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Let $U$ be a bounded open domain in $\mathbb{R}^n$. If $W_0^{k,p}(U)$ is the completion of $C_c^{\infty}(U)$, the space of smooth and compactly supported functions on $U$, with respect to the $W^{k,p}(U)$ norm, does the inclusion $$W^{k_2,p}(U)\cap W_0^{k_1,p}(U)\subset W_0^{\max\{k_1,k_2\},p}(U)$$ hold?

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  • $\begingroup$ Note that $W_0^{0,p}$ is just $L^p$, so $W^{1,p} \cap W_0^{0,p}$ is still $W^{1,p}$ and is not contained in $W_0^{1,p}$. PhoemueX's answer shows that the same happens for general $k_1<k_2$. $\endgroup$ – Michał Miśkiewicz May 7 '17 at 11:15
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No, this is not true in general. To see this, consider the one-dimensional case $U = (0,1) $. Then, if $k_2 > k_1$, $$ W^{k_2, p} \cap W_0^{k_1, p} = \{ f \in W^{k_2, p} \,:\, \partial^\alpha f (x)=0 \text { for } |\alpha| \leq k_1 - 1\text { and } x \in \{0,1\} \} $$ is not a subset of $$ W_0^{k_2,p} = \{ f \in W^{k_2, p} \,:\, \partial^\alpha f (x)=0 \text { for } |\alpha| \leq k_2 - 1 \text { and } x \in \{0,1\} \}, $$ as can already be seen by considering suitable polynomials.

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