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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.

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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

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}}} $?

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    Show us what you did and don't ask 10 times the same question. Write the definition of what you are talking about, most of the answer is in it. What is the norm in $W^{-1,2}$ ? – 2017-02-22

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