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 $\alpha \in (0,1)$?



Reference request for fractional Poincare inequality
**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 $\Vert f\Vert\_p=0$, $f=0$.


