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 | 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.

**[**Earl A. Coddington and Norman Levinson,**CL**]*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[**Eugene B. Fabes,**F**]*Singular integrals and partial differential equations of parabolic type*, Studia Math.**28**(1966/1967), 81–131. MR**0213744****[**Alexander Nagel, Néstor Rivière, and Stephen Wainger,**NRW1**]*On Hilbert transforms along curves*, Bull. Amer. Math. Soc.**80**(1974), 106–108. MR**0450899**, 10.1090/S0002-9904-1974-13374-4**[**Alexander Nagel, Néstor M. Rivière, and Stephen Wainger,**NRW2**]*On Hilbert transforms along curves. II*, Amer. J. Math.**98**(1976), no. 2, 395–403. MR**0450900****[**Alexander Nagel and Stephen Wainger,**NW**]*Hilbert transforms associated with plane curves*, Trans. Amer. Math. Soc.**223**(1976), 235–252. MR**0423010**, 10.1090/S0002-9947-1976-0423010-8**[**W. Nestlerode,**N**]*estimates for singular integrals and maximal functions associated with highly monotone curves*, Ph.D. Dissertation, University of Wisconsin, Madison, Wisc., 1980.**[**N. M. Rivière,**R**]*Singular integrals and multiplier operators*, Ark. Mat.**9**(1971), 243–278. MR**0440268****[**Elias M. Stein,**S1**]*Maximal functions. II. Homogeneous curves*, Proc. Nat. Acad. Sci. U.S.A.**73**(1976), no. 7, 2176–2177. MR**0420117****[**Elias M. Stein,**S2**]*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[**Elias M. Stein and Stephen Wainger,**SW1**]*The estimation of an integral arising in multiplier transformations.*, Studia Math.**35**(1970), 101–104. MR**0265995****[**Elias M. Stein and Stephen Wainger,**SW2**]*Problems in harmonic analysis related to curvature*, Bull. Amer. Math. Soc.**84**(1978), no. 6, 1239–1295. MR**508453**, 10.1090/S0002-9904-1978-14554-6**[**Elias M. Stein and Guido Weiss,**SWe**]*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972**

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Additional Information

DOI:
http://dx.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