Singular integrals with exponential weights

Prestini, E.

doi:10.1090/S0002-9939-96-03272-8

Singular integrals with exponential weights
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Proc. Amer. Math. Soc. 124 (1996), 1171-1175 Request permission

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 $p=2$.

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