Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A remark on singular Calderón-Zygmund theory

Authors: Michael Christ and Elias M. Stein
Journal: Proc. Amer. Math. Soc. 99 (1987), 71-75
MSC: Primary 42B25
MathSciNet review: 866432
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that in $ {{\mathbf{R}}^n}$ the operator

$\displaystyle Hf\left( x \right) = pv\int_{ - \infty }^{ + \infty } {f\left( {{x_1} - t, \ldots {x_n} - {t^n}} \right){t^{ - 1}}dt} $

maps $ L\left( {\log L} \right)$ to weak $ {L^1}$ locally. A slight variant of the Calderón-Zygmund procedure provides a new approach to the previously known $ {L^p}$ boundedness of $ H,1 < p < \infty $. Relatively sharp bounds are obtained as $ p \to {1^ + }$, and extrapolation produces the result for $ L\left( {\log L} \right)$.

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

  • [8] H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger, and D. Weinberg, $ {L^p}$ estimates for maximal functions and Hilbert transforms along flat convex curves in $ {{\mathbf{R}}^2}$, Bull. Amer. Math. Soc. 14 (1986), 263-267. MR 828823 (87f:42044)
  • [C1] M. Christ, Hilbert transforms along curves. I: Nilpotent groups, Ann. of Math. (2) 122 (1985), 575-596. MR 819558 (87f:42039a)
  • [C2] -, Hilbert transforms along curves. II: A flat case, Duke Math. J. 52 (1985), 887-894. MR 816390 (87f:42039b)
  • [C3] -, Differentiation along variable curves and related singular integral operators, announcement.
  • [C4] -, Weak type $ (1,1)$ bounds for rough operators, preprint.
  • [CW] R. R. Coifman and G. Weiss, Analyse noncommutative sur certains espaces homogenes, Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin and New York, 1971. MR 0499948 (58:17690)
  • [G] A. Greenleaf, Singular integral operators with conical singularities, preprint.
  • [JRdF] J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. MR 837527 (87f:42046)
  • [PS] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms, Acta Math. (to appear). MR 857680 (88i:42028a)
  • [S1] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [SW] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-1295. MR 508453 (80k:42023)
  • [Y] S. Yano, An extrapolation theorem, J. Math. Soc. Japan 3 (1951), 296-305. MR 0048619 (14:41c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B25

Retrieve articles in all journals with MSC: 42B25

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society