Let $\mu$ be a probability vector of $\mathbb{R}^n.$ Then we can define the weighted $\ell^p$ $(1 \leq p < \infty)$ norms by $$ \|x\|_{p,\mu} = \left(\sum_{i=1}^n |x_i|^p \mu_i\right)^{1/p},$$ assuming that all the components of $\mu$ are positive. The $\|x\|_{\infty,\mu}$ norm agrees with the usual max norm.
Now let $A$ denote a linear operator $\mathbb{R}^n \rightarrow \mathbb{R}^n.$ Assume that $$ \|Ax\|_{p_0, \mu} \leq M \|x\|_{p_0, \mu}$$ and $$ \|Ax\|_{p_1, \mu} \leq M \|x\|_{p_1, \mu},$$ $1 \leq p_0 < p_1 \leq \infty,$ for all $x \in \mathbb{R}^n.$ Then the Riesz interpolation result does hold? I mean that $$ \|Ax\|_{p_\theta, \mu}\leq M\|x\|_{p_\theta, \mu} $$ follows for every $p_\theta = {\theta \over p_0} + {1-\theta \over p_1}$ $(0 \leq \theta \leq 1)$ and $x \in \mathbb{R}^n$?
As far as I know, Riesz first proved his interpolation result in the real finite dimensional setting with the usual $\ell_p$ norms. That result remains true with the weighted $\ell_p$ norms (in the real setting)?