Weighted norm inequalities for positive operators restricted on the cone ofλ-quasiconcave functions

Abstract

AbstractLetρbe a monotone quasinorm defined on${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). LetTbe a monotone quasilinear operator on${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone ofλ-quasiconcave functions$$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$where$1\les p\les \infty $andvis a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 <p< 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u,v,w) for which the three weighted Hardy-type inequality$$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$holds for allλ-quasiconcave functions and all 0 <p,q⩽ ∞.