AI 中文总结
该研究确定了有限维ℓ_p^d与ℓ_q^d空间对应的张量畸变比值ρ(ℓ_p^d,ℓ_q^d)随维度增长的渐近阶，给出了分区域的精确指数公式并说明了各区域的推导依据。
AI 中文摘要
对于有限维巴拿赫空间E和F，令ε和π分别表示单射和投射张量范数，定义ρ(E,F):=sup_{0≠z∈E⊗F}π(z)/ε(z)。对所有1<p,q<∞，确定了当维度d→∞时ρ(ℓ_p^d,ℓ_q^d)的增长性：ρ(ℓ_p^d,ℓ_q^d)≍_{p,q}d^{γ(p,q)}，其中比较常数仅依赖p、q和标量域，γ(p,q)的表达式为：当1<p,q<2时，γ(p,q)=3/2−max{1/p,1/q}；当2<p,q<∞时，γ(p,q)=1/2+min{1/p,1/q}；其他情况，γ(p,q)=min{1/p+1/q,2−1/p−1/q}。低于2的区域的估计来自各向异性哈代-李特尔伍德不等式与哈达玛矩阵，高于2的区域则由有限维对偶性推导得到，混合区域的公式在所有维度下均精确成立。
英文摘要
For finite-dimensional Banach spaces $E$ and $F$, set \[ ρ(E,F):=\sup_{0\neq z\in E\otimes F}\frac{π(z)}{\varepsilon(z)}. \] The square growth of $ρ(\ell_p^d,\ell_q^d)$ was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If $r=\min\{n,m\}$ and $η(t)=\min\{1/t,1/t'\}$, then, whenever $p$ and $q$ lie on the same side of $2$, \[ ρ(\ell_p^n,\ell_q^m)\asymp_{p,q} r^{1/2}\min\{n^{η(p)},m^{η(q)}\}. \] Thus the classical square exponent splits into two dimensional scales. In the mixed range $1\le p\le2\le q\le\infty$, writing $a=1/p$ and $b=1/q$, we obtain the lower estimate \[ ρ_{p,q}(n,m)\gtrsim_{p,q} \max\!\left\{r^{\min\{a+b,2-a-b\}}, m^{b-1/2}r^{1/2}\min\{n^{1-a},m^{1/2}\}\right\}, \] and the upper estimate \[ ρ_{p,q}(n,m)\lesssim_{p,q} \min\!\left\{n^{1-a}r^{1-b},m^b r^a,r\right\}. \] Moreover, if $(n-m)(a+b-1)\le0$, then \[ ρ_{p,q}(n,m)\asymp_{p,q} r^{\min\{a+b,\,2-a-b\}}. \] In the interior mixed range $1<p<2<q<p'$, if \[ α_{p,q}:= \frac{(a+b-1)(\frac12-b)}{a-\frac12}, \] then \[ ρ_{p,q}(n,m)\lesssim_{p,q}m^{1-α_{p,q}}, \qquad \sup_{n\ge1}ρ_{p,q}(n,m)\asymp_{p,q}m^{1-α_{p,q}}. \] The corresponding statements in the reversed mixed range follow by duality and symmetry.
