Iterated Hardy-type inequalities involving suprema

Abstract

In this paper, the boundedness of the composition of the supremal operators defined, for a non-negative measurable functions f on (0,infinity), by S(u)g(t) := ess sup(0<tau <= t) u(tau)g(tau), t is an element of(0,infinity), and S*(u)g(t) := ess sup(t <=tau<infinity) u(tau)g(tau), t is an element of(0,infinity), where u is a fixed continous weight on (0,infinity), with the Hardy and Copson operators between weighted Lebesgue spaces L-p(v) and L-q(w) are characterized. The complete solution of the restricted inequalities, that is, inequalities parallel to S-u(f)parallel to(q,w(0,infinity)) <= c parallel to f parallel to(p,v,(0,infinity)), and parallel to S-u(f)parallel to(q,w(0,infinity)) <= c parallel to f parallel to(p,v,(0,infinity)), being satisfied on the cones of monotone functions f on (0,infinity), are given. Moreover, the complete characterization of the inequality parallel to T(u,b)f parallel to(q,w,(0,infinity)) <= c parallel to f parallel to(p,v,(0,infinity)), being satisfied for every non-negative and non-increasing functions f on (0,infinity), is given for 0 < p, q < infinity, as well. Here the operator T-u,T-b is defined for a measurable non-negative function f 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, t is an element of (0,infinity), where u, b are two weight functions on (0,infinity) such that u is continuous on (0,infinity) and the function B(t) := integral(t)(0)b(s) ds satisfies 0 < B(t) < infinity for every t is an element of (0,infinity).