On approach regions for the conjugate Poisson integral and singular integrals

Let ū denote the conjugate Poisson integral of a function f ∈ L^p (ℝ). We give conditions on a region Ω so that lim_{(v,ε) → (0,0) (v,ε) ∈ Ω} ũ(x + v, ε) = H f (x), the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that sup_{(v,r) ∈ Ω |t| > r} | ∫ k(x + v - t) f (t) dt | is a bounded operator on L^p, 1 < p < ∞, and is weak (1, 1).