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

http://dx.doi.org/10.4153/CMB-2011-122-7
Canad. Math. Bull. 55(2012), 555-570

  Published:2011-06-17
 Printed: Sep 2012

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 $\mathrm{OP}S^{m}_{\rho, \delta}$.