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
本文研究一般区间上加权双线性 Hardy 算子的紧性，证明其紧性等价于有界性与端点尾部拟范数消失，并给出本质范数、有限秩距离与非紧性测度的精确恒等式，以及显式紧性判据和逼近数收敛结果。
AI 中文摘要
设 $I=(a,b)$ 为端点有限或无限的开区间。对于 $1<p_1,p_2<\infty$ 和 $0<q<\infty$，我们研究从 $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ 到 $L^q(u;I)$ 的加权双线性 Hardy 算子 $\mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g)$，包括拟 Banach 目标值域 $0<q<1$。令 $\vartheta=\min\{1,q\}$，目标空间和有界双线性映射空间通过其完备的幂度量来处理。在权函数的显式局部可积性假设下，紧性等价于有界性和端点尾部拟范数的消失。当全算子有界时，定向组合尾部极限的 $\vartheta$ 次幂恰好等于目标度量中的幂次本质距离、有限秩距离和 Hausdorff 非紧性测度。等价地，在所述的拟范数半径归一化下，未取幂的量一致。分段常数的中间窗口逼近子给出双线性逼近数的定量界。有限秩距离恒等式进而蕴含这些数收敛到本质（拟）范数。将已发表的范数等价有界性特征应用于端点截断的输出权函数，在凸、混合、中等次临界和深度次临界区域中产生显式的紧性判据。使用代数伴随指数来使 $q<1$ 时出现的负幂显式化。四个工作示例包括一个非对称混合范围阈值和一个非局部凸的 $L^{1/2}(0,1)$ 示例，其中 $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$。
英文摘要
Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\vartheta=\min\{1,q\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$.
Comments38 pages; 4 worked examples; Appendix A
