$\int_{-\infty}^{\infty}\ldots\,\mathrm{d}a$ by a complex contour integral
$\int_{\Gamma_{\varepsilon}}\ldots\,\mathrm{d}\omega$ for a complex path
$\Gamma_{\varepsilon}$ which is parametrised by

$\gamma_{\varepsilon}(s)=s+\mathrm{i}\varepsilon\left(\left|\lambda\right|+(r^{2}+s^{2})^{m}\right)^{1/2m},\quad
s\in(-\infty,\infty)$

for some (sufficiently small) parameter $\varepsilon>0$.

Figure 3. The path $\Gamma_{\varepsilon}$.

Hence,

$\displaystyle\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},\lambda)$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{\Gamma_{\varepsilon}}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}\frac{1}{(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{2}}$
$\displaystyle\qquad\cdot\boldsymbol{F}_{1}(\lambda,a,r,\boldsymbol{\varphi},y)\boldsymbol{F}_{2}(\lambda,a,r,\boldsymbol{\varphi})\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\sigma(\boldsymbol{\varphi}).$

We proceed just as in the proof of [17, Proposition 6.5]. First, we find that

$\left|\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|\gamma_{\varepsilon}(s)}\right|=\mathrm{e}^{-\left|\boldsymbol{x}^{\prime}\right|\operatorname{Im}\,(\gamma_{\varepsilon}(s))}=\mathrm{e}^{-\varepsilon\left|\boldsymbol{x}^{\prime}\right|(\left|\lambda\right|+(r^{2}+s^{2})^{m})^{1/2m}}.$

Moreover, we may estimate the function $\boldsymbol{F}_{1}$ as follows:

$\left\|\boldsymbol{F}_{1}(\lambda,\gamma_{\varepsilon}(s),r,\boldsymbol{\varphi},y)\right\|_{\operatorname{\mathcal{B}}(E^{n})}\leq
M\mathrm{e}^{-c_{1}y}.$

In fact, from $\sup\\{\operatorname{Re}\,\mu:\mu\in
S_{-}(\boldsymbol{b},\sigma)\\}<-c_{1}<0$ it follows that for every
$(\boldsymbol{b},\sigma)\in\mathbb{S}^{n-1}\times\Sigma_{\pi-\phi}$ with
$\left|\sigma\right|\leq 1$, there is a constant
$M_{\boldsymbol{b},\sigma}\geq 1$ such that

$\left\|\mathrm{e}^{\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}P_{-}(\boldsymbol{b},\sigma)\right\|_{\operatorname{\mathcal{B}}(E^{n})}\leq
M_{\boldsymbol{b},\sigma}\mathrm{e}^{-c_{1}y}\quad\text{for all }y\geq 0.$

Here,
$\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\in\operatorname{\mathcal{B}}(E^{2m})$
is a continuous linear operator and depends continuously on the parameters
$(\boldsymbol{b},\sigma)\in\mathbb{S}^{n-1}\times\Sigma_{\pi-\phi}$.
Therefore, each (uniformly continuous) semigroup
$\big{(}\mathrm{e}^{\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\big{)}_{y\geq
0}\subseteq\operatorname{\mathcal{B}}(E^{n})$ satisfies the spectral-
determined growth bound condition and, hence, by (9) its restriction to the
image under the spectral projection $P_{-}(\boldsymbol{b},\sigma)$ on the
stable part has a growth bound which is strictly less than $-c_{1}$. Moreover,
as $\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)$ depends holomorphically on the
parameters $(\boldsymbol{b},\sigma)$, so do the (uniformly continuous)
semigroups
$\big{(}\mathrm{e}^{\mathrm{i}A_{0}(\boldsymbol{b},\sigma)y}\big{)}_{y\geq
0}$. Thus, the constant $M_{\boldsymbol{b},\sigma}$ can be chosen uniformly as
$M\geq 1$ for all
$(\boldsymbol{b},\sigma)\in\mathbb{S}^{n-1}\times\Sigma_{\pi-\phi}$ with
$\left|\sigma\right|\leq 1$. This establishes the estimate on
$\boldsymbol{F}_{1}$.
For the function $\boldsymbol{F}_{2}$, observe that

