Potential operators, maximal functions, and generalizations of \(A_{\infty}\)
In the setting of a space \({\mathcal S}\) of homogeneous type with a doubling measure \(\mu\), the authors prove weighted norm estimates of the type \[ \int_{\mathcal S}|Tf|^p w d\mu\leq C\int_{\mathcal S} M_\varphi (f)^p w d\mu \] and \[ \Big(\int_{\mathcal S} (M_\varphi (f)w)^q d\mu\Big)^{1/q}\leq C \Big (\int_{\mathcal S}|f|^p v d\mu\Big)^{1/p} \] for certain fairy general integral potential operators \(T\) similar to the Riesz fractional integrals \(I_\alpha\) when \({\mathcal S}={\mathbb R}^n\), and for certain related maximal functions \( M_\varphi\) that replace the usual maximal functions \( M_\alpha\) on \({\mathbb R}^n\). The conditions on the weights are weaker than the \(A_\infty\) condition, the Lebesgue measure being replaced by certain content functionals.
scientific article; zbMATH DE number 1925046