Consider the Gagliardo Niremberg interpolation inequality :
(Gagliardo Niremberg interpolation inequality)Let $q,r$ be any numbers satisfying $1 \leq q, r \leq \infty $ and let $j,m$ be any integers satisfying $0 \leq j < m$. If u is any function in $C^{m}_{0}(R^n)$ then
$$ || D^j u ||_p \leq C || D^m u ||_{r}^{a} ||u ||_{q}^{1-a}$$
where $$\frac{1}{p} = \frac{j}{n} + a (\frac{1}{r} - \frac{m}{n}) + (1-a) \frac{1}{q}$$
where $C$ is a constant depending only in n,m,j,q,r,a with the following excepetion:
If $m -j - \frac{n}{r}$ is a nonnegative integer, then the inequality is asserted for $j/m \leq a <1$.
With the Gagliardo Niremberg interpolation inequality i am trying to prove this:
$$ || u||_{2 \sigma + 2}^{2 \sigma + 2} \leq C || u||_{H^1(R^n)}^{\sigma N} || u||_2^{2 + \sigma(2-N)}, \forall u \in H^{1}(R^n)$$ where $0 <\sigma < \frac{2}{N-2}$.
Someone can give me a hand?
thanks in advance!