We study the operator H f(x) = 2-x integral 0/+ infinity 2 yf(y)/x-y dy on Lorentz spaces on R+ with respect to the measure 4x 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 L2,9 (R+, 4x dx), 1 < mu < + infinity, and it maps the Lorentz space L2,1 (R+, 4x dx) into a space that we call WEAK-L2,1 (R+, 4x dx). We also prove that H maps L1(R+, 4x dx) into WEAK-L1(R+, 4x dx) + L2(R+, 4x dx)

Colzani, L., Vignati, M. (1992). The Hilbert transform with exponential weights. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 114(2), 451-457 [10.1090/S0002-9939-1992-1075944-6].

The Hilbert transform with exponential weights

COLZANI, LEONARDO;
1992

Abstract

We study the operator H f(x) = 2-x integral 0/+ infinity 2 yf(y)/x-y dy on Lorentz spaces on R+ with respect to the measure 4x 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 L2,9 (R+, 4x dx), 1 < mu < + infinity, and it maps the Lorentz space L2,1 (R+, 4x dx) into a space that we call WEAK-L2,1 (R+, 4x dx). We also prove that H maps L1(R+, 4x dx) into WEAK-L1(R+, 4x dx) + L2(R+, 4x dx)
Articolo in rivista - Articolo scientifico
Hubert transform, Hyperbolic spaces, Lorentz spaces
English
1992
114
2
451
457
none
Colzani, L., Vignati, M. (1992). The Hilbert transform with exponential weights. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 114(2), 451-457 [10.1090/S0002-9939-1992-1075944-6].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/18956
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