Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $p_1, p_2\geq 1$. Consider the product space $W:=W_0^{1, p_1}(\Omega)\times W_0^{1, p_2}(\Omega)$ equipped with the norm
$$\Vert (u, v)\Vert_W = \Vert u\Vert_{W_0^{1, p_1}} + \Vert v\Vert_{W_0^{1, p_2}} \quad \mbox{ for all } (u, v)\in W.$$
My question is the following one. If I consider the norm given by
$$\left(\Vert u\Vert_{W_0{1, p_1}}^{\max(p_1, p_2)} + \Vert v\Vert_{W_0{1, p_2}}^{\max(p_1, p_2)}\right)^{\frac{1}{\max(p_1, p_2)}},$$
it is equivalent to the norm defined above? And what about if I replace $\max(p_1, p_2)$ with $\min(p_1, p_2)$? The equivalence is also preserved?
Could anyone please help or give some references? Thank you in advance!
Best Answer
All norms on a product $X \times Y$ that have the form $$\lVert (x,y)\rVert = N(\lVert x\rVert_X, \lVert y\rVert_Y)$$ where $N \colon \mathbb{R}^2 \to \mathbb{R}$ is a norm are equivalent. This easily follows from the equivalence of all norms on $\mathbb{R}^2$.
The analogous result holds for products of an arbitrary finite number of normed spaces. All norms of the form $$N(\lVert x_1\rVert_{X_1}, \dotsc, \lVert x_n\rVert_{X_n})$$ where $N$ is a norm on $\mathbb{R}^n$ are equivalent.