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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 $p=2$.

References [Enhancements On Off] (What's this?)

  • 1. Colzani L., Crespi A., Travaglini G., Vignati M., Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in euclidean and non euclidean spaces, Trans. Amer. Math. Soc. 338 (1993), 43--55. MR 93j:42009
  • 2. Colzani L., Vignati M., Hilbert transform with exponential weights, Proc. Amer. Math. Soc. 114, n.2 (1992), 451--457. MR 92e:44004
  • 3. Carleson L., On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135--157. MR 33:7774
  • 4. Hunt R., On the convergence of Fourier series, Orthogonal Expansions and Continuous Analogues, Proc. Conference Edwardsville (Illinois 1967, SIU Press 1968), 235--255. MR 38:6296
  • 5. Hunt R., Young W.S., A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274--277. MR 49:3419
  • 6. Meaney C., Prestini E., Almost everywhere convergence of inverse spherical transform on $SL(2, \mathbb{R})$, Ark. Mat. 32 (1994), 195--211. MR 95c:22015
  • 7. Meaney C., Prestini E., Almost everywhere convergence of inverse spherical transforms on noncompact symmetric spaces, J. Funct. Anal. (to appear).
  • 8. S[??]jolin P., Convergence almost everywhere of certain singular integrals and multiple Fourier series, Ark. Math. 9 (1971), 65--90. MR 49:998
  • 9. Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (Princeton N.J. 1970). MR 44:7280
  • 10. Bergh J., Löfström J., Interpolation Spaces, Springer-Verlag (Berlin 1976). MR 58:2349

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Additional Information

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

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