On the solvability of linear partial differential equations in spaces of hyperfunctions
On the solvability of linear partial differential equations in spaces of hyperfunctions
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超函数空间中线性偏微分方程的可解性
DOI：
10.1007/bf02385666
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
J.
中科院分区：
文献类型：
--
作者：
P. Cordaro;J.
从光滑函数和分布空间中的线性偏微分方程的理论（参见Hsrmander [11]，[12]）中可以知道，微分方程的可解性与具有紧奇异支集的齐次伴随方程的解的不存在性有关，这可以用于从伴随算子的微局部研究中获得半整体存在性结果。在本文中，我们表明，一个类似的策略5是可能的超函数的框架。实际上，本文将在C~中具有低正则性的极大真实的流形上的超函数的框架内考虑无相容条件的微分方程组的更一般的情况。论文的第一部分可以被认为是Sehapira [26]，[27]的延续，其中展示了泛函分析如何用于微分算子的超函数理论。我们首先回顾的事实，hyperfunctionsolvability是不敏感的几何边界的域（定理1.2），并表明，有限维障碍，可解性从来没有发生（定理1.3）。然后，我们根据伴随算子的先验不等式的有效性来刻画微分算子的超函数可解性（定理1.4）。这一节的主要结果可能是定理1.6，其中指出，伴随算子的解析奇点的非限制是超函数可解的充分条件。这与定理1.2相似。在第2节中，我们给出了几个例子，说明如何应用第1节中的泛函分析语句来获得看似新的存在定理或经典存在定理的新证明，作为已有的，有时是深层次的微局部结果的推论。涉及完整系统、次解析结构或R”上的主型解析微分方程等问题。定理2.2
It is well known from the theory of linear partial differential equations in spaces of smooth functions and distributions, see HSrmander [11],[12], that the solvability of a differential equation is related to the non-existence of a solution of the homogeneous adjoint equation with compact singular support, and that this may be used to obtain semi-global existence results from the microlocal study of the adjoint o1> erator. In this paper we show that a similar strateg5 is possible in the framework of hyperfunctions. Actually, we shall consider in this paper the more general case of a system of differential equations without compatibility conditions in the framework of hyperfunctions on a maximally real manifold in C~ with low regularity. The first section of the paper may be considered as a continuation of Sehapira [26],[27], in which it was shown how functional analysis can be used in the hyperfunction theory of differential operators. We first recall the fact that hyperfunetion solvability is insensitive to the geometry of the boundary of the domain (Theorem 1.2) and show that finite dimensional obstruction to solvability never occurs (Theorem 1.3). Then we characterize the hyperfunction solvability of a differential operator in terms of the validity of an a priori inequality for the adjoint operator (Theorem 1.4). The main result of this section is perhaps Theorem 1.6 which states that the non-confinement of analytic singularities for the adjoint operator is a sufficient condition for the hyperfunetion solvability. This is similar to Theorem 1.2. 4 of H6rmander [11].In Section 2 we give several examples of how the functional analysis statements of Section 1 apply to obtain seemingly new existence theorems or new proofs of classical existence theorems, as corollaries of already available, sometimes deep, microlocal results. Such topics as holonomic systems, hypo-analytic structures or analytic differential equations of principal type on R" are touched on. Theorem 2.2