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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Singular integrals and maximal functions associated with highly monotone curves

Author: W. C. Nestlerode
Journal: Trans. Amer. Math. Soc. 267 (1981), 435-444
MSC: Primary 42B25; Secondary 42B20
MathSciNet review: 626482
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Abstract: Let $ \gamma :[ - 1,1] \to {{\mathbf{R}}^n}$ be an odd curve. Set

$\displaystyle {H_\gamma }f(x) = {\text{PV}}\int {f(x - \gamma (t))\,(dt/t)} $


$\displaystyle {M_\gamma }f(x) = \sup {h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert\,dt} $

. We introduce a class of highly monotone curves in $ {{\mathbf{R}}^n}$, $ n \geqslant 2$, for which we prove that $ {H_\gamma }$ and $ {M_\gamma }$ are bounded operators on $ {L^2}({{\mathbf{R}}^n})$. These results are known if $ \gamma $ has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin.

Related to this problem, we prove a generalization of van der Corput's estimate of trigonometric integrals.

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Article copyright: © Copyright 1981 American Mathematical Society

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