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 $1<p<\infty$, \[ \|T_Ω^\ast\|_{(1,p)\text{-}\mathrm{sparse}} \lesssim_d p'\|Ω\|_{L^\infty(S^{d-1})}. \] 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 $q^{\prime }\le p<\infty$, with the right-hand side replaced by \[ C_{d,q}p' \|Ω\|_{L^{q,1}\log L(S^{d-1})}. \] 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 $p^{\prime }$ as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent $p=q^{\prime }$. 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 $A_1\cap R{H}_{q^{\prime }}$ 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 $p$.

40 pages. Main results improved, the sparse bound is now $p^{\prime }$