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1996 | 70 | 1 | 133-150
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The aim of this paper is to study singular integrals T generated by holomorphic kernels 𝛷 defined on a natural neighbourhood of the set ${zζ^{-1}: z, ζ ∈ 𝛤, z ≠ ζ}$, where 𝛤 is a star-shaped Lipschitz curve, $𝛤 ={ exp(iz) : z = x+iA(x), A' ∈ L^{∞}[-π,π], A(-π ) =A(π)}$. Under suitable conditions on F and z, the operators are given by (1) $TF(z)= p.v. ∫_𝛤 𝛷(zη^{-1})F(η)(dη/η).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^{2} (𝛤,|d𝛤|)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.
Słowa kluczowe
Rocznik
Tom
70
Numer
1
Strony
133-150
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-11-08
poprawiono
1995-08-30
Twórcy
autor
  • School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
autor
  • Department of Mathematics, The University of New England, Armidale, NSW 2351, Australia
autor
  • Department of Mathematics, Hangzhou University, Hangzhou, Zhejiang P.R. China
Bibliografia
  • [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.
  • [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.
  • [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.
  • [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.
  • [JK] D. Jerison and C. Kenig, Hardy spaces, $A_∞$, and singular integrals on chord-arc domains, Math. Scand. 50 (1982), 221-247.
  • [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 $H^∞$-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.
  • [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.
  • [Q] T. Qian, Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves, preprint.
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv70i1p133bwm
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