Quantitative Weighted Bound for Spectral Multiplier of Laplace Transform Type

Let L be a non-negative and self-adjoint operator that is bounded on L2(Rn). Under the generalized Muckenhoupt weights Apρ,θ, we establish more refined quantitative weighted estimates for the Laplace-transform-type spectral multiplier M(L): the weighted bound is improved from [w]Apρ,θmax{1,1/(p−1)} to [w]Apρ,θ1/p([w]A∞ρ,θ1/p′+[w1−p′]A∞ρ,θ1/p), and the weak-type endpoint estimate at p=1 is provided. For the composition M(L1)M(L2), we obtain the strong-type and weak-type weighted estimates, revealing the dependence on [w]Apρ,θ, [w]A∞ρ,θ and [w1−p′]A∞ρ,θ. These results refine the weight constant dependency and extend composition theory to spectral multipliers.