Weighted inequalities for Hardy-type operators involving suprema

Let u, b be two weight functions on (0, infinity). Assume that u is continuous t on (0, infinity) and that b is such that the function B(t) := integral(t)(0) b(s) ds satisfies 0 < B(t) < infinity for every t is an element of (0, infinity). Let the operator T-u,T-b be given at a measurable non-negative function g on (0, infinity) by (T(u,b)g)(t) = sup(t <=tau<infinity) u(tau)/B(tau) integral(tau)(0) g(s)b(s) ds. We give necessary and sufficient conditions on weights v, w on (0, infinity) for which there exists a positive constant C such that the inequality (integral(infinity)(0)[(T(u,b)g)(t)](q) w(t)dt)(1/q) <= C (integral(infinity)(0) [g(t)](p)(v)(t)dt)(1/p) holds for every measurable non-negative function g on (0, infinity), where p, q is an element of (0, infinity) satisfy certain restrictions. We also characterize weights v, w on (0, infinity) for which there exists a positive constant C such that the inequality (integral(infinity)(0) [(T-u,T-b phi)(t)](q) w(t) dt)(1/q) <= C(integral(infinity)(0) [phi(t)](p) v(t) dt)(1/p) holds for every non-negative and non-increasing function phi on (0, infinity). The crucial tool in our approach to the latter problem is the reduction of the given inequality to the pair of analogous inequalities involving more manageable operators, namely the classical Hardy-type integral operator and the operator (R-u phi)(t) = sup(t <=tau<infinity) u(tau)phi(tau). Such estimates have recently been found indispensable in the study of problems involving fractional maximal operators and optimal Sobolev embeddings.