Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type

TL;DR

This paper establishes boundedness of multivariable spectral multipliers on Hardy and Lebesgue spaces over product spaces of homogeneous type, extending Marcinkiewicz multiplier theory to a multivariable setting with applications to elliptic operators.

Contribution

It introduces Marcinkiewicz-type spectral multiplier results for product spaces with doubling measures, under geometric and operator estimates, broadening the scope of spectral multiplier theory.

Findings

Boundedness of spectral multipliers under Marcinkiewicz conditions

Extension of Hardy and Lebesgue space analysis to product spaces

Applications to elliptic operators and Riesz transforms

Abstract

Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space $X_1\times X_2$. We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner--Riesz means.