Product of $W^{1,p}_0$ functions

Question

Let $p>n$, and let $f,g\in W^{1,p}_0(\mathbb{R}^n)$ be two sobolev functions. Prove that $fg\in W^{1,p}_0(\mathbb{R}^n)$.

I was able to prove the Leibniz formula for weak derivativatives, but still I do not understand why the function should be in $L^p$. Probabily it is Morrey inequality or something like that. But I cannot figure out.

Answer 1

Ok, I got it. Since:

\begin{equation} \lVert u\rVert_{\mathcal{C}^{0,1-\frac{n}{p}}(\mathbb{R}^n)}\leq\lVert u\rVert_{W^{1,p}(\mathbb{R}^n)}, \end{equation}

then we know that if $u\in W^{1,p}(\mathbb{R}^n)$ then it is bounded. Therefore, one takes two sequences in $\mathcal{C}^\infty(\mathbb{R}^n)$ such that: $\lVert f-f_n\rVert_{W^{1,p}(\mathbb{R}^n)}\rightarrow 0$ and $\lVert g-g_n\rVert_{W^{1,p}(\mathbb{R}^n)}\rightarrow 0$ as $n\rightarrow \infty$. Recall that since $\{g_n\}_{n\in\mathbb{N}}$ is cauchy in $W^{1,p}(\mathbb{R}^n)$, then it is Cauchy in $L^\infty(\mathbb{R}^n)$ and therefore there exists a constant $C\in(0,\infty)$ such that $\lVert g_n\rVert_\infty\leq C$. Thus: \begin{equation} \lVert f_ng_n-fg\rVert_{W^{1,p}(\mathbb{R}^n)}\leq\lVert (f_n-f)g_n\rVert_{W^{1,p}(\mathbb{R}^n)}+\lVert f(g_n-g)\rVert_{W^{1,p}(\mathbb{R}^n)}\leq C\lVert f_n-f\rVert_{W^{1,p}(\mathbb{R}^n)}+\lVert f\rVert_\infty\lVert g_n-g\rVert_{W^{1,p}(\mathbb{R}^n)}. \end{equation}

Taking $n\rightarrow \infty$ we prove our claim.

Answer 2

Just a slight correction to @Diesirae92's answer, for posteriority's sake:

we can't really write

$$||f(g_n - g)||_{W^{1,p}(\mathbb{R}^n)} \leq ||f||_{\infty} ||(g_n - g)||_{W^{1,p}(\mathbb{R}^n)}$$

as the $W^{1,p}$ also involves the derivative, which may not be bounded.

Instead, what we should write is \begin{equation} ||f(g_n - g)||_{W^{1,p}(\mathbb{R}^n)} = ||f(g_n - g)||_{L^p} + \sum_i || \frac{\partial}{\partial x_i} f(g_n - g)||_{L^p} \end{equation}

We can take care of the first term as done above, by taking out the infinity norm of $f$, but the second one needs more work:

$$|| \frac{\partial}{\partial x_i} f(g_n - g)||_{L^p}\leq || \frac{\partial f}{\partial x_i} (g_n - g)||_{L^p} + ||f \frac{\partial}{\partial x_i}(g_n - g)||_{L^p} \leq ||g_n - g||_{L^\infty} ||\frac{\partial f}{\partial x_i}||_{L^p} + ||f||_{L^\infty} ||\frac{\partial}{\partial x_i}(g_n - g)||_{L^p}$$

and this goes to $0$, as $||g_n - g||_{L^\infty} \rightarrow 0$ (by continuity of the embedding $W^{1,p} \hookrightarrow L^\infty$) and $||\frac{\partial f}{\partial x_i}||_{L^p} \leq \infty$, and $|f||_{L^\infty} \leq \infty$ and $||\frac{\partial}{\partial x_i}(g_n - g)||_{L^p} \rightarrow 0$.

Doing the same for the other term, $||(f_n - f)g_n||_{W^{1,p}}$, we get the result