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

DOI:
https://doi.org/10.1090/S0002-9947-1981-0621989-4

MathSciNet review:
621989

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and . It is proved that for , the Schwartz class, and for an approximately homogeneous curve , , .

A homogeneous curve is one which satisfies a differential equation , , where is a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curve has the form , where is a carefully specified "error", such that is also restricted for . The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0621989-4

Keywords:
Approximately homogeneous curve,
nonisotropic dilation,
Hilbert transform analogue,
maximal function analogue,
trigonometric estimate,
-function

Article copyright:
© Copyright 1981
American Mathematical Society