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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Hilbert transforms associated with plane curves


Authors: Alexander Nagel and Stephen Wainger
Journal: Trans. Amer. Math. Soc. 223 (1976), 235-252
MSC: Primary 44A25; Secondary 42A40, 47G05
MathSciNet review: 0423010
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Abstract: Let $ (t,\gamma (t))$ be a plane curve. Set $ {H_\gamma }f(x,y) =$   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\Vert {{H_\gamma }f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p},5/3 < p < 5/2$. Various examples are given to show that some condition on the curve $ (t,\gamma (t))$ is necessary.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0423010-8
PII: S 0002-9947(1976)0423010-8
Article copyright: © Copyright 1976 American Mathematical Society