Maximal Calderon–Zygmund operators and Weyl multipliers

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Abstract

Let $T_k$, $k=1,2,…,N$, be a sequence of bounded operators on $L^p$, $1, and $T^*(f)=\max_{1\le k\le N}|T_k(f)|$. For some choices of $T_k$ it is of interest the problem of finding the optimal constant $c(N)$ for the bound

$$
\|T^*\|_{L^p\to L^p}\lesssim c(N) \max_{1\le k\le N}\|T_k\|_{L^p\to L^p}.
$$
We consider this problem for Calderon–Zygmund operators. It was prove by first two authors that $c(N)\lesssim \log N$ when $T_k$ are general Calderon–Zygmund operators with uniformly bounded parameters. In this note we consider Calderon–Zygmund operators with kernels, having certain dyadic decomposition. We prove for such operators $c(N)\lesssim\sqrt{\log N}$. Applying this bound, we prove that the sequence $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged dyadic block trigonometric polynomials.

About the authors

Grigori Artashesovich Karagulyan

Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia; Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Republic of Armenia

Email: g.karagulyan@ysu.am
Doctor of physico-mathematical sciences, Professor

Michael Thoreau Lacey

School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

Email: lacey@math.gatech.edu
Doctor of physico-mathematical sciences, Professor

Khazhakanoush V.. Navoyan

Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia

Email: khvnavoyan@gmail.com

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