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# Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications

Published:2011-06-17
Printed: Sep 2012
• Nicholas Michalowski,
School of Mathematics and the Maxwell Institute of Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland
• David J. Rule,
Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland
• Wolfgang Staubach,
Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland
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## Abstract

In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ 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 $\mathrm{OP}S^{m}_{\rho, \delta}$.
 Keywords: weighted norm inequality, pseudodifferential operator, commutator estimates
 MSC Classifications: 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.) 42B25 - Maximal functions, Littlewood-Paley theory 35S05 - Pseudodifferential operators 47G30 - Pseudodifferential operators [See also 35Sxx, 58Jxx]

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