arXiv subjects
Publications and source records attributed to Jingguo Lai.
We give a necessary and sufficient condition for the two weight $L^p$-estimates for paraproducts in non-homogeneous settings, $1 2$ both "direct" and "adjoint" testing conditions are needed. An interesting feature is that the "adjoint" testing condition is that it is a testing condition not for the adjoint of the paraproduct, but for some auxiliary operator.
We give a new characterization of the two weight inequality for a vector-valued positive operator. Our characterization has a different flavor than the one of Scurry's and Hänninen's. The proof can be essentially derived from the scalar-valued case.
We investigate the relation between Carleson sequence and balayage, and use this to give an easy proof of the equivalence of the L1-norms of the maximal function and the square function in non-honogeneous martingale settings.
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work for the tree-like structure. In this paper, we give a self-complete proof of the coincidence of the Bellman functions on the more general infinitely refining filtered probability space, see Definition 1.3. The proof depends on a remodeling of the Bellman function of the dyadic Carleson Embedding Theorem.