Let $d \ge 2$ be a positive integer. For $x=(x\_1,\dotsc,x\_{d-1},x\_d)$, we write $x'=(x\_1,\dotsc,x\_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x\_d) \mid x\_d>0\}$ denote the $d$-dimensional upper half-space.


Then, we obtain the following result.



> 
> Let $u \in C^2\_c(\mathbb{R}^{d-1})$, $v \in C\_c^1(\mathbb{R}^{d-1})$.
> Then, there exists $w \in C\_b^2(\mathbb{H}^d\cup \partial \mathbb{H}^d)$ such that
> $w(x',0)=u(x')$ and $\partial\_d w(x',0)=v(x')$ for every $x' \in \mathbb{R}^{d-1}$.
> 
> 
> 


This assertion in proved in Lemma B.1 in [Baur and Grothaus - Construction and Strong Feller Property of Distorted Elliptic Diffusion with Reflecting Boundary](https://doi.org/10.1007/s11118-013-9355-8).
In the proof, it is important to introduce a bounded linear operator.


Let $\varphi \in C\_c^2(\mathbb{R}^{d-1})$ such that $\int\_{\mathbb{R}^{d-1}}\varphi(z)\,dz=1.$ We define a linear operator $P\_1\colon C\_c^2(\mathbb{R}^{d-1}) \to C\_b^2(\mathbb{H}^d\cup \partial \mathbb{H}^d)$ by
\begin{align\*}
P\_1h(x)=x\_d \int\_{\mathbb{R}^{d-1}}\varphi(\zeta)h(x'+x\_d\zeta)\,d\zeta,\quad x=(x',x\_d).
\end{align\*}
The authors of the above paper states that the operator $P\_1$ is extended to a bounded linear operator from $C\_c^1(\mathbb{R}^{d-1})$ to $C\_b^2(\mathbb{R}^d \cap \{x\_d \ge 0\})$. This means that there exists $C>0$ such that for every $h \in C\_c^2(\mathbb{R}^{d-1})$,
\begin{align}
&\sup\_{x \in \mathbb{R}^d \cap \{x\_d \ge 0\}} \{|P\_1h(x)|+|\nabla P\_1h(x)|+\|\nabla^2 P\_1h(x)\|\} \\
&\le C \sup\_{z \in \mathbb{R}^{d-1}}\{|h(z)|+|\nabla h(z)|\} \tag{A}
\end{align}
Here, $\|\nabla^2 P\_1h(x)\|$ denotes the Hilbert—Schmidt norm of the Hessian matrix $\nabla^2P\_1h$.


However, I could not understand why the estimate (A) holds. if you know the proof of this estimate, please let me know.



Boundedness of an extension operator
Let me give an estimate for the first derivatives of $v:=P\_1 h$ in terms of the sup-norm of $h$.
Setting $z=x'+x\_d \xi$ we get
$$v(x)=\int\_{\mathbb {R}^{d-1}}\phi \left (\frac{z-x'}{x\_d}\right )h(z) x\_d^{2-d} \, dz.
$$
Then
$$v\_{x\_d}(x)=(2-d)\int\_{\mathbb {R}^{d-1}}\phi \left (\frac{z-x'}{x\_d}\right )h(z) x\_d^{1-d} \, dz+\int\_{\mathbb {R}^{d-1}}\psi \left (\frac{z-x'}{x\_d}\right )h(z) x\_d^{1-d} \, dz
$$
with $\psi(y)=-y\cdot \nabla \phi (y),\ y \in \mathbb R^{d-1}$. Now the estimate follows since the scaling $x\_d^{1-d}$ makes constant the $L^1$ norms of the mollifiers.


The estimates for the tangential derivatives are similar. For second order derivatives, one first differentiate $h$ and then uses similar arguments. Note however that $v$ is not bounded for large $x\_d$ (take formally $h=1$).


