The Hilbert transform with exponential weights

Abstract:

We study the operator \[ \mathcal {H}f(x) = {2^{ - x}}\int _0^{ + \infty } {\frac {{{2^y}f(y)}}{{x - y}}dy} \] 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-${L^1}({\mathbb {R}_ + },{4^x}dx) + {L^2}({\mathbb {R}_ + },{4^x}dx)$.