The phase diagram of injective-to-projective tensor distortion

Abstract

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 $\rho ({\ell }_p^d,{\ell }_q^d)$ 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 $\eta (t)=\min \{1/t,1/t^{\prime }\}$, 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 $1\le p\le 2\le q\le \infty$, 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 $(n-m)(a+b-1)\le 0$, then \[ ρ_{p,q}(n,m)\asymp_{p,q} r^{\min\{a+b,\,2-a-b\}}. \] In the interior mixed range $1<p<2<q<p^{\prime }$, 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.