# Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves

Studia Mathematica (1997)

- Volume: 123, Issue: 3, page 195-216
- ISSN: 0039-3223

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topQian, Tao. "Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves." Studia Mathematica 123.3 (1997): 195-216. <http://eudml.org/doc/216389>.

@article{Qian1997,

abstract = {The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the $L^2$-boundedness of the Cauchy integral operator on Lipschitz curves. The operator theory has a counterpart in Fourier multiplier theory, as well as a counterpart in functional calculus of the differential operator 1/i d/dz on the curves.},

author = {Qian, Tao},

journal = {Studia Mathematica},

keywords = {singular integrals; Fourier multipliers; Cauchy integral operator; Lipschitz curves},

language = {eng},

number = {3},

pages = {195-216},

title = {Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves},

url = {http://eudml.org/doc/216389},

volume = {123},

year = {1997},

}

TY - JOUR

AU - Qian, Tao

TI - Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves

JO - Studia Mathematica

PY - 1997

VL - 123

IS - 3

SP - 195

EP - 216

AB - The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the $L^2$-boundedness of the Cauchy integral operator on Lipschitz curves. The operator theory has a counterpart in Fourier multiplier theory, as well as a counterpart in functional calculus of the differential operator 1/i d/dz on the curves.

LA - eng

KW - singular integrals; Fourier multipliers; Cauchy integral operator; Lipschitz curves

UR - http://eudml.org/doc/216389

ER -

## References

top- [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.
- [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
- [FJR] E. B. Fabes, M. Jodeit, Jr. and N. M. Rivière, Potential techniques for boundary value problems on ${C}^{1}$ domains, Acta Math. 141 (1978), 165-186. Zbl0402.31009
- [GQW] G. Gaudry, T. Qian and S.-L. Wang, Boundedness of singular integral operators with holomorphic kernels on star-shaped Lipschitz curves, Colloq. Math. 70 (1996), 133-150. Zbl0860.42013
- [LMcQ] C. Li, A. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994), 665-721. Zbl0817.42008
- [LMcS] C. Li, A. McIntosh and S. Semmes, Convolution singular integral operators on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455-481. Zbl0763.42009
- [Mc] A. McIntosh, Operators which have an ${H}_{\infty}$-functional calculus, in: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Canberra, 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, Singular integrals along Lipschitz curves with holomorphic kernels, Approx. Theory Appl. 6 (1990), 40-57. Zbl0737.42017
- [McQ3] A. McIntosh and T. Qian, Fourier multipliers on Lipschitz curves, Trans. Amer. Math. Soc. 333 (1992), 157-176. Zbl0766.42005
- [Q1] T. Qian, Singular integrals on the m-torus and its Lipschitz perturbations, in: Clifford Algebras in Analysis and Related Topics, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1995, 94-108.
- [Q2] T. Qian, Transference from Lipschitz graphs to periodic Lipschitz graphs, in: Miniconference on Analysis and Applications (Brisbane, 1993), Proc. Centre Math. Appl. Austral. Nat. Univ. 33, Canberra, 1994, 189-194.
- [Q3] T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, preprint.
- [Q4] T. Qian, A holomorphic extension result, Complex Variables Theory Appl. 32 (1996), 59-77.
- [Q5] T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space and generalizations to ${\mathbb{R}}^{n}$, in: Proc. Conf. on Clifford and Quaternionic Analysis and Numerical Methods, June 1996, to appear.
- [S] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
- [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
- [V] G. Verchota, Layer potentials and regularity for the Dirichlet problems for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. Zbl0589.31005
- [Z] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, London and New York, 1968. Zbl0157.38204