Control of pseudodifferential operators by maximal functions via weighted inequalities

Abstract

We establish general weighted L 2 L^2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S ρ , δ m S^m_{\rho ,\delta } . Such inequalities allow one 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.