About equivalent definitions of $H^{-1}$ Sobolev norm Recently, I have been troubled by two notions of norms for the $H^{-1}$ Sobolev space. To make the discussion concrete, I would like to clarify that we define $H^{-1}(\Omega) := H^1(\Omega)^*$, where $\Omega$ is a given smooth, bounded domain. Note that this definition is different from the one showing up in the classical PDE textbook by Evans, who defines the space as the dual of $H^1_0(\Omega)$.
Indeed, it is clear that one can define the norm on $H^{-1}(\Omega)$ using the operator norm, namely for any $f \in H^{-1}(\Omega)$, we have
$$
|f|_{H^{-1}}= \sup_{g \in H^1(\Omega), |g|_{H^1} = 1}\left|\int_\Omega fg dx\right|.
$$
On the other hand, one may use the inverse Dirichlet Laplacian to give:
$$
|f|_{H^{-1}}= \int_\Omega f(-\Delta)^{-1}f dx,
$$
where $(-\Delta)^{-1}$ denotes the inverse Dirichlet Laplacian on $\Omega$, which is a positive operator and thus defines a norm.
My question is: is it true that the two definitions above coincide? If so, could anyone kindly gives an explanation or points me to a reference? Thanks a lot in advance!
EDIT: Indeed, one direction is clear by elliptic regularity, as we observe that $(-\Delta)^{-1}f \in H^1_0$ for any $f \in H^{-1}$.
 A: I think now I have a definite answer to this question. Conclusions first: the two definitions above DO NOT coincide. In fact, the norm
$$
|f|_{-1} = \int f(-\Delta_D)^{-1}f dx
$$
actually defines not a norm on $H^{-1}$, but on $H^{-1}_0 := (H^1_0)^*$. Moreover, this definition is equivalent to the operator norm
$$
|f|_{op} = \sup_{g \in H^1_0, |g|_1 = 1}\left|\int fg dx\right|,
$$
where I used $|\cdot|_1$ to denote the $H^1$ norm. Here is the proof: (in the proof, I will write $A \lesssim B$ if there exists some generic constant $C$ only depending on the domain $\Omega$ such that $A \le CB$)

*

*$|f|_{-1} \lesssim |f|_{op}$: by elliptic regularity, we immediately have the following estimate:
$$
\|(-\Delta_D)^{-1}f\|_1 \lesssim \|f\|_{-1}.
$$
Moreover, we note that $(-\Delta_D)^{-1}f \in H^1_0$ due to definition of Dirichlet Laplacian. We simply take $g = C(-\Delta_D)^{-1}f$, where $C$ is a normalizing coefficient such that $|g|_1 = 1$. This direction is thus done.


*$|f|_{op} \lesssim |f|_{-1}$: for this direction, by spectral properties of $-\Delta$ on a smooth, bounded domain $\Omega$, we let $\lambda_k^2 > 0$ be the eigenvalues and $e_k$ be the corresponding eigenfunctions. Then we have the following equivalent definitions of $H^1_0$ norm and $|\cdot|_{-1}$: for $g \in H^1_0$, $f \in H^{-1}_0$,
$$
|g|_1^2 \sim \sum_k \lambda_k^2|\hat{g}_k|^2,\; |f|_{-1}^2 = \sum_k \lambda_k^{-2}|\hat{f}_k|^2,
$$
where $\hat{g}_k,\hat{f}_k$ denote the Fourier coefficients with respect to basis $(e_k)_k$. (Note that I have implicitly used Poincare inequality when I define $|g|_1$. This is valid as $g$ has zero trace.) Now we pick arbitrary $g \in H^1_0$, $|g|_1 = 1$. Observe that by Cauchy-Schwarz:
$$
\left|\int fg dx\right| \le \sum_k |\lambda_k||\lambda_k|^{-1}|\hat{g}_k\hat{f}_k| \le (\sum_k \lambda_k^2|\hat{g}_k|^2)^{1/2}(\sum_k \lambda_k^{-2}|\hat{f}_k|^2)^{1/2}\le |f|_{-1}.
$$
We are done by taking supremum over $g$.
