On approach regions for the conjugate Poisson integral and singular integrals

Let (u) over tilde denote the conjugate Poisson integral of a function f is an element of L(p)(R). We give conditions on a region Omega so that [GRAPHICS] the Hilbert transform of f at x, for a.e. x. We also consider more general Calderon-Zygmund singular integrals and give conditions on a set Omega so that [GRAPHICS] is a bounded operator on L(p), 1 < p < infinity, and is weak (1, 1).