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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.

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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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