Norm characterization on Sobolev Spaces $W^{k,p}(\mathbb{R}^n)$.

Question

I need to prove that $u \in W^{k,p}(\mathbb{R}^n)$ if and only if $u \in L^p(\mathbb{R}^n)$ with $D^l u \in L^p(\mathbb{R}^n)$ for every multi-index $l$ with $|l| = k$. This is related to the fact that $$\lVert u \rVert_p +\sum_{|l| = k}\lVert D^lu \rVert_p$$ is an equivalent norm for the space. Any advice?

Answer 1

Added later: Upon revisiting this question I realised the proof I originally wrote was incorrect; crucially one cannot just iteratively apply the case $\ell=1, k=2$ to deduce the general case. My answer has been amended to more closely follow the proof in the cited text.

This is true, and is a consequence of the Gagliardo-Nirenberg interpolation inequality which can be found in many sources. We will follow Chapter 5 of the following reference:

Adams, Robert A.; Fournier, John J. F., Sobolev spaces, Pure and Applied Mathematics 140. New York, NY: Academic Press (ISBN 0-12-044143-8/hbk). xiii, 305 p. (2003). ZBL1098.46001.

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The key ingredient is the one-dimensional inequality $$ |f'(0)|^p \leq \frac{2^{p-1}9^p}{\delta} \left(\delta^{-p}\int_0^1 |f(t)|^p \,\mathrm{d}t+ \delta^p\int_0^1 |f''(t)|^p \,\mathrm{d}t \right ), $$ for all $f \in C^2([0,\delta])$ with $\delta>0,$ and $1 \leq p < \infty.$

By rescaling we can assume $\delta=1.$ We use the fundamental theorem of calculus $$ f'(0) = f'(z) - \int_0^z f''(t) \,\mathrm{d}t, $$ and for $0 \leq x<y \leq 1$ the mean-value theorem gives $z \in (x,y)$ such that $$ f'(z) = \frac{u(y)-u(x)}{y-x}. $$ Chaining these we deduce that $$ |f'(0)| \leq |x-y|(|f(x)|+|f(y)|) \int_0^1 |f''(t)| \,\mathrm{d}t.$$ From here we can integrate along $x \in (0,1/3),$ $y \in (1/3,1)$ and apply Hölder to conclude.

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Now given $u \in L^p(\Bbb R^n)$ whose $k$th order derivatives are in $L^p,$ by mollifying we can assume $u$ is smooth. We apply the above estimate to $f(r) = u(x+r\omega)$ for $\omega \in S^{n-1},$ and integrate over all $\omega \in S^{n-1}$ and $x \in B_R$ to obtain \begin{equation*}\begin{split} &\int_{B_R} |\nabla u|^p \,\mathrm{d}x \\ &\leq \frac{C(n,p)}{\delta} \int_{B_R} \int_{S^{n-1}} \left( \delta^{-p}\int_0^\delta |u(x+t\omega)|^p \,\mathrm{d} t +\delta^p\int_0^{\delta} |\nabla^2u(x+t\omega)|^p \,\mathrm{d}t \right) \,\mathrm{d} \omega\,\mathrm{d} x \\ &\leq C(n,p) \left(\delta^{-p} \int_{B_{R}} |u|^p \,\mathrm{d}x + \delta^p\int_{B_{R}} |\nabla^2u|^p \,\mathrm{d} x \right). \end{split}\end{equation*} I've been a bit brief here, but the last line uses an application of Fubini by (temporarily) assuming $u$ and its derivatives vanish outside of $B_R.$ We refer the reader to Lemma 5.4 of the cited text for the full details.

From here the idea is to apply similar estimates to $\nabla^j u$ and chaining the inequalities. This requires an induction argument, together with a carefully choosing the parameters $\delta$ so the relevant terms can be absorbed, as is done in Lemma 5.6. This will eventually give $$ \int_{B_R} |\nabla^j u|^p \,\mathrm{d}x \leq C(n,p,k) \left( \int_{B_{R}} |u|^p \,\mathrm{d}x + \int_{B_{R}} |\nabla^ku|^p \,\mathrm{d} x \right) $$ for each $1 < j < k.$ Note one could have proved this in the case $R=1,$ and deduced the general case by scaling. To conclude, we send $R \to \infty.$