Let $W^{1,p} (\mathbb{R}^n)$ be the sobolev space, and $W^{-1,p^{\prime}} (\mathbb{R}^n)$ be the it's dual space.
($1<p<\infty, and \ \frac{1}{p} + \frac{1}{p^{\prime}} = 1$)
I know $||f||_{L^2} \leq ||f||^{\frac{1}{2}}_{W^{1,2}} ||f||^{\frac{1}{2}}_{W^{-1,2}} \ \forall f \in W^{1,2}$.
Let $1<q<\infty, q \neq 2$.
Is there a $1 \leq r \leq \infty , C>0, 0<\theta<1$ such that
$||f||_{L^r} \leq C ||f||^{\theta}_{W^{1,q}} ||f||^{1- \theta}_{W^{-1,q^{\prime}}} $?