After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm.
Fix dimension to be $3$. H-L-S says that for $1<p<q<\infty$ and $0<\nu<3$ such that $$\frac{1}{q}=\frac{1}{p}-\frac{3-\nu}{3}$$ we have $$||\int |v-v_*|^{-\nu} f(v_*) d v_*||_{L^q_v}\leq ||f||_{L^p_v}$$
I am interested in the case when $q=\infty$ but since I can't apply this theorem directly, I tried to use Sobolev embedding first:
We know $||g||_{L^\infty}\leq ||g||_{W^{1,3}}$(by Sobolev embedding) and we have
$$||g||_{W^{1,3}}=||\partial_v g||_{L^3}+||g||_{L^3}$$
If $g=\int |v-v_*|^{-\nu}f(v_*) d v_*$ then we can now apply H-L-S to the second term and for the first term we note that $$\partial_v \int |v-v_*|^{-\nu} f(v_*) d v_* \leq \int |v-v_*|^{-\nu-1} \langle v_*\rangle f dv_*$$
Here $\langle v\rangle=\sqrt{1+v_i^2}$
If we assume $\nu<2$ and $\langle v\rangle f \in L^1$ then we can apply H-L-S to this to get $$||\int |v-v_*|^{-\nu} f||_{L^\infty}\leq ||f||_{L^p}+||\langle v\rangle f||_{L^q}$$ where $\frac{1}{3}=\frac{1}{p}-\frac{3-\nu}{3}$ and $\frac{1}{3}=\frac{1}{q}-\frac{2-\nu}{3}$
Question:
1) Does what I do make sense?
2) Is it possible to get this for a fractional Sobolev space that is $L^\infty \hookrightarrow W^{s,p}$ such that $s\in(0,1)$ and $s> \frac{3}{p}$? A reference if possible would be much appreciated.
I think I figured out a partial answer to the question. Please take a look