Multivariate Spectral Multipliers

2014/07/09 by Błażej Wróbel, Wróbel, Błażej · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1407.2393

openalex publication_date 2014/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This thesis is devoted to the study of multivariate (joint) spectral multipliers for systems of strongly commuting non-negative self-adjoint operators, L=(L₁,…,L_d), on L²(X,ν), where (X,ν) is a measure space. By strong commutativity we mean that the operators L_r, r=1,…,d, admit a joint spectral resolution E(λ). In that case, for a bounded function m\colon [0,∞)^d→ ℂ, the multiplier operator m(L) is defined on L²(X,ν) by m(L)=∫_[0,∞)^dm(λ)dE(λ). By spectral theory, m(L) is then bounded on L²(X,ν). The purpose of the dissertation is to investigate under which assumptions on the multiplier function m it is possible to extend m(L) to a bounded operator on L^p(X,ν), 1

Citations

• Algebras of differential operators on Lie groups and spectral multipliers
• Estimates for translation invariant operators in Lp spaces
• Convolution operator and maximal function for the Dunkl transform

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• Joint spectral multipliers for mixed systems of operators

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