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

Abstract

Let M be the Doob maximal operator on a filtered measure space and let v be an Ap weight with 1p+∞. We try proving that ∥Mf∥Lp(v)≤p′[v]Ap1p−1∥f∥Lp(v), where 1/p+1/p′=1. Although we do not find an approach which gives the constant p′, we obtain that ∥Mf∥Lp(v)≤p1p−1p′[v]Ap1p−1∥f∥Lp(v), with limp→+∞p1p−1=1.

Keywords

• filtered measure space
• Doob maximal operator
• weighted inequality
• principal set