A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Let $M$ be the Doob maximal operator on a filtered measure space and let $v$ be an $A_p$ weight with $1<p<+\infty$. We try proving that \begin{equation}\lVert M f\rVert _{L ^{p}(v) }\leq p^{\prime}[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)},\end{equation} where $1/p+1/p^{\prime}=1.$ Although we do not find an approach which gives the constant $p^{\prime},$ we obtain that \begin{equation}\lVert M f\rVert _{L ^{p}(v) }\leq p^{\frac{1}{p-1}}p^{\prime}[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)}, \end{equation} with $\lim\limits_{p\rightarrow+\infty}p^{\frac{1}{p-1}}=1.$