Recently I have read the paper *Whitney's problem on extendability of functions and an intrinsic metric* written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one proposition of this paper. Now let me state the background of this paper.

Let $\Omega$ be a bounded connected open set in $\mathbb{R}^n$. Consider the following Sobolev function space
\begin{equation}\tag{#}\label{W1infty}
W^1_\infty(\Omega)=\{f\in C(\Omega):\forall \alpha\in \mathbb{Z}^n_+,|\alpha|=1,f^{(\alpha)}\in L^\infty(\Omega)\}.
\end{equation}
(This definition follows from E. M. Stein, *Singular Integrals and Differentiability Properties of Functions*, Princeton University Press, 1970, Chap. V, Section 6.2.)
Here $|\alpha|=\sum_{i=1}^n \alpha_i$ for $\alpha=(\alpha_1,\cdots,\alpha_n)\in \mathbb{Z}^n_+$, $f^{(\alpha)}$ denotes the corresponding distributional partial derivative, $C(\Omega)$ denotes the space of continuous functions, $L^\infty(\Omega)$ denotes the space of essentially bounded functions on $\Omega$.

Let $W^1_\infty(\mathbb{R}^n)|_\Omega$ denote the space of restrictions to $\Omega$ of functions from $W^1_\infty(\mathbb{R}^n)$.

For $x,y\in\Omega$, let
\begin{equation*}
d_\Omega(x,y)=\text{infimum of lengths of polygonal paths in $\Omega$ joining $x$ and $y$}.
\end{equation*}
$d_\Omega(x,y)$ is called the *intrinsic metric* in $\Omega$.

Now fix $x\in \Omega$, consider the function $f(y)=d_\Omega(x,y)$. Then $f\in W^1_\infty(\Omega)$ and \begin{equation}\tag{$*$}\label{1} \sup_{\substack{z\in \Omega\\ |\alpha|=1}} |f^{(\alpha)}(z)|=1. \end{equation} If $W^1_\infty(\Omega)\subset W^1_\infty(\mathbb{R}^n)|_\Omega$, then $f$ is extendible to $\tilde{f}\in W^1_\infty(\mathbb{R}^n)$ and the open mapping theorem guarantees that on can choose $\tilde{f}$ such that \begin{equation}\tag{$**$}\label{2} \sup_{\substack{z\in \mathbb{R}^n\\ |\alpha|=1}} |f^{(\alpha)}(z)| \leqslant C\sup_{\substack{z\in \Omega\\ |\alpha|=1}} |f^{(\alpha)}(z)|=C. \end{equation} So, \begin{align} d_\Omega(x,y)&=|f(y)-f(x)|=|\tilde{f}(y)-\tilde{f}(x)|\notag\\ &\leqslant \sup_{\substack{z\in \mathbb{R}^n\\ |\alpha|=1}}|\tilde{f}^{(\alpha)}(z)|\cdot |x-y| \leqslant C|x-y|.\tag{$***$}\label{3} \end{align}

**My questions are:**

Why $f\in W^1_\infty(\Omega)$ and \eqref{1} holds?

How to use the open mapping theorem and why \eqref{2} holds?

Why the first inequality of \eqref{3} holds?

About Question 1, I know $f$ is Lipschitz with constant $1$ under the intrinsic metric, that is, \begin{equation*} |f(z_1)-f(z_2)|\leqslant d_\Omega(z_1,z_2),\quad \forall z_1,z_2\in\Omega. \end{equation*} But I don't know whether this property can help us deal with this question.

****************************************

Here I provide more materials of those questions.

This paper defined the Sobolev space as follows: \begin{equation*} W^{k+1}_\infty(\Omega)=\{f\in C^k(\Omega): \forall \alpha\in \mathbb{Z}^n_+,|\alpha|=k+1,f^{(\alpha)}\in L^\infty(\Omega)\}, \end{equation*} where $C^k(\Omega)$ denotes the space of $k$ times continuously differentiable functions. Others are same as before. Previously, I was also confused of $C^k(\Omega)$, but unfortunately, this paper does not provide additional information about it. Since I only need $k=0$, I wrote it as \eqref{W1infty}.

The definition of Sobolev space in E. M. Stein's book (pp. 121-122, Chap. V, Sect. 2): for any nonnegative integer $k$ and $1\leqslant p\leqslant \infty$, the Sobolev space $L^p_k(\mathbb{R}^n)$ is defined as the space of functions $f$, with $f\in L^p(\mathbb{R}^n)$ and where all $f^{(\alpha)}$ exist and $f^{(\alpha)}\in L^p(\mathbb{R}^n)$ in the distributional derivative sense, whenever $|\alpha|\leqslant k$. The space of functions can be normed by the expression \begin{equation*} \|f\|_{L^p_k(\mathbb{R}^n)}=\sum_{|\alpha|\leqslant k}\|f^{(\alpha)}\|_{L^p(\mathbb{R}^n)}. \end{equation*}

The Stein's book also writes the following (pp. 159, Chap. V. Sect. 6.2). $f$ belongs to $L^\infty_k(\mathbb{R}^n), k\geqslant 1$ if and only if $f$ can be modified on a set of measure zero so that $f$ has continuous partial derivatives of total order $\leqslant k-1$. Moreover, whenever $g=f^{(\alpha)}$, $|\alpha|\leqslant k-1$, then \begin{equation*} \sup_x |g(x)|<\infty,\quad \text{and}\quad \sup_{x,x'}\dfrac{|g(x)-g(x')|}{|x-x'|}<\infty. \end{equation*}