# Trace theorems for arbitrary differentiability $k$, with embedding constants under control as $k\to\infty$

The usual trace theorem (with non-optimal exponents, but I don't care for those at the moment) says that

$$W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega)$$ for Lipschitz domains.

When $\Omega=\mathbb{R}^n_+$ is an upper half space, we even have

$$W^{k,p}(\mathbb{R}^n_+)\hookrightarrow W^{k-1,p}(\mathbb{R}^{n-1})\quad\forall k\in\mathbb{N}$$ and even more, the embedding constant can be chosen independent of $k$!

However, I have already problems controlling the embedding constants in the case where $\Gamma\subset\partial\Omega\subset\mathbb{R}^n$ is (part of) a graph of a polynomial. It is easily shown that

$$\|f\|_{W^{k-1,p}(\Gamma)}\leq C(k)\|f\|_{W^{k,p}(\Omega)}$$ for finite $C(k)$, (works for any smooth boundary) and one way to bound $C(k)$ is: If $\phi\colon U\subset\mathbb{R}^{n-1}\to\Gamma$ is a polynomial of degree $m$ that parametrizes $\Gamma$, then we have to check that $f\circ \phi\in W^{k-1,p}(U)$. All boils down (by chain rule) to showing $$\|\sum_{|\alpha|=0}^{k-1}((D^{\alpha}f)\circ\phi) H_{\alpha,\phi}\|_{L^p(U)}\leq C(k)\|f\|_{W^{k,p}(\Omega)}$$ where $H_{\alpha,\phi}$ is a sum of products of powers of derivatives of $\phi$ up to order $|\alpha|$. Using that all derivatives of $\phi$ vanish for $|\alpha|\geq m$, I can show with very crude arguments that $\|H_{\alpha,\phi}\|_{L^\infty(U)}\leq C_0^{|\alpha|^{C(n,m)}}$ , and conclude by applying the usual trace theorem (say with embedding constant $C_1$) to all $D^\alpha f$ and the $L^\infty$ bound to $H_{\alpha,\phi}$: $$\|\sum_{|\alpha|=0}^{k}((D^{\alpha}f)\circ\phi) H_{\alpha,\phi}\|_{L^p(U)}\leq C_0^{k^{C(n,m)}}\sum_{|\alpha|=0}^{k-1}\|((D^{\alpha}f)\circ\phi) \|_{L^p(U)}\\ \leq C_0^{k^{C(n,m)}}C_1\sum_{|\alpha|=0}^{k-1}\|f\|_{W^{|\alpha|+1,p}(\Omega)}\\ \leq C_1 k^dC_0^{k^{C(n,m)}}\|f\|_{W^{k,p}(\Omega)}$$

This should also work if $\Gamma$ is a rigid transformation of a (part of a ) graph of a polynomial (by IFT); however, then the calculations get even messier.

My questions: Are there better bounds on $C(k)$ than the one I described? Are there bounds for more general classes of boundaries than piecewise polynomial boundaries? And related to all this: is there a more intrinsic way (than using parametrizations) that allows one to capture statements like

"the boundary has only $m$ non-vanishing derivatives" (true for pieceiwse polynomials), or

"all derivatives of the boundary are uniformly bounded" (think of a piecewise $sin(x)$ boundary in $R^2$)

Intuitively, I would think a ball is just as good as a piecewise polynomial boundary, but I fail to completely to show any bound in this case.