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 $(+)$.