Definitions of negative order Sobolev spaces

Question

I am having a problem with the definition of the space $W^{-k,p}$. I use Adams's definition

$$ W^{-k,p} = \left\{T \in D'(\Omega) \ \middle| \sum \limits_{0 \leq |i| \leq k} (-1)^{|i|} \int_{\Omega} v_i D^{i} \phi \,dx \ \forall \phi \in D(\Omega), 0 \leq |i| \leq k\right\}, $$

and he says in his book Sobolev spaces that the dual space $(W_0^{k,q})'$ is isometrically isomorphic to $W^{-k,p}$ for all $1 \leq q < \infty$, where $q$ is the conjugate index of $p$. My question is if this is true also for $q = \infty$. I would really appreciate if you could tell me a source where this is proven/ claimed. This would really help me because I am trying to show that $W^{-1,1} \subset H^{-k}$ for some $k$. Do you have any other idea how I could show this embedding without using $W^{-1,1}$ as a dual space. Thank you very much in advance.

Answer 1

$W^{-k,p}$ is the dual of $W_0^{k,q}$ if $p>1$. For $p=1$ this is not a natural definition. You should use the alternative definition that $W^{-1,1}$ is the set of all distributions of the form $f_0+\sum_m{\partial f_m\over\partial x_m}$, where $f_0,f_m\in L^1$. Using this definition for embeddings into $H^{-k}$ spaces presents no problem.