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1997 | 123 | 3 | 195-216
Tytuł artykułu

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

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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.
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autor
  • Department of Mathematics, The New England University, Armidale, New South Wales 2351, Australia , tao@neumann.une.edu.au
Bibliografia
  • [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).
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [LMcS] C. Li, A. McIntosh and S. Semmes, Convolution singular integral operators on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455-481.
  • [Mc] A. McIntosh, Operators which have an $H_∞$-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.
  • [McQ2] A. McIntosh and T. Qian, Singular integrals along Lipschitz curves with holomorphic kernels, Approx. Theory Appl. 6 (1990), 40-57.
  • [McQ3] A. McIntosh and T. Qian, Fourier multipliers on Lipschitz curves, Trans. Amer. Math. Soc. 333 (1992), 157-176.
  • [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 $ℝ^n$, 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.
  • [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.
  • [Z] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, London and New York, 1968.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv123i3p195bwm
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