Singular integral operators with non-smooth kernels on irregular domains.

Xuan Thinh Duong; Alan McIntosh

Revista Matemática Iberoamericana (1999)

  • Volume: 15, Issue: 2, page 233-263
  • ISSN: 0213-2230

Abstract

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Let χ be a space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 < p ≤ 2; our condition is weaker then the usual Hörmander integral condition.ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(Ω) for 1 < p ≤ 2.iii) We establish sufficient conditions for the maximal truncated operator T* which is defined by T*u(x) = supε>0 |Tεu(x)|, to be Lp bounded, 1 < p < ∞. Applications include weak (1,1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

How to cite

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Duong, Xuan Thinh, and McIntosh, Alan. "Singular integral operators with non-smooth kernels on irregular domains.." Revista Matemática Iberoamericana 15.2 (1999): 233-263. <http://eudml.org/doc/39575>.

@article{Duong1999,
abstract = {Let χ be a space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 &lt; p ≤ 2; our condition is weaker then the usual Hörmander integral condition.ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(Ω) for 1 &lt; p ≤ 2.iii) We establish sufficient conditions for the maximal truncated operator T* which is defined by T*u(x) = supε&gt;0 |Tεu(x)|, to be Lp bounded, 1 &lt; p &lt; ∞. Applications include weak (1,1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.},
author = {Duong, Xuan Thinh, McIntosh, Alan},
journal = {Revista Matemática Iberoamericana},
keywords = {Espacio de medida; Espacios LP; Operadores lineales; Operadores integrales; Operadores de tipo débil; singular integrals; bounded operators; Hörmander condition; maximal functions; Riesz transforms},
language = {eng},
number = {2},
pages = {233-263},
title = {Singular integral operators with non-smooth kernels on irregular domains.},
url = {http://eudml.org/doc/39575},
volume = {15},
year = {1999},
}

TY - JOUR
AU - Duong, Xuan Thinh
AU - McIntosh, Alan
TI - Singular integral operators with non-smooth kernels on irregular domains.
JO - Revista Matemática Iberoamericana
PY - 1999
VL - 15
IS - 2
SP - 233
EP - 263
AB - Let χ be a space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 &lt; p ≤ 2; our condition is weaker then the usual Hörmander integral condition.ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(Ω) for 1 &lt; p ≤ 2.iii) We establish sufficient conditions for the maximal truncated operator T* which is defined by T*u(x) = supε&gt;0 |Tεu(x)|, to be Lp bounded, 1 &lt; p &lt; ∞. Applications include weak (1,1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.
LA - eng
KW - Espacio de medida; Espacios LP; Operadores lineales; Operadores integrales; Operadores de tipo débil; singular integrals; bounded operators; Hörmander condition; maximal functions; Riesz transforms
UR - http://eudml.org/doc/39575
ER -

Citations in EuDML Documents

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  1. Pascal Auscher, Au-delà des opérateurs de Calderón-Zygmund  : avancées récentes sur la théorie L p
  2. Sönke Blunck, A Hörmander-type spectral multiplier theorem for operators without heat kernel
  3. Steve Hofmann, Marius Mitrea, Sylvie Monniaux, Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds
  4. Pascal Auscher, Besma Ben Ali, Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials
  5. Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, Steve Hofmann, Riesz transform on manifolds and heat kernel regularity

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