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

Let X₁ and X₂ be metric spaces equipped with doubling measures and let L₁ and L₂ be nonnegative self-adjoint operators acting on L²(X₁) and L²(X₂) respectively. We study multivariable spectral multipliers F(L₁, L₂) acting on the Cartesian product of X₁ and X₂. Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators L₁ and L₂, we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator F(L₁, L₂) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space X₁× X₂. 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.