Two-weighted estimates for positive operators and Doob maximal operators on filtered measure spaces

Abstract

We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in R n due to Lacey, Sawyer and Uriarte-Tuero [30]. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen-Pérez type and Lerner-Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.

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

“…The construction of principal sets first appeared in Tanaka and Terasawa [2], and Chen, Zhu, Zuo and Jiao [5,14] found the conditional sparsity of the construction, which is new and useful. We will use the construction of principal sets.…”

“…Furthermore, we have χ P 0 ≤ ηE i (χ E(P 0 ) )χ P 0 , which is called the conditional sparsity of principal sets with η (see [5,14]).…”

“…Remark 1. The content of Theorem 1 is (5). In order to prove (5), we use different approaches as follows:…”

“…Using the construction of principal sets [2] and the conditional sparsity [5], we have C = a 2 η (p −1) p , where a, η are the constants in the construction of principal sets (Appendix A).…”

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

“…At the end of Section 4, we check our work with graphing device in Figure 1. The construction of principal sets first appeared in Tanaka and Terasawa [13], and Chen, Zhu, Zuo and Jiao [1,2] find a new property of the construction which was called conditional sparsity. Our results are mainly based on the construction of principal sets and the conditional sparsity.…”