient condition and new necessary conditions for the surjectivity. Especially in one dimensional case, they become a sufficient and necessary condition. To this aim we use the Fourier analysis as in L. Ehrenpreis [E-2] and T. Kawai [Ka-1].

For a hyperfunction mu(x) with compact support, we consider the convolution equation (E) mu*f = g with holomorphic functions f and g defined on a domain of the form U x root-1R(n). Under a natural condition (the condition (S)), we will prove the existence of solution of (E) and the analytic continuation of homogeneous equation mu*f = 0. Defining the characteristic set Char(mu*), these imply that directions to which a solution cannot be continued are estimated by Char(mu*).

The aim of the present article is to describe second microlocal singularities along linear involutive submanifolds for hyperfunctions or microfunctions in the language of boundary values of holomorphic functions. To this aim, we introduce generalized V-profiles and strict tuboids associated to generalized V-profiles. This enables us to show the equivalence between second analytic wave front sets due to J. Sjostrand and second singular spectrums due to M. Kashiwara.