Boundedness of an extension operator

Question

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. 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.

Answer 1

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 $h=1$).