Sobolev embedding for the $L^q$ norm.

Question

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.

Answer 1

This is immediate from the Gagliardo-Nirenberg inequality, or from the Sobolev inequality with fractional exponents, which says in particular that $$ \|f\|_{L^q} \leq C\|f\|_{H^s} \leq C'\|f\|_{H^1} , \qquad f\in C^1_0(\mathbb{R}^2), $$ for $2\leq q<\infty$ and $s=1-\frac2q$.