# Reference request for fractional Poincare inequality

Suppose we consider in $$\mathbb R^n$$, then how to show $$\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$$, where $$s>0$$ is noninteger and $$\alpha \in (0,1)$$?

• Can you be more precise: what are $p$ and $q$ and what do you mean by $\nabla^s f$? – Piotr Hajlasz Nov 21 '18 at 14:31

Such an inequality cannot be true unless $$f=0$$. That can be proved by a standard homogeneity argument. Suppose that $$\Vert \nabla^sf\Vert_p<\infty$$. Replacing $$f$$ by $$tf$$, where $$t>0$$ we have $$t\Vert f\Vert_p=\Vert tf\Vert_p\leq C\Vert \nabla^s(tf)\Vert_p^\alpha=t^\alpha C\Vert \nabla^sf\Vert_p^\alpha, \quad \Vert f\Vert_p\leq t^{\alpha-1} C\Vert \nabla^sf\Vert_p^{\alpha},$$ Since $$\alpha-1<0$$, letting $$t\to\infty$$, the right hand side will converge to $$0$$ and hence $$\Vert f\Vert_p=0$$, $$f=0$$.
I guess that your question is ill-formulated: you have to respect the homogeneity on both sides of the inequality. Let us say that for $$f$$ in the Schwartz space you always have $$\Vert f\Vert_{W^{t,q}(\mathbb R^n)}\lesssim \Vert \vert D\vert^s f \Vert_{L^{p}(\mathbb R^n)}\quad\text{(and also}\quad \Vert f\Vert_{L^{n/(n-1)}(\mathbb R^n)}\lesssim \Vert D f\Vert_{L^{1}(\mathbb R^n)}),$$ provided $$\frac{s-t}{n}=\frac{1}{p}-\frac{1}{q},\quad 1 Applying the second inequality above to $$f=u^2$$, you obtain for $$p>1$$, $$\Vert u\Vert_{L^{2n/(n-1)}(\mathbb R^n)}^2\lesssim \Vert uD u\Vert_{L^{1}(\mathbb R^n)}\lesssim \Vert u\Vert_{L^{p'}(\mathbb R^n)} \Vert D u\Vert_{L^{p}(\mathbb R^n)} \lesssim \Vert u\Vert_{L^{p'}(\mathbb R^n)} \Vert \vert D\vert u\Vert_{L^{p}(\mathbb R^n)},$$ so that $$\Vert u\Vert_{L^{2n/(n-1)}(\mathbb R^n)} \lesssim \Vert u\Vert_{L^{p'}(\mathbb R^n)}^{1/2} \Vert \vert D\vert u\Vert_{L^{p}(\mathbb R^n)}^{1/2},$$ an inequality resembling yours. Choosing $$f= u^\alpha$$ leads to more general interpolation inequalities.