发表机构
JIS University（JIS大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半直线上加权双线性Hardy-Steklov算子的紧致性，通过输出局部化原理将紧致性与尾部不变量关联，并在五个加权有界性区域中刻画本质距离，揭示临界边界处的结构转变。
AI 中文摘要
我们研究了半直线上加权双线性Hardy-Steklov算子，其中$1<p_1,p_2<\infty$且$0<q<\infty$。算子理论的核心是一个精确的输出局部化原理。对于有界双线性映射到$L^q$，紧致中间截断将紧致性与有向输出尾部不变量的消失等同起来，并以常数1将该不变量与本质距离及基于半径的Hausdorff非紧性测度等同。在局部有限秩逼近性的附加假设下，同一不变量也等于有限秩距离，并控制秩逼近结论。我们对Hardy-Steklov映射验证了这些假设，然后将尾部不变量穿过五个加权有界性区域进行转化。在区域$I,\mathrm{II}_1,\mathrm{II}_2,\mathrm{III}_1$中，有界性必须辅以上端点和下端点的消失条件，且本质距离与较大的端点缺陷相当。在严格次临界区域$\mathrm{III}_2$中，仅有界性本身就蕴含紧致性。这产生了等式边界$P=Q$与可和区域$P>Q$之间的尖锐结构转变。在整个过程中，拟Banach范围$0<q<1$使用幂度量$d_q(f,g)=\\|f-g\\|_q^q$处理。
英文摘要
We study weighted bilinear Hardy-Steklov operators on the half-line for $1<p_1,p_2<\infty$ and $0<q<\infty$. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into $L^q$, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions $I,\mathrm{II}_1,\mathrm{II}_2,\mathrm{III}_1$, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region $\mathrm{III}_2$, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary $P=Q$ and the summable regime $P>Q$. The quasi-Banach range $0<q<1$ is treated throughout with the powered metric $d_q(f,g)=\|f-g\|_q^q$.
Comments31 pages, no figures
