Multivariable spectral multipliers and quasielliptic operators

TL;DR

This paper establishes that multivariable spectral multipliers satisfying Hörmander conditions are Calderón-Zygmund operators and applies these results to analyze quasielliptic operators on fractal spaces, showing their Riesz operators are L^p continuous.

Contribution

It proves multivariable spectral multipliers are Calderón-Zygmund operators under Hörmander conditions and extends analysis to quasielliptic operators on fractals.

Findings

Spectral multipliers satisfying Hörmander conditions are Calderón-Zygmund operators.

Riesz operators for quasielliptic operators are bounded on L^p spaces.

Application to fractal spaces demonstrates the operators' properties on complex geometries.

Abstract

We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies H\"ormander type differentiability condition then the operator $F(L_1,L_2)$ is of Calder\'on-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators corresponding to quasielliptic operators are continuous on $L^p$ spaces.