Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let (X_1) and (X_2) be metric spaces equipped with doubling measures and let (L_1) and (L_2) be nonnegative self-adjoint 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_1times 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.