Suppose $f:\mathbb{R}^n\to \mathbb{R}$ and let $Df=(f_{x_1},f_{x_2},..., f_{x_n})$, the gradient of $f$.
A special case of the Gagliardo-Nirenberg inequality says that
$$||f||_{p^*}\leq C||Df||_{p}$$
But I can show that
$$||f||_{p^*}\leq C\left(\sum_{i=1}^n||\frac{\partial}{\partial x_i}f||_{p}^p\right)^{1/p}$$
I am a bit confused about what the definition of $||Df||_{p}$ is. Is
$$||Df||_{p}=\left(\sum_{i=1}^n||\frac{\partial}{\partial{x_i}}f||_{p}^p\right)^{1/p}$$
true? I had always thought that
$$||Df||^p_{p}=\int{|Df|^p\,dx}=\int{\left((f_{x_1})^2+(f_{x_2})^2+...+(f_{x_n})^2\right)^{p/2}}$$
Thanks!
Both norms are equivalent: As all norms on $\mathbb R^n$ are equivalent, there is a constant $c>0$ such that for all $v\in \mathbb R^n$ $$ c \left( \sum_{i=1}^n |v_i|^p \right)^{1/p} \le \left( \sum_{i=1}^n |v_i|^2 \right)^{1/2} \le c^{-1} \left( \sum_{i=1}^n |v_i|^p \right)^{1/p}. $$ Now replace $v_i$ by $f_{x_i}(x) = (\frac\partial{\partial x_i}f)(x)$, integrate over $x\in \Omega$, and see that indeed both $L^p$-norms of $Df$ are equivalent.