Control of pseudodifferential operators by maximal functions via weighted inequalities

Abstract

We establish general weighted $L^2$ inequalities for pseudodifferential operators associated to the Hörmander symbol classes $S_{\rho ,\delta }^m$. Such inequalities allow to control these operators by fractional "non-tangential" maximal functions, and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar-Stein almost orthogonality principle and a quantitative version of the symbolic calculus.

arXiv commentRevised version. To appear in Trans. Amer. Math. Soc