Abstract:
Let $T_k$, $k=1,2,\dots,N$, be a sequence of bounded operators on $L^p$, $1<p<\infty$, and let $T^*(f)=\max_{1\le k\le N}|T_k(f)|$. For some choices of $T_k$ the problem of finding the optimal constant $c(N)$ for the bound
$$
\|T^*\|_{L^p\to L^p}\lesssim c(N)
\max_{1\le k\le N}\|T_k\|_{L^p\to L^p}
$$
is of interest. We consider this problem for Calderón–Zygmund operators. It was proved by the two first-named authors that $c(N)\lesssim \log N$ when the $T_k$ are general Calderón–Zygmund operators with uniformly bounded parameters. In this note we consider
Calderón–Zygmund operators with kernels having a certain dyadic decomposition. We prove that $c(N)\lesssim\sqrt{\log N}$ for such operators. Applying this bound, we prove that the sequence $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged dyadic block trigonometric polynomials.
Bibliography: 46 titles.
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