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.
- Marcinkiewicz-type spectral multipliers
- Hardy spaces
- Nonnegative self-adjoint operators
- Restriction type estimates
- Finite propagation speed property