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

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Let X₁ and X₂ be metric spaces equipped with doubling measures and let L₁ and L₂ be nonnegative self-adjoint second-order 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.