Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$

Let $$\Omega \subset \mathbb{R}^3$$ be a lipschitz domain. We then have the trace operator $$\tau : H^1(\Omega) \to L^2(\partial \Omega)$$ and can define the space $$H^{1/2}(\partial \Omega) := \tau(H^1(\Omega)),$$ which is equipped with the norm $$\lVert g\rVert_{H^{1/2}(\partial \Omega)}:=\inf_{v\in H^1(\Omega), \tau(v)=g} \lVert v\rVert_{H^1(\Omega)}.$$

Finally, let $$H^{-1/2}(\partial \Omega)$$ denote the dual space of $$H^{1/2}(\partial \Omega)$$. We can view $$L^2(\partial \Omega)$$ as a subspace of $$H^{-1/2}(\partial \Omega)$$ via the linear map $$\begin{split} L:L^2(\partial \Omega)&\to H^{-1/2}(\partial \Omega),\\ f &\mapsto L_f, \end{split}$$ where $$L_f(\tau(\phi)):= \int_{\partial \Omega} f\cdot \tau(\phi)\,dS,\quad \phi \in H^1(\Omega).$$

$$L$$ is injective by the fundamental lemma of the calculus of variations and also bounded: The inequality $$\lvert L_f(\tau(\phi))\rvert \leq \lVert f\rVert_{L^2(\partial \Omega)} \lVert \tau(\phi)\rVert_{L^2(\partial \Omega)}\leq C \lVert f\rVert_{L^2(\partial \Omega)} \lVert\phi\rVert_{H^1(\Omega)}$$ implies that $$\lvert L_f(\tau(\phi))\vert\leq C\lVert f\rVert_{L^2(\partial \Omega)}\lVert\tau(\phi)\rVert_{H^{1/2}(\partial \Omega)},$$ so $$\lVert L_f\rVert_{H^{-1/2}(\partial \Omega)}\leq C\lVert f\rVert_{L^2(\partial \Omega)}.$$

Does there exists a constant $$C>0$$ such that $$(+)\quad\lvert\lvert f\rvert\rvert_{L^2(\partial \Omega)}\leq C \lVert L_f\rVert_{H^{-1/2}(\partial \Omega)}\quad \forall f\in L^2(\partial \Omega)$$ or a similar inequality holds?

My question is motivated by the following: For $$u\in L^2(\Omega)^3$$, a function $$v\in L^2(\Omega)^3$$ is called the weak curl of $$u$$ if $$\int_{\Omega} u\cdot \text{curl }\phi\,dx = \int_{\Omega} v\cdot \phi\,dx\quad\forall \phi\in C_0^{\infty}(\Omega)^3.$$ We write $$v=\text{curl } u$$. We can introduce the hilbert space $$H(\text{curl}, \Omega)$$ of all functions $$u\in L^2(\Omega)^3$$ for which a weak curl exists with the norm $$\lvert\lvert u\rvert\rvert_{H(\text{curl},\Omega)}^2:= \lvert\lvert u\rvert\rvert_{L^2(\Omega)^3}^2 + \lvert\lvert \text{curl } u\rvert\rvert_{L^2(\Omega)^3}^2.$$ One can then show the following theorem (for a reference, see Theorem 3.29 in "Finite Element Methods for Maxwell's Equations" by Monk):  Let $$\Omega\subset \mathbb{R}^3$$ be a bounded lipschitz domain. Then the trace map $$\gamma_t$$ which is defined classically by $$\gamma_t(v) := \nu \times (v\rvert_{\partial \Omega}),\quad v\in C^{\infty}(\overline{\Omega})^3$$ (where $$\nu$$ is the outward normal vector of $$\Omega$$) can be extended by continuity to a continuous linear map $$\gamma_t:H(\text{curl},\Omega)\to H^{-1/2}(\partial \Omega)^3$$. Furthermore, the following Green's theorem holds for any $$v\in H(\text{curl},\Omega)$$ and $$\phi\in H^1(\Omega)^3$$: $$\gamma_t(v) (\phi) = \int_{\Omega} \text{curl } v \cdot \phi - v\cdot \text{curl } \phi\,dx.$$  By the discussion above, we have $$L^2(\partial \Omega)^3 \subset H^{-1/2}(\partial \Omega)^3$$. In other words, for some $$v\in H(\text{curl},\Omega)$$, the functional $$\gamma_t(v)$$ can be expressed by a function $$f\in L^2(\partial \Omega)^3$$, i.e. $$\gamma_t(v)(\phi) = \int_{\partial \Omega} f\cdot \tau(\phi)\,dx\quad \forall \phi \in H^1(\Omega)^3.$$ Since $$\gamma_t$$ is bounded, we have $$\lvert\lvert \gamma_t(v)\rvert\rvert_{H^{-1/2}(\partial \Omega)^3}\leq C\lvert\lvert v\rvert\rvert_{H(\text{curl},\Omega)}.$$ My question is whether we have a similar inequality for $$\lvert\lvert f\rvert\rvert_{L^2(\partial \Omega)^3}$$: $$\lvert\lvert f\rvert\rvert_{L^2(\partial \Omega)^3} \leq C\lvert\lvert v\rvert\rvert_{H(\text{curl},\Omega)}.$$ This would follow immediatly from $$(+)$$.

• Welcome to MO! Since I assume you plan to do something with this, could you maybe specify what that is; in particular, what exactly you mean by "bounded" here? (Or maybe kind of equivalently, how you want to understand $L^{-1}$.) Oct 10, 2023 at 14:31
• Thank you for your comment. I have changed my question and added some context. Oct 11, 2023 at 17:25
• Starting from "In other words", why can $\gamma_t(v)$ be expressed by $f \in L^2(\partial\Omega)$ if it is generically in $H^{-1/2}(\partial\Omega)$? Oct 16, 2023 at 14:32
• By "for some $v$" i mean that for certain $v$ (but not for all) the functional $\gamma_t(v)$ can be expressed by a function $f$, for example $f=0$ for $v=0$ (if that was your question) Oct 16, 2023 at 20:35

Lemma. If $$A \colon X \supseteq D \to Y$$ is a closed operator between Banach spaces with domain $$D$$, then $$\|Ax\|_Y \lesssim \|x\|_X \quad \text{for all}~x\in D$$ if and only if $$D$$ is a closed subspace of $$X$$. In particular, if $$A$$ is in fact densely defined, then the foregoing inequality is equivalent to $$D = X$$.
The "if" direction follows with the closed graph theorem. For the "only if" part, observe that if $$D \supseteq (x_k) \to x$$ in $$X$$, then by the assumed inequality, $$(Ax_k)$$ is a Cauchy sequence in $$Y$$ and thus convergent, hence the closedness of $$A$$ implies that $$x \in D$$.
That being said, consider $$\gamma_t$$ as an unbounded operator $$D \subseteq H(\text{curl},\Omega) \to L^2(\partial\Omega)$$ with the domain $$D = \Bigl\{ v \in H(\text{curl},\Omega) \colon \gamma_t(v) \in L^2(\partial\Omega)\Bigr\}.$$
Using the Green type formula, you can show that this is in fact a closed operator. Moreover, since, say, continuous differentiable functions on the closure of $$\Omega$$ are included in $$D$$, it is also densely defined. Hence, with the Lemma, it follows that the desired inequality $$\|\gamma_t(v)\|_{L^2(\partial\Omega)} \lesssim \|v\|_{H(\text{curl},\Omega)} \qquad \text{for all}~v \in D$$ is equivalent to $$D = H(\text{curl},\Omega)$$ which is, I suppose, certainly not correct.