Ultradistributions on $$ {\mathbb {R}}_{+}^{n}$$ and solvability and hypoellipticity through series expansions of ultradistributions

IF 0.9 3区 数学 Q2 MATHEMATICS
Stevan Pilipović, Ɖorđe Vučković
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引用次数: 0

Abstract

In the first part we analyze space \({\mathcal {G}}^*({\mathbb {R}}^{n}_+)\) and its dual through Laguerre expansions when these spaces correspond to a general sequence \(\{M_p\}_{p\in {\mathbb {N}}_0}\), where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form \(Lu=f,\; L=\sum _{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),\) where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on \({\mathbb {R}}^n_+\) and \({\mathbb {R}}^m\); \(A_j, j=1,...,k\), \(E_y\) and \(P(x,D_x)\) are operators whose eigenfunctions form orthonormal basis of corresponding \(L^2-\)space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.

$$ {\mathbb {R}}_{+}^{n}$ 上的超分布以及通过超分布的数列展开实现的可解性和次椭圆性
在第一部分中,我们通过拉盖尔展开分析空间({\mathcal {G}}^*({\mathbb {R}}^{n}_+)\) 及其对偶,当这些空间对应于一般序列 \(\{M_p\}_{p\in{\mathbb {N}}}_0}\) 时,其中 * 是 Beurling 和 Roumieu 空间情况的通用符号。在第二部分中,我们要求解的方程的形式是 (Lu=f,\;L=sum_{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),()其中 f 属于无边界紧凑流形上超分布空间以及 \({\mathbb {R}^n_+\) 和 \({\mathbb {R}^m\) 上超分布空间的张量积;(A_j, j=1,....,k\)、(E_y\)和(P(x,D_x)\)是其特征函数构成相应的(L^2-\)空间正交基础的算子。解的序列空间表示使我们能够研究超分布指定空间中的可解性和次椭圆性。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
59
期刊介绍: The Journal of Pseudo-Differential Operators and Applications is a forum for high quality papers in the mathematics, applications and numerical analysis of pseudo-differential operators. Pseudo-differential operators are understood in a very broad sense embracing but not limited to harmonic analysis, functional analysis, operator theory and algebras, partial differential equations, geometry, mathematical physics and novel applications in engineering, geophysics and medical sciences.
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