I am looking for a proof of the following interpolation inequality:
$ ||u||_{p} \leq ||u||^{N(p-1)/(2+N(p-1))}_{pN/(N-2)} ||u||_{1}^{2/(2+ N(p-1))} $
Where $||\cdot||_{p} = ||\cdot||_{L^p(\mathbb{R})}$. If anyone could provide a proof, or even a source that I could read myself, I would greatly apppreciate it. Thank you.
It follows from the well-known inequality (which is itself a simple consequence of Hölder's inequality): Let $u \in L^p \cap L^q$ for some $p,q \in [1,\infty]$. For $\lambda \in [0,1]$ we define $r \in [1,\infty]$ via $$ \frac1r = \lambda \, \frac1p + (1-\lambda) \, \frac1q,$$ i.e. we take a convex combination of the reciprocals. Then, $$ \|u\|_r \le \|u\|_p^\lambda \, \|u\|_q^{1-\lambda}.$$