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)
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