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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 weighted norm inequality, pseudodifferential operator, commutator estimates
MSC Classifications: 42B20, 42B25, 35S05, 47G30 show english descriptions Singular and oscillatory integrals (Calderon-Zygmund, etc.)
Maximal functions, Littlewood-Paley theory
Pseudodifferential operators
Pseudodifferential operators [See also 35Sxx, 58Jxx]
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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