Let $ n \in \mathbb{N}$ and $ p \in [1,\infty]$ be fixed and endow $ \mathbb{R}^n$ with the $ p$ -norm $ \|\cdot\|_p$ . For every matrix $ A \in \mathbb{R}^{n \times n}$ we denote the operator norm of $ A$ as an operator on $ \mathbb{R}^n$ by $ \|A\|_p$ , too. Moreover, let $ |A|$ denote the matrix whose entries are the absolute values of the entries of $ A$ .

The number $ \| \,|A|\,\|_p$ is sometimes called the **regular norm** of $ A$ (in particular in Banach lattice theory, where a complete norm on the space of regular operators is constructed this way).

We clearly have $ \|A\|_p \le \| \,|A|\,\|_p$ , and equality holds for $ p = 1$ and $ p = \infty$ . For general $ p$ , Mark Meckes’ explained in his answer to this question that the estimate \begin{align*} \| \,|A|\,\|_p \le n^{\frac{2}{p}(1 – \frac{1}{p})} \|A\|_p \tag{E} \end{align*} holds as a consequence of the Riesz-Thorin theorem. This estimate is sharp for $ p = 2$ ; this is mentioned in Mark Meckes’ post quoted above, and it can also be found in [Schaefer: Banach Lattices and Positive Operators (1974), Example 1 on page 231]. Obviously, $ (\text{E})$ is also sharp for $ p = 1$ und $ p = \infty$ .

My question is:

**Question 1.** Is the estimate $ (\text{E})$ sharp for $ p \in (1,\infty) \setminus \{2\}$ ?

In case that the answer to Question 1 is “no”, I would like to ask:

**Question 2.** What is the best known constant (in dependence of $ n$ ) in $ (\text{E})$ ? Is the optimal constant known?

Note: A related question concerned with the Schatten $ p$ -norm of a matrix can be found here.