Let me give an estimate only for the first derivatives of $v:=P_1 h$ in terms of the sup-norm of $h$, since the argument for second derivatives is similar. 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 other derivatives are similar.

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 $h=1$).

Let me give an estimate only for the first derivatives of $v:=P_1 h$ in terms of the sup-norm of $h$, since the argument for second derivatives is similar. 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.

Let me give an estimate only for the first derivatives of $v:=P_1 h$ in terms of the sup-norm of $h$, since the argument for second derivatives is similar. 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 other derivatives are similar.