On weighted norm inequalities for positive linear operators

Abstract

Let T T be a positive linear operator defined for nonnegative functions on a σ \sigma -finite measure space ( X , m , μ ) \left ( {X,m,\mu } \right ) . Given 1 > p > ∞ 1 > p > \infty and a nonnegative weight function w w on X X , it is shown that there exists a nonnegative weight function v v , finite μ \mu -almost everywhere on X X , such that (1) \[ ∫ X ( T f ) p w d μ ≤ ∫ X f p v d μ , for all f ≤ 0 \int _X {{{\left ( {Tf} \right )}^p}wd\mu \leq \int _X {{f^p}vd\mu } } ,\quad {\text {for all }}f\leq 0 \] , if and only if there exists ϕ \phi positive μ \mu -almost everywhere on X X with (2) \[ ∫ X ( T ϕ ) p w d μ > ∞ . \int \limits _X {{{\left ( {T\phi } \right )}^p}wd\mu > \infty .} \] In case (2) holds, we may take v = ϕ 1 − p T ∗ [ ( T ϕ ) p − 1 w ] v = {\phi ^{1 - p}}{T^*}\left [ {{{\left ( {T\phi } \right )}^{p - 1}}w} \right ] in (1). This partially answers a question of B. Muckenhoupt in [5]. Applications to some specific operators are also given.