While trying to prove some extrapolation theorem for $B_{p}$ weights, I tried to prove the following:
Suppose that $0<p\leq 1 $ and $w(x)$ is a non-negative and belongs to $B_{p}$, that is \begin{equation} \int_{r}^{\infty }\frac{w(x)}{x^{p}}dx\leq \frac{C_{1}}{r^{p}}% \int_{0}^{r}w(x)dx. \end{equation} then, the following inequality \begin{equation} \int_{0}^{\infty }w(x)\left( \frac{1}{x}\int_{0}^{x}f(t)dt\right) ^{p}dx\leq C_{2}\int_{0}^{\infty }w(x)f^{p}(x)dx \end{equation} holds for all non-negative, non-increasing functions $f(x)$.
Actually, I couldn't complete it since the exponent now $(p-1)$ is negative.
How can I prove the above result?
I had just used the monotonicity of the function $f(t)$ to get it out from the inner integration and then the result follows directly. So, thanks for all your support and close the question, please.