# On approach regions for the conjugate Poisson integral and singular integrals

Studia Mathematica (1996)

• Volume: 120, Issue: 2, page 169-182
• ISSN: 0039-3223

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## Abstract

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Let ũ denote the conjugate Poisson integral of a function $f\in {L}^{p}\left(ℝ\right)$. We give conditions on a region Ω so that $li{m}_{\left(v,\epsilon \right)\to \left(0,0\right)}\left(v,\epsilon \right)\in \Omega ũ\left(x+v,\epsilon \right)=Hf\left(x\right)$, 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 $su{p}_{\left(v,r\right)\in \Omega }|{ʃ}_{|t|>r}k\left(x+v-t\right)f\left(t\right)dt|$ is a bounded operator on ${L}^{p}$, 1 < p < ∞, and is weak (1,1).

## How to cite

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Ferrando, S., Jones, R., and Reinhold, K.. "On approach regions for the conjugate Poisson integral and singular integrals." Studia Mathematica 120.2 (1996): 169-182. <http://eudml.org/doc/216328>.

@article{Ferrando1996,
abstract = {Let ũ denote the conjugate Poisson integral of a function $f ∈ L^\{p\}(ℝ)$. We give conditions on a region Ω so that $lim_\{(v,ε)→(0,0)\}_\{(v,ε)∈Ω\} ũ(x+v,ε) = Hf(x)$, 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 $sup_\{(v,r)∈Ω\} |ʃ_\{|t|>r\} k(x+v-t)f(t)dt|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).},
author = {Ferrando, S., Jones, R., Reinhold, K.},
journal = {Studia Mathematica},
keywords = {cone condition; conjugate Poisson integral; singular integrals; ergodic Hilbert transform; approach regions; Calderón-Zygmund singular integral; maximal transform; ergodic theory},
language = {eng},
number = {2},
pages = {169-182},
title = {On approach regions for the conjugate Poisson integral and singular integrals},
url = {http://eudml.org/doc/216328},
volume = {120},
year = {1996},
}

TY - JOUR
AU - Ferrando, S.
AU - Jones, R.
AU - Reinhold, K.
TI - On approach regions for the conjugate Poisson integral and singular integrals
JO - Studia Mathematica
PY - 1996
VL - 120
IS - 2
SP - 169
EP - 182
AB - Let ũ denote the conjugate Poisson integral of a function $f ∈ L^{p}(ℝ)$. We give conditions on a region Ω so that $lim_{(v,ε)→(0,0)}_{(v,ε)∈Ω} ũ(x+v,ε) = Hf(x)$, 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 $sup_{(v,r)∈Ω} |ʃ_{|t|>r} k(x+v-t)f(t)dt|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).
LA - eng
KW - cone condition; conjugate Poisson integral; singular integrals; ergodic Hilbert transform; approach regions; Calderón-Zygmund singular integral; maximal transform; ergodic theory
UR - http://eudml.org/doc/216328
ER -

## References

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1. [1] M. A. Akcoglu and Y. Déniel, Moving weighted averages, Canad. J. Math. 45 (1993), 449-469.
2. [2] A. P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349-353. Zbl0185.21806
3. [3] P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335-400. Zbl37.0283.01
4. [4] S. Ferrando, Moving ergodic theorems for superadditive processes, Ph.D. thesis, Univ. of Toronto, 1994. Zbl0837.28013
5. [5] A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. Math. 54 (1984), 83-106. Zbl0546.42017
6. [6] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
7. [7] E. M. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1993.

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