On weighted norm inequalities for positive linear operators

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