Let $W^{k,p}$, for nonnegative integer $k$ and real number $1\leq p\leq\infty$ denote the usual Sobolev space. If we replace $k$ by a noninteger $s\geq 0$, then $W^{s,p}$ denotes the Bessel potential definition of the fractional Sobolev space.
Given any function $f \in W^{1,\infty}(\mathbb{R}^n) \cap W^{s-\frac{1}{p},p}(\mathbb{R}^n)$, for $s>1/p$, then it is well-known that I can find functions $f_1 \in W^{1,\infty}(\mathbb{R}^{n+1})$ and $f_2\in W^{s,p}(\mathbb{R}^{n+1})$ such that $$f_1(\cdot,0) = f_2(\cdot, 0) = f$$ and $$\|f_1\|_{W^{1,\infty}(\mathbb{R}^{n+1})}\lesssim \|f\|_{W^{1,\infty}(\mathbb{R}^{n})} \quad \text{and} \quad \|f_2\|_{W^{s,p}(\mathbb{R}^{n+1})}\lesssim \|f\|_{W^{s-\frac{1}{p},p}(\mathbb{R}^{n})}.$$ Essentially, this is the surjectivity of the trace operator. A priori, the functions $f_1$ and $f_2$ are not the same. Therefore, my question is the following.
Question. Given any $f\in W^{1,\infty}(\mathbb{R}^n) \cap W^{s-\frac{1}{p},p}(\mathbb{R}^n)$, is it possible to find an extension $g\in W^{1,\infty}(\mathbb{R}^{n+1}) \cap W^{s,p}(\mathbb{R}^{n+1})$ whose trace is $f$ and $$\|g\|_{W^{1,\infty}(\mathbb{R}^{n+1})}\lesssim \|f\|_{W^{1,\infty}(\mathbb{R}^{n})} \quad \text{and} \quad \|g\|_{W^{s,p}(\mathbb{R}^{n+1})}\lesssim \|f\|_{W^{s-\frac{1}{p},p}(\mathbb{R}^{n})}.$$
The issue I run into answering my question is that the constructions I have in mind for each function space are different. The case $W^{1,\infty}$ is easy. Just take a bump function $\chi$ identically one in a neighborhood of the origin in $\mathbb{R}$ and set $$f_1(x,x_{n+1}) := f(x)\chi(x_{n+1}), \qquad (x,x_{n+1})\in\mathbb{R}^{n}\times\mathbb{R}.$$ The case $W^{s,p}$ is harder, and the construction I know (see Section 2.7.2 of H. Triebel's Theory of Function Spaces I) is based on Littlewood-Paley theory. Unfortunately, I am not aware of a Littlewood-Paley characterization of $W^{1,\infty}$ that would allow such a construction to work.