$\left\|\boldsymbol{F}_{2}(\lambda,a,r,\varphi)\right\|_{\operatorname{\mathcal{B}}(E^{n})}=\left\|\boldsymbol{M}((\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-1/2m}\boldsymbol{Q}^{\mathsf{T}}(a,r\boldsymbol{\varphi}),\lambda(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-1})\right\|_{\operatorname{\mathcal{B}}(E^{n})},$

where
$(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-1/2m}\boldsymbol{Q}^{\mathsf{T}}(a,r\boldsymbol{\varphi})\in\operatorname{\mathbb{R}}^{n}$
has norm less or equal $1$ and
$\lambda(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-1}\in\Sigma_{\pi-\phi}$ has
modulus less or equal $1$. Since the map
$(\boldsymbol{b},\sigma)\mapsto\boldsymbol{M}(\boldsymbol{b},\sigma)\in\operatorname{\mathcal{B}}(E^{n})$
is holomorphic,

$\sup_{(\lambda,a,r,\boldsymbol{\varphi})}\left\|\boldsymbol{F}_{2}(\lambda,a,r,\boldsymbol{\varphi})\right\|_{\operatorname{\mathcal{B}}(E^{n})}<\infty,$

where we take the supremum over all admissible values for
$(\lambda,a,r,\boldsymbol{\varphi})$. For the calculations following next, let
us remark that the functions

$\rho(\lambda,s,r)=\big{(}\lambda+(s^{2}+r^{2})^{m}\big{)}^{1/2m}\quad\text{and}\quad\tilde{\rho}(\lambda,s,r)=\big{(}\lambda^{1/m}+s^{2}+r^{2}\big{)}^{1/2}$

on
$\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}$
(the latter having been the choice employed in [17]) are comparable in the
sense that there is a constant $c>0$ such that

$\frac{1}{c}\tilde{\rho}(\lambda,r,s)\leq\rho(\lambda,r,s)\leq
c\tilde{\rho}(\lambda,r,s).$

This can be seen, e.g., by employing the binomial formula to estimate

$\tilde{\rho}(\lambda,s,r)^{2m}=\sum_{k=0}^{m}\binom{m}{k}\left|\lambda\right|^{k/m}(s^{2}+r^{2})^{m-k}\leq
C(\left|\lambda\right|+(s^{2}+r^{2})^{m})=\rho(\lambda,s,r)^{2m}$

and the trivial estimate

$\rho(\lambda,s,r)^{2}=\big{(}\left|\lambda\right|+(s^{2}+r^{2})^{m}\big{)}^{1/m}\leq\big{(}2\left|\lambda\right|\big{)}^{1/m}+\big{(}2(s^{2}+r^{2})^{m}\big{)}^{1/m}=2^{1/m}\tilde{\rho}(\lambda,s,r)^{2}.$

We may, therefore, readily estimate
$\mathrm{D}^{\boldsymbol{\alpha}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y)$
for every multi-index
$\boldsymbol{\alpha}=(\boldsymbol{\alpha}^{\prime},\alpha_{n+1})\in\operatorname{\mathbb{N}}_{0}^{n+1}$:

