Results on weighted norm inequalities for multipliers

Abstract

Weighted L p {L^p} -norm inequalities are derived for multiplier operators on Euclidean space. The multipliers are assumed to satisfy conditions of the Hörmander-Mikhlin type, and the weight functions are generally required to satisfy conditions more restrictive than A p {A_p} which depend on the degree of differentiability of the multiplier. For weights which are powers of | x | \left | x \right | , sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function f # {f^\# } of C. Fefferman and E. Stein rather than on Littlewood-Paley theory. The method also yields results for singular integral operators.