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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Singular integrals with exponential weights

Author: E. Prestini
Journal: Proc. Amer. Math. Soc. 124 (1996), 1171-1175
MSC (1991): Primary 42A50; Secondary 43A80
MathSciNet review: 1317046
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Abstract: We study the operators

\begin{equation*}\overline{V} f (t)= \frac{1}{w(t)} V(f(r) w(r)) (t) \end{equation*}

where $V$ is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator.

Under suitable conditions on the weight $w(t)$ of exponential type, we prove boundedness of $\overline{V}$ from $L^{p}$ spaces, defined on $[1, +\infty )$ with respect to the measure $w^{2}(t) dt,$ to $L^{p} + L^{2},\ 1 < p\leq 2,$ with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from $L^{p}$ to $L^{p}, 1 < p < \infty ,$ if and only if $p=2$.

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E. Prestini
Affiliation: Department of Mathematics, University of Rome, Tor Vergata, 00133 Rome, Italy

Received by editor(s): October 7, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society