Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions

Abstract

It is well known that the Hilbert tranformation and the conjugate function operator restricted to even (odd) functions define bounded linear operators on weighted L p {L^p} spaces under more general conditions than is the case for the unrestricted operators. In analogy with recent results of Hunt, Muckenhoupt and Wheeden for the Hilbert transform and the conjugate function operator, we obtain necessary and sufficient conditions in order that these restricted operators should satisfy weighted weak-type inequalities and hence also necessary and sufficient conditions in order that these operators should be bounded on weighted L p {L^p} spaces for 1 > p > ∞ 1 > p > \infty .