On weighted norm inequalities for positive linear operators

Abstract:

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