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Transactions of the American Mathematical Society

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



The Hilbert transform and maximal function for approximately homogeneous curves

Author: David A. Weinberg
Journal: Trans. Amer. Math. Soc. 267 (1981), 295-306
MSC: Primary 42B20; Secondary 42B25, 44A15
MathSciNet review: 621989
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Abstract: Let $ {\mathcal{H}_\gamma }f(x) = {\text{p}}{\text{.v}}{\text{.}}\int_{ - 1}^1 {f(x - \gamma (t))dt/t} $ and $ {\mathfrak{M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert dt} $. It is proved that for $ f \in \mathcal{S}({{\mathbf{R}}^n})$, the Schwartz class, and for an approximately homogeneous curve $ \gamma (t) \in {{\mathbf{R}}^n}$, $ {\left\Vert {{\mathcal{H}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$, $ {\left\Vert {{\mathfrak{M}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$.

A homogeneous curve is one which satisfies a differential equation $ {\gamma '_1}(t) = (A/t){\gamma _1}(t)$, $ 0 < t < \infty $, where $ A$ is a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curve $ \gamma (t)$ has the form $ {\gamma _1}(t) + {\gamma _2}(t)$, where $ {\gamma _2}(t)$ is a carefully specified "error", such that $ \gamma _2^{(j)}$ is also restricted for $ j = 2, \ldots ,n + 1$. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.

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

  • [CL] E. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • [F] E. Fabes, Singular integrals and partial differential equations of parabolic type, Studia Math. 28 (1966), 81-131. MR 0213744 (35:4601)
  • [NRW1] A. Nagel, N. M. Rivière and S. Wainger, On Hilbert transforms along curves, Bull. Amer. Math. Soc. 8 (1974), 106-108. MR 0450899 (56:9191a)
  • [NRW2] -, On Hilbert transforms along curves. II, Amer. J. Math. 98 (1976), 395-403. MR 0450900 (56:9191b)
  • [NW] A. Nagel and S. Wainger, Hilbert transforms associated with plane curves, Trans. Amer. Math. Soc. 223 (1976), 235-252. MR 0423010 (54:10994)
  • [N] W. Nestlerode, $ {L^2}$ estimates for singular integrals and maximal functions associated with highly monotone curves, Ph.D. Dissertation, University of Wisconsin, Madison, Wisc., 1980.
  • [R] N. M. Rivière, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243-278. MR 0440268 (55:13146)
  • [S1] E. M. Stein, Maximal functions: homogeneous curves, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2176-2177. MR 0420117 (54:8133b)
  • [S2] -, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [SW1] E. M. Stein and S. Wainger, The estimation of an integral arising in multiplier transformations, Studia Math. 35 (1970), 101-104. MR 0265995 (42:904)
  • [SW2] -, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-1295. MR 508453 (80k:42023)
  • [SWe] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)

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Keywords: Approximately homogeneous curve, nonisotropic dilation, Hilbert transform analogue, maximal function analogue, trigonometric estimate, $ g$-function
Article copyright: © Copyright 1981 American Mathematical Society

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