Consider the special case of the Gagliardo--Nirenberg inequality in R^4
$$
\|u\|_p \le C_p \|\Delta u\|_2^{1-4/p} \|u\|_4^{4/p}
$$
with $p \ge 4$. What is the value of the best constant $C_p$?
Jacopo Schino
Thank you Zhong Xuexiu for your answer. Unfortunately, I need to estimate the value of such a constant in order to determine if a certain series converges or not, and, more importantly, I need it for $p \geq 4$ (at least when $p$ is an even integer).
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