On logarithmic bounds of maximal sparse operators

Abstract: Given sparse collections of measurable sets $\mathcal S_k$, $k=1,2,\ldots ,N$, 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*} &| \Lambda | {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 \Lambda \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 Calder\'on-Zygmund operators in the plane. Prior results of this type had a fixed choice of Calder\'on-Zygmund operator for each direction.