发表机构
Center for Applied Mathematics, Tianjin University（天津大学应用数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对极大粗糙奇异积分T_Ω^*，证明了定量非对称稀疏界，改进了已有稀疏界的平均结构，将极大稀疏理论扩展至无界角核，其证明结合了多种分析方法。
AI 中文摘要
设Ω属于L¹(S^{d-1})且均值为零，T_Ω^*为对应的粗糙齐次奇异积分的极大截断算子。我们证明了T_Ω^*的定量非对称稀疏界：若Ω属于L^∞(S^{d-1})，则对所有1<p<∞，有‖T_Ω^*‖_{(1,p)-稀疏}≲_d (p')⁴‖Ω‖_{L^∞(S^{d-1})}；对于无界角核，若1<q<∞且Ω属于L^{q,1}log L(S^{d-1})，当q'≤p<∞时，同样的估计成立，右端替换为C_{d,q}(p')⁴‖Ω‖_{L^{q,1}log L(S^{d-1})}。这表明稀疏形式的第一个项是真正的L¹平均，改进了此前已知的极大粗糙截断的对称及Orlicz- bump稀疏界的平均结构，并将极大稀疏理论扩展至无界角核。证明结合了局部稀疏控制原理、极大截断的物理空间线性化、Rademacher-Menshov不等式及粗糙核的微局部分解。
英文摘要
Let $Ω\in L^1(S^{d-1})$ have vanishing average, and let $T_Ω^\ast$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for $T_Ω^\ast$. If $Ω\in L^\infty(S^{d-1})$, 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 $Ω\in L^{q,1}\log L(S^{d-1})$, then the same estimate holds for $q'\leq p<\infty$, with the right-hand side replaced by \[ C_{d,q}p' \|Ω\|_{L^{q,1}\log L(S^{d-1})}. \] 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 $p'$ 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 $p$.
Comments40 pages. Main results improved, the sparse bound is now $p'$
