Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Question

Let $D$ be an open subset of $\mathbb{R}^n$ , $p$ and $q$ be in $(1,\infty)$ such that $p^ {-1} +q^ {-1} = 1$. Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$ and $$\dfrac{\partial(uv)}{\partial x_{i}}=\dfrac{\partial u}{\partial x_{i}}v+u\dfrac{\partial v}{\partial x_{i}}\;\forall i\in\left\{ 1,\ldots,n\right\} $$

Help me some hints to prove.

Thanks a lot.

Answer 1

You need to show that

1. $uv\in L^1(D)$, and

2. $\dfrac{\partial (uv)}{\partial x_i}\in L^1(D)$, for all $i=1,\ldots,n$.

Both are consequences of Holder's inequality and of the your formula expanding $\dfrac{\partial (uv)}{\partial x_i}$.