# 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

## Access Full Article

top## Abstract

top## How to cite

topGaudry, 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

top- [D] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. 17 (1984), 157-189. Zbl0537.42016
- [DJS] G. David, J.-L. Journé et S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), 1-56. Zbl0604.42014
- [CGQ] M. G. Cowling, G. I. Gaudry and T. Qian, A note on martingales with respect to complex measures, in: Miniconf. on Operators in Analysis, Macquarie Univ., Sept. 1989, Proc. Centre Math. Anal. Austral. Nat. Univ. 24, Austral. Nat. Univ., Canberra, 1989, 10-27.
- [CJS] R. R. Coifman, P. W. Jones and S. Semmes, Two elementary proofs of the ${L}^{2}$ boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), 553-564. Zbl0713.42010
- [CM1] R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Calderón's theorem, and analysis on Lipschitz curves, in: Lecture Notes in Math. 779, Springer 1980, 104-122.
- [CM2] R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57 (1978). Zbl0483.35082
- [CMcM] R. Coifman, A. McIntosh et Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur ${L}^{2}$ pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361-387.
- [deL] K. de Leeuw, On ${L}_{p}$ multipliers, Ann. of Math. 81 (1965), 364-379.
- [EG] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Springer, 1970.
- [GLQ] A martingale proof of ${L}_{2}$ boundedness of Clifford-valued singular integrals, Ann. Mat. Pura Appl. 165 (1993), 369-394. Zbl0814.42009
- [JK] D. Jerison and C. Kenig, Hardy spaces, ${A}_{\infty}$, and singular integrals on chord-arc domains, Math. Scand. 50 (1982), 221-247. Zbl0509.30025
- [K] C. E. Kenig, Weighted ${H}^{p}$ spaces on Lipschitz domains, Amer. J. Math. 102 (1980), 129-163.
- [LMcS] C. Li, A. McIntosh and S. Semmes, Convolution singular integral operators on Lipschitz surfaces, to appear.
- [LMcQ] C. Li, A. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana, to appear.
- [Mc] A. McIntosh, Operators which have an ${H}^{\infty}$-functional calculus, in: Miniconf. on Operator Theory and Partial Differential Equations, 1986, Proc. Centre Math. Anal. Austral. Nat. Univ. Canberra, 14 Austral. Nat. Univ., 1986, 210-231.
- [McQ1] A. McIntosh and T. Qian, Convolution singular integral operators on Lipschitz curves, in: Lecture Notes in Math. 1494, Springer, 1991, 142-162. Zbl0791.42012
- [McQ2] A. McIntosh and T. Qian, A note on singular integrals with holomorphic kernels, Approx. Theory Appl. 6 (1990), 40-54.
- [McQ3] A. McIntosh and T. Qian, Fourier multipliers on Lipschitz curves, Trans. Amer. Math. Soc. 333 (1992), 157-176. Zbl0766.42005
- [Q] T. Qian, Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves, preprint. Zbl0924.42012
- [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
- [T] T. Tao, Convolution operators generated by right-monogenic and harmonic kernels, M.Sc. thesis, Flinders Univ. of South Australia, 1992; Paper of same title, submitted.
- [Z] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, 1968. Zbl0157.38204

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.