Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fourier multipliers on Lipschitz curves

Authors: Alan McIntosh and Tao Qian
Journal: Trans. Amer. Math. Soc. 333 (1992), 157-176
MSC: Primary 42B15; Secondary 47B35, 47G99
MathSciNet review: 1062194
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop the theory of Fourier multipliers acting on $ {L_p}(\gamma )$ where $ \gamma $ is a Lipschitz curve of the form $ \gamma = \{ x + ig(x)\} $ with $ \left\Vert g\right\Vert _\infty < \infty $ and $ \left\Vert g\prime\right\Vert _\infty < \infty $ . The aim is to better understand convolution singular integrals $ B$ defined naturally on such curves by

$\displaystyle Bu(z) = {\text{p.v.}}\int_\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } $

for almost all $ z \in \gamma $ .

References [Enhancements On Off] (What's this?)

  • [1] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur 𝐿² pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387 (French). MR 672839,
  • [2] R. R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 104–122. MR 576041
  • [3] M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with an $ {H_\infty }$-functional calculus (in preparation).
  • [4] Alan McIntosh, Operators which have an 𝐻_{∞} functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
  • [5] Alan McIntosh and Tao Qian, Fourier theory on Lipschitz curves, Miniconference on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 15, Austral. Nat. Univ., Canberra, 1987, pp. 157–166. MR 935598
  • [6] Alan McIntosh and Tao Qian, Convolution singular integral operators on Lipschitz curves, Harmonic analysis (Tianjin, 1988) Lecture Notes in Math., vol. 1494, Springer, Berlin, 1991, pp. 142–162. MR 1187074,
  • [7] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B15, 47B35, 47G99

Retrieve articles in all journals with MSC: 42B15, 47B35, 47G99

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society