Sharp $$L_p$$ estimates for paraproducts on general measure spaces

The paper contains the identification of the $$L_p$$ L p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.

Weighted exponential inequality for differentially subordinate martingales

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition $$A_{\infty }$$ A ∞ . The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted $$L^p$$ L p estimates for differential subordinates of bounded martingales.

Application of Criteria Aggregation Techniques for the Selection of Innovative Products

The peculiarity of innovative products requires taking into account a large number of criteria that should be aggregated into generalized ones and building a convolution tree with qualitative, quantitative and fuzzy rules. The paper proposes a methodology for the optimal partitioning of criteria scales into generalized gradations for using combined methods of multicriteria analysis of alternatives. The transition to fewer criteria leads to a signifi cant increase in the dimension of the scales of generalized criteria. Scales have to be converted to new scales with fewer gradations. To solve the problem of minimizing the information loss occurring when converting the scales picked Bellman function and applied method of dynamic programming. Computational experiments have shown the effectiveness of the proposed approach.

Bellman Function Sitting on a Tree

In this note we give a proof-by-formula of certain important embedding inequalities on a dyadic tree. We also consider the case of a bi-tree, where a different approach is explained.