发表机构
Universidade Federal da Paraíba; Universidad Nacional de Colombia（帕拉伊巴联邦大学; 哥伦比亚国立大学）
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 Grothendieck 不等式在 $\ell_p^r$ 空间中的推广，证明有效维数 $d=\min\{r,m,n\}$ 决定普适指数 $d^{|1/p-1/2|}$，并给出精确的矩形端点公式与有界性阈值。
AI 中文摘要
当 Grothendieck 不等式中的 Hilbert 空间向量被替换为 $\ell_p^r$ 和 $\ell_{p'}^r$ 中的向量时，该不等式会变成什么？对于 $m\times n$ 矩阵，我们证明普适的维数损失并非仅由 $r$ 决定，而是由有效维数 \\[ d=\min\{r,m,n\} \\] 决定。更精确地说，最优常数在忽略与维数无关的因子后，上下界由 \\[ d^{|1/p-1/2|} \\] 给出。上界依赖于逐点的 Hilbert 化稳定性质：对每个矩阵 $A$，\\[ G_2^{(r)}(A)=G_2^{(d)}(A)。\\] 将此稳定性与 Lewis 的欧氏距离估计相结合，即得到有效维数的 Grothendieck–Hölder 界。在普适幂律之外，我们保留了更精细的矩形几何。在 $p=\infty$ 时，我们证明了精确的压缩公式 \\[ \Gamma_\infty^\K(r;m,n) = \rho\\!\left( \ell_1^m(\K), \ell_1^{\min\{r,n\}}(\K) \right)，\\] 并由此推导出在整个 Hölder 尺度上对端点敏感的的下界。在复数域上，Hilbert 问题与精确矩形端点之间的插值给出相应的上界。因此，有效维数决定普适指数，而张量范数端点保留额外的纵横比信息。当 $p$ 趋近于 $2$ 时，我们确定了有界性阈值：它由 \\[ |1/p-1/2|\log d \\] 控制。对于复值两行矩阵，端点张量比是显式的，而一般的矩形估计为每个 $1\le p\le\infty$ 给出定量的双侧界。
英文摘要
What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely, the optimal constant is bounded above and below, up to dimension-free factors, by \[ d^{|1/p-1/2|}. \] The upper bound rests on a pointwise Hilbertian stabilization: for every matrix $A$, \[ G_2^{(r)}(A)=G_2^{(d)}(A). \] Combining this stabilization with Lewis's Euclidean distortion estimate gives the effective-dimensional Grothendieck--Hölder bound. Beyond the universal power law, we retain the finer rectangular geometry. At $p=\infty$ we prove the exact compression formula \[ Γ_\infty^\K(r;m,n) = ρ\!\left( \ell_1^m(\K), \ell_1^{\min\{r,n\}}(\K) \right), \] and derive from it endpoint-sensitive lower bounds throughout the full Hölder scale. Over the complex field, interpolation between the Hilbertian problem and the exact rectangular endpoints yields corresponding upper bounds. Thus the effective dimension determines the universal exponent, whereas the tensor-norm endpoint retains additional aspect-ratio information. We determine the boundedness threshold when $p$ approaches $2$: it is governed by \[ |1/p-1/2|\log d. \] For complex two-row matrices, the endpoint tensor ratios are explicit, and the general rectangular estimates give quantitative two-sided bounds for every $1\le p\le\infty$.
