The phase diagram of injective-to-projective tensor distortion

RESUMO

For finite-dimensional Banach spaces $E$ and $F$, set \[ \rho(E,F):=\sup_{0\neq z\in E\otimes F}\frac{\pi(z)}{\varepsilon(z)}. \] 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 $\eta(t)=\min\{1/t,1/t'\}$, then, whenever $p$ and $q$ lie on the same side of $2$, \[ \rho(\ell_p^n,\ell_q^m)\asymp_{p,q} r^{1/2}\min\{n^{\eta(p)},m^{\eta(q)}\}. \] 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 \[ \rho_{p,q}(n,m)\gtrsim_{p,q} \max\!\left\{r^{\min\{a+b,2-a-b\}}, m^{b-1/2}r^{1/2}\min\{n^{1-a},m^{1/2}\}\right\}, \] and the upper estimate \[ \rho_{p,q}(n,m)\lesssim_{p,q} \min\!\left\{n^{1-a}r^{1-b},m^b r^a,r\right\}. \] Moreover, if $(n-m)(a+b-1)\le0$, then \[ \rho_{p,q}(n,m)\asymp_{p,q} r^{\min\{a+b,\,2-a-b\}}. \] In the interior mixed range $1<p<2<q<p'$, if \[ \alpha_{p,q}:= \frac{(a+b-1)(\frac12-b)}{a-\frac12}, \] then \[ \rho_{p,q}(n,m)\lesssim_{p,q}m^{1-\alpha_{p,q}}, \qquad \sup_{n\ge1}\rho_{p,q}(n,m)\asymp_{p,q}m^{1-\alpha_{p,q}}. \] The corresponding statements in the reversed mixed range follow by duality and symmetry.