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arXiv 1602.04017math.FA
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$\mathbb{R}^d_+$上超分布的$G$型空间与带径向符号的Weyl伪微分算子

$G$-type Spaces of Ultradistributions over $\mathbb{R}^d_+$ and the Weyl Pseudo-differential Operators with Radial Symbols

Smiljana Jaksić, Stevan Pilipović, Bojan Prangoski

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摘要：

第一部分的文件是专门的$G$型空间，即空间$G^α_α(\mathbb R^d_+)$，$α\geq 1$和他们的可描述为类似的Gelfand-Shilov空间和他们的可描述，但与全新的理由所获得的结果。通过系数的指数和次指数增长性，讨论了$G^α_α(\mathbb R^d_+)$，$α\geq 1$中元素的Laguerre型展开式及其对这些空间的逼近.给出了完整的拓扑描述，并利用$G_α^α(\mathbb{R}^d_+)$、$α\geq 1$的核性证明了核定理.第二部分讨论了属于$G$型空间的带径向符号的Weyl算子类。证明了这类伪微分算子在Gelfand-Shilov型空间上的连续性及其连续性.这样就把Weyl伪微分算子类推广到了具有指数和次指数增长性的径向符号算子类.

英文摘要：

The first part of the paper is devoted to the $G$-type spaces i.e. the spaces $G^α_α(\mathbb R^d_+)$, $α\geq 1$ and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justification of obtained results. The Laguerre type expansions of the elements in $G^α_α(\mathbb R^d_+)$, $α\geq 1$ and their duals characterise these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological description and by the nuclearity of $G_α^α(\mathbb{R}^d_+)$, $α\geq 1$ the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the $G$-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.

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