$\| u \|_{W^{k,p}(K)} \leq c$ for every $K \subset U$ implies $u \in W^{k,p}(U)$

Question

Let $U \subset \mathbb{R}^n$. I want to show that a particular function $u$ lies in $W^{k,p}(U)$. I could already show that $\| u \|_{W^{k,p}(K)} \leq c$ for every relatively compact set $K \subset U$. Now I wonder if this already implies that $u \in W^{k,p}(U)$?

Answer 1

If the constant $c$ can be chosen independent of $K$, then the claim is true: Let $(K_j)$ be a sequence of compact subsets of $U$ such that $\cup_jK_j=U$ and $K_j\subset K_{j+1}$ for all $j$.

Take a multiindex $\alpha$ with $|\alpha|\le k$. Then the functions $|\chi_{K_j}D^\alpha u|$ converge monotonically to $|D^\alpha u|$. By the monotone convergence theorem $$ c\ge \int_{K_j}|D^\alpha u|dx =\int_U |\chi_{K_j}D^\alpha u| dx \to \int_U |D^\alpha u| dx, $$ hence $D^\alpha u\in L^p$.