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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 $1<p<q<\infty$ 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 enter image description here

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