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

$$\begin{aligned}&\Vert \Lambda \Vert _{L^p(X) \mapsto L^{p,\infty }(X)} \lesssim \log N\cdot \Vert M_{\mathcal {S}}\Vert _{L^p(X) \mapsto L^{p,\infty }(X)},\,1\le p<\infty , \\&||\Lambda ||_{L^p(X) \mapsto L^p(X)} \lesssim (\log N)^{\max \{1,1/(p-1)\}}\cdot \Vert M_{\mathcal {S}}\Vert _{L^p(X) \mapsto L^p(X)},\, 1<p<\infty , \end{aligned}$$

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.