Potential operators on cones of nonincreasing functions
Summary: Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators \((I_\alpha f)(x) = \int^\infty_0(f(t)/|x - t|^{1-\alpha})\,dt\) and \((\mathcal I_{\alpha_1,\alpha_2} f)(x, y) = \int^\infty_0 \int^\infty_0 (f(t, \tau)/|x - t|^{1-\alpha_1}|y - \tau|^{1-\alpha_2})\,dt\,d\tau\) on the cone of nonincreasing functions are derived. In the case of \(\mathcal I_{\alpha_1,\alpha_2}\), we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between \(1\) and \(\infty\).
scientific article; zbMATH DE number 6032363
scientific article