$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y)\right\|_{\operatorname{\mathcal{B}}(E^{2m})}$
$\displaystyle=\left\|\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\alpha}^{\prime}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}^{\prime}}\mathrm{D}_{y}^{\alpha_{n+1}}(\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda))\,\mathrm{d}\boldsymbol{\xi}^{\prime}\right\|_{\operatorname{\mathcal{B}}(E^{n})}$
$\displaystyle=\left\|\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{\Gamma_{\varepsilon}}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}(\rho\boldsymbol{b})^{\boldsymbol{\alpha}^{\prime}}\frac{1}{\left|\lambda\right|+(s^{2}+r^{2})^{m}}(\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma))^{\alpha_{n+1}}\boldsymbol{F}_{1}\boldsymbol{F}_{2}\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\mathbb{S}^{n-2}(\boldsymbol{\varphi})\right\|$
$\displaystyle\leq\frac{C}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\,\mathrm{d}\mathbb{S}^{n-2}(\boldsymbol{\varphi})\int_{0}^{\infty}r^{n-2}\int_{\Gamma_{\varepsilon}}\frac{1}{(\left|\lambda\right|+(s^{2}+r^{2})^{m})^{\frac{2m-\left|\boldsymbol{\alpha}\right|}{2m}}}\mathrm{e}^{-(\left|\lambda\right|+(s^{2}+r^{2})^{m})^{1/2m}\cdot(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)}|\mathrm{d}a|\,\mathrm{d}r$
$\displaystyle\leq
C_{\varepsilon_{0}}\int_{0}^{\infty}r^{n-2}\int_{0}^{\infty}\mathrm{e}^{-(\left|\lambda\right|+(s^{2}+r^{2})^{m})^{1/2m}\cdot(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)}\cdot\frac{1}{(\left|\lambda\right|+(s^{2}+r^{2})^{m})^{\frac{2m-\left|\boldsymbol{\alpha}\right|}{2m}}}\,\mathrm{d}s\,\mathrm{d}r$
$\displaystyle\leq
C_{\varepsilon_{0}}\int_{0}^{\infty}r^{n-2}\int_{r}^{\infty}\mathrm{e}^{-c(\left|\lambda\right|^{1/m}+(\tau-r)^{2}+r^{2})^{1/2}\cdot(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)}\cdot\frac{1}{(\left|\lambda\right|^{1/2m}+\tau)^{2m-\left|\boldsymbol{\alpha}\right|}}\,\mathrm{d}\tau\,\mathrm{d}r$
$\displaystyle\leq
C_{\varepsilon_{0}}\int_{0}^{\infty}r^{n-2}\int_{r}^{\infty}\mathrm{e}^{-c\varepsilon\cdot(\left|\lambda\right|^{1/2m}+\tau)\cdot\left|\boldsymbol{x}^{\prime}\right|}\mathrm{e}^{-cc_{1}(\left|\lambda\right|^{1/2m}+\tau)y}\cdot\frac{1}{(\left|\lambda\right|^{1/2m}+\tau)^{2m-\left|\boldsymbol{\alpha}\right|}}\,\mathrm{d}\tau\,\mathrm{d}r$
$\displaystyle\leq
C_{\varepsilon_{0}}\int_{0}^{\infty}r^{n-2}\int_{r}^{\infty}\mathrm{e}^{-c(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)\cdot(\left|\lambda\right|^{1/2m}+\tau)}\frac{1}{(\left|\lambda\right|^{1/2m}+\tau)^{2m-\left|\boldsymbol{\alpha}\right|}}\,\mathrm{d}\tau\,\mathrm{d}r$
$\displaystyle=C_{\varepsilon_{0}}\left|\lambda\right|^{\frac{1}{2m}-\frac{2m-\left|\boldsymbol{\alpha}\right|}{2m}+\frac{n-1}{2m}}\int_{0}^{\infty}s^{n-2}\int_{s}^{\infty}\mathrm{e}^{-c(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)\left|\lambda\right|^{1/2m}(1+\tilde{\tau})}\frac{1}{(1+\tilde{\tau})^{2m-\left|\boldsymbol{\alpha}\right|}}\,\mathrm{d}\tilde{\tau}\,\mathrm{d}s$
$\displaystyle=C_{\varepsilon_{0}}\left|\lambda\right|^{\frac{n}{2m}-1+\frac{\left|\alpha\right|}{2m}}\frac{1}{n-1}\int_{0}^{\infty}\frac{\tilde{\tau}^{n-1}}{(1+\tilde{\tau})^{2m-\left|\boldsymbol{\alpha}\right|}}\mathrm{e}^{-c\left|\lambda\right|^{1/2m}(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y)(1+\tilde{\tau})}\,\mathrm{d}\tilde{\tau}$
$\displaystyle=\frac{C_{\varepsilon_{0}}}{n-1}\left|\lambda\right|^{\frac{n}{m}-1+\frac{\left|\alpha\right|}{2m}}p_{2m,\left|\boldsymbol{\alpha}\right|-1}^{n+1}(c\left|\lambda\right|^{1/2m}(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+c_{1}y))$
$\displaystyle\leq
C_{\varepsilon_{0}}^{\prime}\left|\lambda\right|^{\frac{n-2m+\left|\boldsymbol{\alpha}\right|}{2m}}p_{2m,\left|\boldsymbol{\alpha}\right|-1}^{n+1}(c_{\varepsilon_{0}}\left|\lambda\right|^{1/2m}(\left|\boldsymbol{x}^{\prime}\right|+y))$

