Imbedding theorem for Sobolev space $D^{k,p}$

Question

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by

$$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as that for the Sobolev space $W^{k,p}$?

Answer 1

For bounded domains $\Omega\in\mathbb{R}^n$, $n\geqslant 2$, satisfying the cone condition, the spaces coincide, i.e., $D^{k,p}(\Omega)=W^{k,p}(\Omega)$.  Examples of nonsmooth bounded domains $\Omega$ for which $D^{1,p}(\Omega)\neq W^{1,p}(\Omega)$, i.e., $D^{1,p}(\Omega)\not\subset L^p(\Omega)$, can be found in "Sobolev spaces" http://bookza.org/book/977600/c2bb9a by V.G. Mazya.  A domain $\Omega$ for which $D^{1,2}(\Omega)\subset L^2(\Omega)$ is called the Nikodym domain — for details see pages 329–330 in "Functional analysis. Theory and applications" by R.E. Edwards (corrected edition http://bookza.org/book/763103/086133). The space $D^{k,p}(\Omega)$ was introduced by Sobolev in 1930-ies, and later studied in his book "Some applications of functional analysis in mathematical physics" originally published in Russian in 1950, while the English translation was published by the AMS in 1991 as Translations of Mathematical Monographs, 90 (see spaces $L_p^l$ therein http://bookza.org/book/685689/f6d2c1). See also spaces $L_2^{(m)}$ in the book "The theory of cubature formulas"  by Sobolev & Vaskevich http://bookza.org/book/2129623/ee52eb.

It might be helpful to remark that among the norms introduced so far on $D^{k,p}(\Omega)$, most suitable in all respects seems to be the norm $$ \|u\|_{D^{k,p}(\Omega)}=\|u\|_{L^p(\omega)}+ \sum_{|\alpha|=k}\|D^{\alpha}u\|_{L^p(\Omega)} $$ with some chosen bounded open subset $\omega\subset\Omega$,  e.g., some open ball.