Hilbert transforms associated with plane curves
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- by Alexander Nagel and Stephen Wainger
- Trans. Amer. Math. Soc. 223 (1976), 235-252
- DOI: https://doi.org/10.1090/S0002-9947-1976-0423010-8
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Abstract:
Let $(t,\gamma (t))$ be a plane curve. Set ${H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t$ for $f \in C_0^\infty ({R^2})$. For a large class of curves, the authors prove ${\left \| {{H_\gamma }f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p},5/3 < p < 5/2$. Various examples are given to show that some condition on the curve $(t,\gamma (t))$ is necessary.References
- Alexander Nagel and Stephen Wainger, $L^{2}$ boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), no. 4, 761–785. MR 450901, DOI 10.2307/2373864
- Elias M. Stein, 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 Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 223 (1976), 235-252
- MSC: Primary 44A25; Secondary 42A40, 47G05
- DOI: https://doi.org/10.1090/S0002-9947-1976-0423010-8
- MathSciNet review: 0423010