Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves

Garth Gaudry; Tao Qian; Silei Wang

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 1, page 133-150
  • ISSN: 0010-1354

Abstract

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The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set z ζ - 1 : z , ζ ∈ , z ≠ ζ , where is a star-shaped Lipschitz curve, = e x p ( i z ) : z = x + i A ( x ) , A ' ∈ L ∞ [ - π , π ] , A ( - π ) = A ( π ) . Under suitable conditions on F and z, the operators are given by (1) T F ( z ) = p . v . ∫ ( z η - 1 ) F ( η ) ( d η / η ) . We identify a class of kernels of the stated type that give rise to bounded operators on L 2 ( , | d | ) . We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

How to cite

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Gaudry, Garth, Qian, Tao, and Wang, Silei. "Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves." Colloquium Mathematicae 70.1 (1996): 133-150. <http://eudml.org/doc/210391>.

@article{Gaudry1996,
abstract = {The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set $\{zζ^\{-1\}: z, ζ ∈ , z ≠ ζ\}$, where is a star-shaped Lipschitz curve, $ =\{ exp(iz) : z = x+iA(x), A^\{\prime \} ∈ L^\{∞\}[-π,π], A(-π ) =A(π)\}$. Under suitable conditions on F and z, the operators are given by (1) $TF(z)= p.v. ∫_ (zη^\{-1\})F(η)(dη/η).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^\{2\} (,|d|)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.},
author = {Gaudry, Garth, Qian, Tao, Wang, Silei},
journal = {Colloquium Mathematicae},
keywords = {singular integrals; holomorphic kernels; star-shaped Lipschitz curve; transference; boundedness},
language = {eng},
number = {1},
pages = {133-150},
title = {Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves},
url = {http://eudml.org/doc/210391},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Gaudry, Garth
AU - Qian, Tao
AU - Wang, Silei
TI - Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 1
SP - 133
EP - 150
AB - The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set ${zζ^{-1}: z, ζ ∈ , z ≠ ζ}$, where is a star-shaped Lipschitz curve, $ ={ exp(iz) : z = x+iA(x), A^{\prime } ∈ L^{∞}[-π,π], A(-π ) =A(π)}$. Under suitable conditions on F and z, the operators are given by (1) $TF(z)= p.v. ∫_ (zη^{-1})F(η)(dη/η).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^{2} (,|d|)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.
LA - eng
KW - singular integrals; holomorphic kernels; star-shaped Lipschitz curve; transference; boundedness
UR - http://eudml.org/doc/210391
ER -

References

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