On logarithmic bounds of maximal sparse operators

Abstract

Given sparse collections of measurable sets ${\mathcal{S}}_k$, $k=1,2,\dots ,N$, in a general measure space $(X,\mathfrak{M},\mu )$, let ${\Lambda }_{{\mathcal{S}}_k}$ be the sparse operator, corresponding to ${\mathcal{S}}_k$. We show that the maximal sparse function $\Lambda f={\max }_{1\le k\le N}{\Lambda }_{{\mathcal{S}}_k}f$ satisfies \begin{align*} &\| Λ\| _{L^p(X) \mapsto L^{p,\infty}(X)} \lesssim \log N\cdot \|M_{\mathcal S}\|_{L^p(X) \mapsto L^{p,\infty}(X)},\,1\le p<\infty, \\ &\lVert Λ\rVert _{L^p(X) \mapsto L^p(X)} \lesssim (\log N)^{\max\{1,1/(p-1)\}}\cdot \|M_{\mathcal S}\|_{L^p(X) \mapsto L^p(X)},\, 1<p<\infty, \end{align*} where $M_{\mathcal{S}}$ is the maximal function corresponding to the collection of sets $\mathcal{S}={\cup }_k{\mathcal{S}}_k$. As a consequence, one can derive norm bounds for maximal functions formed from taking measurable selections of one-dimensional Calderón-Zygmund operators in the plane. Prior results of this type had a fixed choice of Calderón-Zygmund operator for each direction.

12 pages