A calculus approach to hyperfunctions I

Abstract

In this paper, we shall give a new characterization of hyperfunctions without algebraic method and apply to give simpler proofs to problems discussed in [3], Chapter 9. In [3], the spaces of hyperfunctions A′(K) with compact support in K ⊂ Rn (n ≧ 1) is considered as the dual of the space A(K) of functions which are real analytic near K. Each element u of A′(K) is characterized as a density of a double layer potential in Rn × R.

The Bochner–Schwartz theorem for Fourier ultra-hyperfunctions

“…Matsuzawa [1][2][3]24] established the correspondence between the space of generalized functions and the space of the solutions of the heat equation with some estimate…”

The action of operator semigroups on the topological dual of the Beurling–Björck space

“…|z| M e kw(y−y ) G(z)tM e kw(y ) ϕ(y ) dz, (14) for some y in the line segment between y and y + tz. Now, we note that w(y − y ) = Ω(|y − y |) Ω(|tz|) Ω(|z|) = w(z), since t < 1.…”

“…In this respect, our work is inspired by a substantial body of work on the realization of various types of generalized functions as initial values of solutions for the classical heat equation. This work, pioneered by T. Matsuzawa [13,14] for hyperfunctions and inspired by the work of L. Hörmander [10], typically uses functional estimates that are variations of those describing real analyticity. In our case, however, the definition of the space S w involves conditions on the function and on its Fourier transform.…”

Distributions of Exponential Growth with Support in a Proper Convex Cone

“…In this section, we shall characterize H (R n , K), the space of distributions of exponential growth, by the heat kernel method introduced by T. Matsuzawa in [12]. We notice that many authors make use of his idea ( [2], [3], [9], [10], [20]).…”

“…In §4 we shall characterize the space H (R n , K) by the heat kernel method, which T. Matsuzawa introduced for the spaces of distributions, ultradistributions and hyperfunctions [4], [12], [13], [14]. The main purpose in this section is to show that the convolution of the heat kernel and a distribution of exponential growth is a smooth solution of the heat equation with some exponential growth condition and conversely such a smooth solution can be represented by the convolution of the heat kernel and a distribution of exponential growth (Theorem 4.4).…”