A note on weighted norm inequalities for fractional maximal operators with non-doubling measures
The aim of the present paper is to study two-weight norm inequalities for fractional maximal functions associated to a non-negative ``\(n-\)dimensional'' Borel measure \(\mu\) on \(\mathbb{R}^d\), which only satisfies the growth condition \(\mu(Q)\leq l(Q)^n\) for any cube \(Q\subset \mathbb{R}^d\), where \(l(Q)\) stands for the side length of \(Q\). The non-centered maximal function \(\displaystyle M_{\alpha}\) is defined as \[ \displaystyle M_{\alpha}f(x)=\sup_{x\in Q}\frac{1}{\mu(5Q)^{1-\alpha}}\int_{Q}|f(y)|\,\text{d}\mu(y) \] for \(\alpha\in[0,1)\) and the supremum taken over all cubes \(Q\) that contain \(x\). The authors investigate the pairs of weights (i.e non-negative and locally integrable functions) for which \(M_{\alpha}\) satisfies a weak or a strong type inequality. More precisely, there are two main results. The first one explores the pairs of weights for which the maximal operator \(M_{\alpha}\) satisfies a weak type inequality. The second one characterizes those pairs of weights for which \(M_{\alpha}\) satisfies a strong type inequality, which can be achieved by certain Orlicz norms localized in cubes.
scientific article; zbMATH DE number 6137975