Sparse bounds for maximal rough singular integrals

Abstract

Let $\Omega\in L^1(S^{d-1})$ have vanishing average, and let $T_\Omega^\ast$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for $T_\Omega^\ast$. If $\Omega\in L^\infty(S^{d-1})$, then, for every $1<p<\infty$, \[ \|T_\Omega^\ast\|_{(1,p)\text{-}\mathrm{sparse}} \lesssim_d p'\|\Omega\|_{L^\infty(S^{d-1})}. \] For unbounded angular kernels, if $1<q<\infty$ and $\Omega\in L^{q,1}\log L(S^{d-1})$, then the same estimate holds for $q'\leq p<\infty$, with the right-hand side replaced by \[ C_{d,q}p' \|\Omega\|_{L^{q,1}\log L(S^{d-1})}. \] These estimates retain a genuine $L^1$ average in the first entry of the sparse form. In the bounded-kernel case, the upper bound has the same linear growth in $p'$ as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent $p=q'$. As a consequence, we obtain weighted weak-type $(1,1)$ estimates for all $A_1$ weights in the bounded-kernel case and for weights in $A_1\cap RH_{q'}$ in the unbounded-kernel case. The proof combines physical-space linearization and microlocal decomposition with localized sparse testing. An amplitude decomposition of the second input, together with the Rademacher--Menshov inequality, yields the quantitative dependence on $p$.