On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm

$$\|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|^{p_1}\right)^{p_2/p_1} \right)^{1/p_2}.$$

On the other hand, given a normed space $(\mathbf{E},\|\cdot\|)$, its modulus of convexity is defined as the function $\delta:[0,2]\to[0,1]$

$$\delta(\varepsilon) = \inf \left\{ 1-\left\|\frac{x-y}{2}\right\|: \,\, x,y\in\partial B(0,1),\,\,\|x-y\|\geq \varepsilon\right\},$$ and we say its modulus of convexity is of power type $q\geq 2$ if there exists $c>0$ such that $\delta(\varepsilon)\geq c \varepsilon^q$.

In the study of the modulus of convexity of $\ell_p$ spaces, there is an interesting behavior of the power type, which turns out to be $q=\max\{2,p\}$. The case $p\geq 2$ can be derived from Clarkson's inequality; and for the case $1\leq p <2$ see e.g., here. I am interested in how the modulus of convexity behaves for mixed-norm spaces $\ell_{p_1,p_2}$. The cases I know are the following:

• In the case $p_1,p_2\geq 2$, I believe this paper gives a tight Clarkson-type inequality, which can lead to the right power type for the modulus of convexity (if I am not mistaken, the power type is $q=\max\{p_1,p_2\}$).

• It is also known that when $1\leq p_1,p_2\leq 2$, then the power type is 2 (see, e.g. this paper, where it is stated somewhat differently).

Finally, my question is: What is the power type of the modulus of convexity for the space $\ell_{p_1,p_2}$ when $p_1<2$, $p_2>2$ (and the reverse case, if it makes any difference)?

• A fairly easy way of seeing that you get the expected is to take advantage of the fact that in Banach lattices you can calculate the moduli of convexity and smoothness in terms of properties of the lattice ($p$ convexity; $p$ concavity). Look at volume 2 of Lindenstrauss-Tzafriri. I think it contains what you need; at least it will refer to the relevant papers. – Bill Johnson Apr 2 '15 at 18:17
• Thank you for your answer. For the problem I am studying, the constant $c$ in the modulus of convexity matters, and the results in Lindenstrauss-Tzafriri do not give such estimate. Is there a more precise bound, e.g., in the case $p_2=\log n$, and $1/p_1 + 1/\log m = 1$? – Cristóbal Guzmán Apr 3 '15 at 22:27
• When $p=\log n$, $\ell_p^n$ is $\ell_\infty^n$ up to a constant $C$ (the base of the logarithm). So you are looking at $\ell_\infty^n(\ell_1^m)$ in a $C$-equivalent norm. – Bill Johnson Apr 5 '15 at 17:02