On Weyl multipliers of non-overlapping Franklin polynomial systems

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping Franklin polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.Comment: 15 page

On good-λ\lambda inequalities for couples of measurable functions

We give a domination condition implying good-$\lambda$ and exponential inequalities for couples of measurable functions. Those inequalities recover several classical and new estimations involving some operators in Harminic Analysis. Among other corollaries we prove a new exponential estimate for Carleson operators. The main results of the paper are considered in a general setting, namely, on abstract measure spaces equipped with a ball-basis.Comment: 18 page

Sharp inequalities involving multiplicative chaos sums

The present note is an essential addition to the author's arxiv paper arXiv:2001.01070, concerning general multiplicative systems of random variables. Using some lemmas and the methodology of \cite{Kar4}, we obtain a general extreme inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system.Comment: The present note is an essential addition to the author's arxiv paper arXiv:2001.0107

On systems of non-overlapping Haar polynomials

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.Comment: 9 page

Asymptotic estimates for double-coverings

A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ is said to be a double-covering if each $a\in \cup_{k=1}^{p}A_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let $\mu_{l,p}$ be the number of equivalency classes of double-coverings with $\#(A_k)=l$, $k=1,2,\ldots,p$. We characterize the asymptotic behavior of the quantity $\mu_{l,p}$ as $p\to \infty$. The results are applied to give an alternative approach to the Bonami-Kiener hypercontraction inequality.Comment: 19 page