Boundedness and Nuclearity of pseudo-differential operators on Homogeneous Trees

DOI10.1007/S13324-022-00691-9arXiv2111.11685WikidataQ114220017 ScholiaQ114220017MaRDI QIDQ6383725

Shyam Swarup Mondal

Publication date: 23 November 2021

Abstract: Let $x$ be the homogeneous tree of degree $q+1(qgeq2)$. In this article, we present symbolic criteria for boundedness of pseudo-differential operators on homogeneous tree $x$. A sufficient condition for weak type $(p,q)$ boundedness is also given. We present necessary and sufficient conditions on the symbols $sigma$ such that the corresponding pseudo-differential operators $T_{s}igma$ from $L^{p_1}(x)$ into $L^{p_2}(x)$ to be nuclear for $1leq{p}_1,p_2<infty$. We show that the adjoint of the nuclear operator from $L^{{p_2}^{\prime }}(x)$ into $L^{{p_1}^{\prime }}(x)$ is again a nuclear operator. Explicit formulas for the symbol of the adjoint and product of the nuclear pseudo differential operators on $x$ are given.

Mathematics Subject Classification ID

Harmonic analysis on homogeneous spaces (43A85) Discrete version of topics in analysis (39A12) Groups acting on trees (20E08) Pseudodifferential operators (47G30)

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