Let $p,q >1$ and $u \in W^{1,p}_{0}(\Omega)$ and $v \in W^{1,q}_{0}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ with smooth boundary.
Suppose that $p,q \in (1,N)$, $q^{'} \in [1,q^{\star}]$ and $p^{'} \in [1,p^{\star}]$ where $1/p + 1/p^{'} = 1$ and $1/q + 1/q^{'} = 1$. Let $\alpha_i,\gamma_i \in (0,\theta_i), i=1,2$ where
$$\theta_1 = min \{1, p-1, \frac{q^{'}}{p^{'}} \} \ \text{and} \ \theta_2 = min \{1, q-1, \frac{p^{'}}{q^{'}} \}.$$
Here I am considering $q^{\star} = Nq/(N-q)$ and $p^{\star} = Np/(N-p)$.
I am reading a paper and the author says that there is a constant $C>0$ such that
$$ \int_{\Omega} v^{\alpha_1} u \leq C \| u\|_{W^{1,p}(\Omega)} \| v\|_{W^{1,q}(\Omega)}^{\alpha_1}$$
Probably this is a simple application of Holder and Poincare inequality. I am trying to use these inequalitites to obtain the result . But I am getting anywhere. Someone could help me?
Thanks in advance
My thought is that it is just Holder + Jensen. Write $$ \int_{\Omega} v^{\alpha_1}u \,d x \leq \|v^{\alpha_1}\|_{L^p(\Omega)}\|u\|_{L^q(\Omega)}. $$ Now note that \begin{align} \|v^{\alpha_1}\|_{L^p(\Omega)}^p = \int_{\Omega}\left(v^{p}\right)^{\alpha_1}\,dx &= |\Omega|\int_{\Omega}\left(v^{p}\right)^{\alpha_1}\,\frac{dx}{|\Omega|}\\ &\leq |\Omega|\left(\int_{\Omega}v^{p}\,\frac{dx}{|\Omega|}\right)^{\alpha_1} = |\Omega|^{1 - \alpha_1}\|v\|_{L^p(\Omega)}^{p\alpha_1} \end{align} where the inequality is Jensen's inequality and the fact that $\alpha_1 < \theta_1 \leq 1$ (so $x \mapsto |x|^{\alpha_1}$ is concave).
So we conclude $$ \int_{\Omega} v^{\alpha_1}u \,d x \leq C\|v\|_{L^p(\Omega)}^{\alpha_1}\|u\|_{L^q(\Omega)} \leq C\|v\|_{W^{1,p}(\Omega)}^{\alpha_1}\|u\|_{W^{1,q}(\Omega)}. $$