My first question is the following: We have the embedding $$W^{1,p} \subset C^{0,\alpha} \ \text{for} \ p>n \quad \text{but} \quad C^{0,\alpha} \nsubseteq W^{1,q} \ \text{for any} \ q.$$ Can an embedding of the following type hold: $$W^{1,2} \cap C^{0,\alpha} \subset W^{1,n}?$$

My second question: In particular, is there an interpolation type inequality of the form $$\|\nabla u\|_p \leq C \|\nabla u\|_2^{b} \|u\|_{C^{0,\alpha}}^c$$ for any $2\leq p \leq n$.

EDIT: I am strongly inclined to claim the interpolation estimate above is false for any $\alpha \in (0,1)$. It would be nice to get an explicit counterexample.


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