Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Singular integrals with exponential weights

Author(s): E. Prestini
Journal: Proc. Amer. Math. Soc. 124 (1996), 1171-1175.
MSC (1991): Primary 42A50; Secondary 43A80
MathSciNet review: 1317046
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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:

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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|