Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.

My question: Knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\cdot g\in W^{1,p}(\Omega)$? In other words, for what $p\in (1,\infty)$ is there a constant $c>0$ such that:

$$\Vert fg\Vert_{W^{1,p}(\Omega)}\leq c\Vert f\Vert_{W^{1,p}(\Omega)}\cdot \Vert g\Vert_{W^{1,p}(\Omega)},\ \forall\ f,g\in W^{1,p}(\Omega)$$

So is, $W^{1,p}(\Omega)$ a Banach algebra?

P.S. I know that when $d=1$ the answer is yes, and it is proved in Brezis book, page 214. I also know that in some lecture notes of Terence Tao (lecture 4 here https://www.math.ucla.edu/~tao/254a.1.01w/) it is stated that this is true when $p>d$ for $\Omega=\mathbb{R}^d$.

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.

My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\cdot g\in W^{1,p}(\Omega)$? 
In other words, for what $p\in (1,\infty)$ is there a constant $c>0$ such that:

$$\Vert fg\Vert_{W^{1,p}(\Omega)}\leq c\Vert f\Vert_{W^{1,p}(\Omega)}\cdot \Vert g\Vert_{W^{1,p}(\Omega)},\quad \forall\ f,g\in W^{1,p}(\Omega)\;?$$

So, is $W^{1,p}(\Omega)$ a Banach algebra?

P.S. I know that when $d=1$ the answer is yes, and it is proved in Brezis book, page 214. I also know that in some lecture notes of Terence Tao (lecture 4 here) it is stated that this is true when $p>d$ for $\Omega=\mathbb{R}^d$.