# Does Gagliardo-Nirenberg inequality in unbounded domain still hold?

Often we have the following Gagliardo-Nirenberg inequality: Let $1\leq p_1, p_2\leq \infty$, $0\leq r<l (r, l\in Z_+)$. Suppose that the number $$\theta=\frac{n/p-n/p_1-r}{n/p_2-n/p_1-l}$$ satisfies the inequality $r/l\leq \theta<1$. Then $$\|u\|_{r, p}\leq C\|u\|_{0, p_1}^{1-\theta}\|u\|_{l, p_2}^\theta.$$

We have known that the inequality holds if the domain is bounded with regular boundary. I wonder if it still holds when the domain is unbounded in $R^n$ with $n\leq 3$? It seems true because some people usually use the fact in proving some theorems.

• If I recall correctly from my analysis class, it is indeed true. You can approximate with compactly supported functions to extend to all of $R^n$ – abnry Mar 25 '15 at 1:45
• Thank you very much! What about $n>3$? Is still true? – Gary Mar 25 '15 at 1:55
• Why do you think there should be a difference between $n\leq 3$ and $n>3$? – Jose27 Mar 25 '15 at 2:33