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Weighted Lp Boundedness of Pseudodifferential Operators and Applications

Published online by Cambridge University Press:  20 November 2018

Nicholas Michalowski
Affiliation:
School of Mathematics and the Maxwell Institute of Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, Scotlande-mail: Nicholas.Michalowski@ed.ac.uk
David J. Rule
Affiliation:
Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotlande-mail: rule@uchicago.edue-mail: W.Staubach@hw.ac.uk
Wolfgang Staubach
Affiliation:
Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotlande-mail: rule@uchicago.edue-mail: W.Staubach@hw.ac.uk
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Abstract

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In this paper we prove weighted norm inequalities with weights in the ${{A}_{p}}$ classes, for pseudodifferential operators with symbols in the class $S_{\rho ,\delta }^{n(\rho -1)}$ that fall outside the scope of Calderón–Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy–Littlewood type maximal functions. Our weighted norm inequalities also yield ${{L}^{p}}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\text{OPS}_{\rho ,\delta }^{m}$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

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