A calculus approach to hyperfunctions. II
A calculus approach to hyperfunctions. II
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超函数的微积分方法。
DOI：
10.1090/s0002-9947-1989-0997676-7
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发表时间：
1989
影响因子：
1.3
通讯作者：
T. Matsuzawa
中科院分区：
文献类型：
--
作者：
T. Matsuzawa
我们考虑任何具有紧化支持的超函数作为热方程解的初值。本文的主要目的是统一分布和超函数的理论，并简化一些重要结果的证明。在之前的论文[9]中，我们讨论了由[3，Chapter 9]中发展的分析所激发的微积分方法的超函数理论。我们已经在b[9]中展示了我们方法的思想。也就是说，我们提出将具有紧支持的超函数作为热方程解的初值。本文的主要目的是利用Schwartz分布理论中的热核极限过程来重新表述超泛函理论的基础。为此，将使用两个基本工具。一种是（1.14）中的热核估计，它是在[1，第4章]中给出的结果的特殊情况下得到的。另一种是[5,6,7]中给出的一系列关于超分布的结构定理。正如[9]中提到的，我们的方法的优点是它可以统一分布和超函数的理论，并且简化了一些重要结果的证明。在吗?我们简要回顾了紧支持下的分布和超函数的定义。特别地，在K c Rn （n > 1）中具有紧支持的超函数（解析泛函）空间A′[K]被认为是K附近实解析函数空间A[K]的对偶。我们将在命题1.1中准备一个有用的热核估计。在吗?2，我们将证明我们可以取u E A'[K]作为热方程（A /ot-A） u (x,t)=O inRn+l =R n x R+ u (-，O) = u的唯一解u的初值。我们将在定理2.1中看到，Schwartz分布和超分布也具有u （-，O）的渐近性质。， t)为1987年6月8日编辑收到的t -+ 0，以及1987年12月8日修改后的版本。1980年数学学科分类（1985年修订）。主要46 f15;二级46F05、46F10、46F12、35K05。？ 1989美国数学学会0002-9947/89 $1.00 + $。每页25元
We consider any hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results via heat kernel. INTRODUCTION In the previous paper [9], we have discussed the theory of the hyperfunctions by a calculus approach motivated by the analysis developed in [3, Chapter 9]. We have shown in [9] the idea of our approach. Namely, we have proposed to take hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to reformulate the foundation of hyperfunction theory by a limiting process via heat kernel in the theory of Schwartz distributions. For this purpose, two fundamental tools will be used. One is the estimate for the heat kernel in (1.14) which is obtained as a special case of the result given in [1, Chapter 4]. The other is a series of structure theorems on ultradistributions given in [5, 6 and 7]. As was mentioned in [9], the advantage of our method is that it can unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results. In ?1, we briefly recall the definitions of distributions and hyperfunctions with the compact support. Especially, the space of hyperfunctions (analytic functionals) A'[K] with the compact support in K c Rn (n > 1) is considered as the dual of the space A[K] of functions which are real analytic near K. We will prepare a useful estimate for the heat kernel in Proposition 1.1. In ?2, we will prove that we can take u E A'[K] as an initial value of a unique solution U of the heat equation (a/ot-A)U(x,t)=O inRn+l =R n x R+ U(-,O) = u. We shall see in Theorem 2.1 that Schwartz distributions and ultradistributions are also characterized by the asymptotic behavior of U(., t) as t -+ 0 at the Received by the editors June 8, 1987 and, in revised form, December 8, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46F15; Secondary 46F05, 46F10, 46F12, 35K05. ? 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page