Sobolev space and equivalence of norms

Question

I'm considering the space $W\{n,p\}[0,1]$ of functions with $n-1$ continuous derivatives $f^{(n-1)}$ is absolutely continuous and $f^{(n)}$ is in $L^p[0,1]$. The usual norm is the sum of the $p$-norms of each derivative from $1$ to $n$ and the $p$ norm of the function.

Now just consider the $p$ norm of the function + $p$ norm of the $n^{th}$ derivative, i.e.$(\int |f(x)|^p)^{1/p}$ + $(\int |f^{(n)}|^p)^{1/p}$ I want to show that this is equivalent to the usual norm defined above.

This requires finding positive constants and sandwiching this new norm. In one direction it's obvious since the new norm is less than the usual norm for every $x$, say $\|x\|_2 \leq \|x\|_1$. I'm not sure how to show the other direction.

Answer 1

Here is a proof for $p>1$. We want to show the existence of $c>0$ such that $$ \|u \|_{W^{n,p}} \le c ( \|u \|_{L^p} + \|D^nu\|_{L^p}) \quad \forall u\in W^{n,p}(0,1). $$ Assume this is not true. Then for every $k$ there is $u_k$ such that $$ \|u_k\|_{W^{n,p}} > k ( \|u_k\|_{L^p} + \|D^nu_k\|_{L^p}). $$ Clearly, $u_k\ne0$. We can rescale the $u_k$'s to achieve $$ 1 = \|u_k\|_{W^{n,p}} > k ( \|u_k\|_{L^p} + \|D^nu_k\|_{L^p}). $$ So $(u_k)$ is bounded in $W^{n,p}$, moreover $u_k \to 0$ in $L^p$ and $D^nu\to 0$ in $L^p$. We can extract a weakly (weak-star for $p=\infty$) converging subsequence (still denoted by same index): $u_k \rightharpoonup^* u$ in $W^{n,p}$. Here, we need $p>1$. By compactness of the embedding, $u_k \to u$ in $W^{n-1,p}$. We already know that the limit is zero: $u=0$, so $\|u_k\|^p_{W^{n-1,p}} \to 0$.

Going back to the initial inequality (for $p<\infty$), proves $$ 1 = \|u_k\|^p_{W^{n,p}} = \|D^nu_k\|_{L^p}^p + \|u_k\|^p_{W^{n-1,p}} \to 0. $$ Similarly, we get for $p=\infty$: $$ 1 = \|u_k\|_{W^{n,\infty}} =\max( \|D^nu_k\|_{L^\infty} , \|u_k\|_{W^{n-1,\infty})} \to 0. $$ In both ways, we get a contradiction.

Answer 2

By the fundamental theorem of calculus, $f^{(k)}(x) = \int_0^x f^{(n+1)}(y) dy$. Thus $$ \|f^{k}\|_{L^p} =\left [\int_0^1 \left(\int_0^x f^{(k+1)}(y) dy\right)^p\right]^{1/p} $$ Minkowski's integral inequality tells us that $$ \left[\int_0^1 \left(\int_0^x f^{(k+1)}(y) dy\right)^p\right]^{1/p} \leq \int_0^1 \left(\int_0^x |f^{(k+1)}(y)|^{p} dy\right)^{1/p}dx \leq \|f^{(k+1)}\|_{L^p} $$ Now we see by induction that $$ \sum_{k=1}^n \|f^{(k)}\|_{L^p} \leq (n-1)\|f^{(n)}\|_{L^p} $$ So the norm equivalence easily follows. Note that the calculations here are much easier because the total measure space is one, as we're working on $[0,1]$.