On approach regions for the conjugate Poisson integral and singular integrals

S. Ferrando; R. Jones; K. Reinhold

ArticleOriginal scientific text

Title

On approach regions for the conjugate Poisson integral and singular integrals

Authors ¹, ², ³

Affiliations

Departamento de Matemáticas, Universidad Nacional de Mar del Plata, Mar del Plata, Bs. As., Argentina
Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, Illinois 60614, U.S.A.
Department of Mathematics, University at Albany, SUNY, 1400 Western Ave., Albany, New York 12222, U.S.A.

Abstract

Let ũ denote the conjugate Poisson integral of a function

f \in L^{p} (ℝ)

. We give conditions on a region Ω so that

lim_{(v, ε) \to (0, 0)}_{(v, ε) \in 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

\supset_{(v, r) \in Ω} | ʃ_{| t | > r} k (x + v - t) f (t) d t |

is a bounded operator on

L^{p}

, 1 < p < ∞, and is weak (1,1).

Keywords

ENG

cone condition, conjugate Poisson integral, singular integrals, ergodic Hilbert transform

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Title

On approach regions for the conjugate Poisson integral and singular integrals

Affiliations

Abstract

Keywords

Bibliography