On logarithmic bounds of maximal sparse operators

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.