Sharp two-weight inequality for fractional maximal operators

The paper is devoted to two-weight estimates for the fractional maximal operators ${\mathcal{M}}^{\alpha }$ on general probability spaces equipped with a tree-like structure. For given $1<p\le q<\mathrm{\infty }$, we study the sharp universal upper bound for the norm $\mathrm{\parallel }{\mathcal{M}}^{\alpha }{\mathrm{\parallel }}_{L^p(v)\to L^q(u)}$, where $(u,v)$ is an arbitrary pair of weights satisfying the Sawyer testing condition. The proof is based on the abstract Bellman function method, which reveals an unexpected connection of the above problem with the sharp version of the classical Sobolev imbedding theorem.