Control of pseudodifferential operators by maximal functions via weighted inequalities

Beltran, David

doi:10.1090/tran/7365

Control of pseudodifferential operators by maximal functions via weighted inequalities
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Trans. Amer. Math. Soc. 371 (2019), 3117-3143 Request permission

Abstract:

We establish general weighted $L^2$ inequalities for pseudodifferential operators associated to the Hörmander symbol classes $S^m_{\rho ,\delta }$. 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.

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