A weighted weak type estimate for the fractional integral operator on spaces of homogeneous type. (English) Zbl 1235.42011

Zhang, Qihui

Hiroshima Math. J. 41, No. 3, 389-407 (2011).

The author considers the fractional integral \(I_\alpha f\) with the kernel \[ Q_\alpha(x,y)=\begin{cases} \eta(x,y)^{\alpha-1}, & \text{ if } x\not=y,\cr \mu(\{x\})^{\alpha-1} & \text{ if } x=y \text{ and } \mu(\{x\})>0 \end{cases} \] in spaces of homogeneous type \(({\mathcal K},d,\mu)\) in the sence of Coifman and Weiss. It is given a sufficient condition on the pair of weights \((u,v)\) so that \(I_\alpha\) is bounded from \(L^p({\mathcal K},v)\) to weak \(L^q({\mathcal K},u)\) with \(1<p\leq q<\infty\).

Reviewer: Lubomira Softova (Aversa)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
43A85	Harmonic analysis on homogeneous spaces

Keywords:

spaces of homogeneous type; weighted norm inequality; fractional integral operators

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