Looking for a "trace inequality" of normal derivative in $L^1$

Question

Let $U = \Omega \times (0,\infty)$ with $\partial U = \Omega \times \{0\}$.

Let $w \in H^1(U)$ with $(w|_{\partial U})^{\frac 1 m} \in L^1(\partial U)$.

I am looking for a trace inequality of the form $$\lVert \frac{\partial w}{\partial\nu} \rVert_{L^1(\partial U)} \leq C\lVert w \rVert_{L^p(\partial U)}$$ for some $p$ where $\nu$ is the unit normal. And yes the norm over the boundary on the right hand side is not a typo.

Answer 1

This cannot hold. For $w \in H_0^1(U)$, the right-hand side is always $0$.

Moreover, the trace of the normal derivative is not well-defined for $w \in H^1(U)$, but your inequality even fails for smooth functions.