G-type Spaces of Ultradistributions over ℝ^d₊ and the Weyl Pseudo-differential Operators with Radial Symbols

2016/02/12 by Smiljana Jaksić, Smiljana Jakšić, Stevan Pilipović +4
Mathematics · #Differential Equations and Boundary Problems #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.FA

paper · pdf · doi:10.48550/arxiv.1602.04017

arxiv created 2016/02/12 · arxiv updated 2016/02/15

Abstract

The first part of the paper is devoted to the G-type spaces i.e. the spaces G^α_α(\mathbb R^d₊), α≥ 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≥ 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(ℝ^d₊), α≥ 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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