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

Wei Chen, Jingya Cui

Published 2021-03-04Version 1

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 $\lVert M f\rVert _{L ^{p}(v) }\leq p'[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)},$ where $1/p+1/p'=1.$ But we do not find an approach which gives the constant $p'.$ Our results are as follows: $\lVert M f\rVert _{L ^{p}(v) }\leq \min\{p^{\frac{1}{p-1}},~a^{\frac{2}{p}}\eta^{(p'-1)}\}p'[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)}, $ where $a>1$ and $\eta=\frac{a}{a-1}$ are the constants in the construction of the principal sets. Furthermore, we show that $$\lim\limits_{p\rightarrow+\infty}p^{\frac{1}{p-1}}=\lim\limits_{p\rightarrow+\infty}(\min\limits_{a>1}a^{\frac{2}{p}}\eta^{(p'-1)})=1.$$