# Hardy-Littlewood in Sobolev Spaces

For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $$q=\infty$$. The dimension is $$3$$ and H-L-S inequality says that for $$1 and $$0<\nu<3$$ such that $$\frac{1}{q}=\frac{1}{p}-\frac{3-\nu}{3}$$ we have $$\left\Vert\int |v-v_*|^{-\nu} f(v_*) d v_*\right\Vert_{L^q_v}\leq \Vert f\Vert_{L^p_v}$$ Question: Is the following inequality true? $$\left\Vert\int |v-v_*|^{-\nu} f(v_*) d v_*\right\Vert_{L^\infty_v}\leq \Vert f\Vert_{H^{s(\nu)}}$$

I would also be interested in how this $$s$$ depends on $$\nu$$, more specifically if we have $$\nu<2$$ can we arrange $$s<1/2$$? Note that if H-L-S did apply in the case of infinity then for $$\nu=2$$, we would have $$\left\Vert\int |v-v_*|^{-\nu} f(v_*) d v_*\right\Vert_{L^\infty_v} \leq {\Vert f\Vert_{L^3_v}}$$ which by sobolev embedding corresponds to $$1/2$$ derivatives. So hopefully for $$\nu<2$$ we can arrange $$s<1/2$$.

P.S.- There is a similar question asked for Lorentz spaces but I would like to know either the equivalent result for sobolev space or if there is an embedding $$H^s \hookrightarrow L^{p,1}$$.

Hardy-Littlewood-Sobolev inequality in Lorentz spaces

Edit: I think I have a partial answer to the question I posed, please take a look