On the geometric mean operator
The authors give a characterization of pairs of weights \((u,v)\) such that the geometric mean operator \(Gf(x)= \exp((1/x) \int_ 0^ x \log f(t) dt)\), defined for \(f>0\) almost everywhere on \((0,\infty)\), is bounded from \(L_{p,v} (0,\infty)\) to \(L_{q,u} (0,\infty)\), where \(0<q< p\leq\infty\). Their proofs are based on the simple observation that in the case \(v\equiv 1\) and \(p>1\) the good weights for \(G\) coincide with those good for the averaging operator \(Af(x)= (1/x) \int_ 0^ x f(t)dt\). Their result applies to a certain independence on \(p\), \(q\) of weighted \(L_ p- L_ q\) inequalities involving the operator \(A\).
scientific article; zbMATH DE number 599926