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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Hilbert transform with exponential weights


Authors: Leonardo Colzani and Marco Vignati
Journal: Proc. Amer. Math. Soc. 114 (1992), 451-457
MSC: Primary 44A15; Secondary 42A50, 43A50
MathSciNet review: 1075944
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Abstract: We study the operator

$\displaystyle \mathcal{H}f(x) = {2^{ - x}}\int_0^{ + \infty } {\frac{{{2^y}f(y)}}{{x - y}}dy} $

on Lorentz spaces on $ {\mathbb{R}_ + }$ with respect to the measure $ {4^x}dx$. This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this operator is bounded on the Lorentz spaces $ {L^{2,9}}({\mathbb{R}_ + },{4^x}dx),1 < q < + \infty $, and it maps the Lorentz space $ {L^{2,1}}({\mathbb{R}_ + },{4^x}dx)$ into a space that we call WEAK- $ {L^{2,1}}({\mathbb{R}_ + },{4^x}dx)$. We also prove that $ \mathcal{H}$ maps $ {L^1}({\mathbb{R}_ + },{4^x}dx)$ into WEAK- $ {L^1}({\mathbb{R}_ + },{4^x}dx) + {L^2}({\mathbb{R}_ + },{4^x}dx)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1075944-6
PII: S 0002-9939(1992)1075944-6
Keywords: Hilbert transform, Lorentz spaces, hyperbolic spaces
Article copyright: © Copyright 1992 American Mathematical Society