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.
Thanks for your reply!