Sparse bounds for maximal rough singular integrals

Let $\mathrm{\Omega }\in L^1(S^{d-1})$ have vanishing average, and let $T_{\mathrm{\Omega }}^{\ast }$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative asymmetric sparse bounds for $T_{\mathrm{\Omega }}^{\ast }$. If $\mathrm{\Omega }\in L^{\mathrm{\infty }}(S^{d-1})$, then, for every $1<p<\mathrm{\infty }$, $$\mathrm{\parallel }{T}_{\mathrm{\Omega }}^{\ast }{\mathrm{\parallel }}_{(1,p)\text{-}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}}{\lesssim }_d(p^{\mathrm{\prime }}{)}^4\mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(S^{d-1})}\mathrm{.}$$ For unbounded angular kernels, if $1<q<\mathrm{\infty }$ and $\mathrm{\Omega }\in L^{q,1}\log L(S^{d-1})$, then the same estimate holds for $q^{\mathrm{\prime }}\le p<\mathrm{\infty }$, with the right-hand side replaced by $$C_{d,q}(p^{\mathrm{\prime }}{)}^4\mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}_{L^{q,1}\log L(S^{d-1})}\mathrm{.}$$ Thus the first entry of the sparse form is a genuine $L^1$ average. This improves the averaging structure of the previously known symmetric and Orlicz-bumped sparse bounds for maximal rough truncations and extends the maximal sparse theory to unbounded angular kernels. The proof combines a localized sparse domination principle with a physical-space linearization of the maximal truncation, the Rademacher-Menshov inequality, and a microlocal decomposition of the rough kernel.