Weighted inequalities for the fractional maximal operator in Lorentz spaces via atomic decomposition of tent spaces (Q1363044)

The author considers the fractional maximal operator \[ M_{\alpha}f(x)= \sup_{x\in Q} | Q| ^{\alpha /n-1}\int_{Q}| f(y)| dy \quad\text{in }{\mathbb{R}}^{n}, \qquad 0\leq\alpha<n. \] The main result in this paper is a necessary condition which is also close to be sufficient on the problem of determining the weight functions \(u\) and \(\sigma\) for which \(M_{\alpha} fd\sigma\) maps the generalized Lorentz space \(\Lambda_\sigma^s (w_1)\) into \(\Lambda_u ^r (w_2)\), \(1<s\leq r<\infty\). \(\Lambda_{d\nu}^r (w)\) is defined as the measurable function that satisfies \( \int_0 ^\infty (f\nu ^*)^r w(t)dt<\infty,\) where \(f_\nu ^*\) is the decreasing rearrangement of the function \(f\) with respect to the measure \(\nu\).