for some constants $C_{\varepsilon_{0}}$, $C_{\varepsilon_{0}}^{\prime}$ and
$c_{\varepsilon_{0}}>0$, for sufficiently small $\varepsilon_{0}>0$, and all
$\varepsilon\in(0,\varepsilon_{0}]$. Thus, we have established the kernel
estimate

$\displaystyle\left|\mathrm{D}^{\boldsymbol{\alpha}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y)\right|$
$\displaystyle\leq
M\left|\lambda\right|^{\frac{n-2m+\left|\boldsymbol{\alpha}\right|}{2m}}p_{2m,\left|\boldsymbol{\alpha}\right|-1}^{n+1}(c\left|\lambda\right|^{\nicefrac{{1}}{{2}}m}(\left|\boldsymbol{x}^{\prime}\right|+\left|y\right|))\quad\text{for
}\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0,\lambda\in\Sigma_{\pi-\phi}.$

∎

######  Proposition 4.8.

Let $\mathcal{A}(\mathrm{D})$ be a parameter-elliptic operator of order $2m$
with angle of ellipticity $\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$.
Let $\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ and assume that the
Lopatinskii–Shapiro condition is satisfied. Then, for every multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$, there are
constants $M,c>0$ such that

$\left|\mathrm{D}^{\boldsymbol{\alpha}}\boldsymbol{K}^{\mathrm{corr},\lambda}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})\right|\leq
M\left|\lambda\right|^{\frac{n+1-2m+\left|\boldsymbol{\alpha}\right|}{2m}}p_{2m,\left|\boldsymbol{\alpha}\right|}^{n+1}\left(c\left|\lambda\right|^{1/(2m)}(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})\right)$

for all
$\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y,\tilde{y}>0$ and
$\lambda\in\Sigma_{\pi-\phi}$, where

$\boldsymbol{K}^{\mathrm{corr},\lambda}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})=\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}^{\prime}}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\boldsymbol{h}^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\xi}^{\prime}.$

###### Proof.

Consider the inverse Fourier transform of the function

$\mathcal{F}w^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},y,\lambda)=-\big{[}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)(\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}})(\boldsymbol{\xi}^{\prime},0)\big{]}_{1}$

corresponding to the correction term for
$\mathcal{B}_{j}(\mathrm{D})\mathcal{F}w^{\mathrm{fs}}|_{\partial\operatorname{\mathbb{R}}_{+}^{n+1}}$,
hence,

$\displaystyle\boldsymbol{K}^{\mathrm{corr},\lambda}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})$
$\displaystyle=-\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\boldsymbol{h}_{j,k}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\xi}^{\prime}$

where
$\boldsymbol{h}_{j}^{\mathrm{corr}}=(0,\ldots,0,h_{j}^{\mathrm{corr}},0,\ldots,0)^{\mathsf{T}}=(0,\ldots,0,\sum_{k=0}^{m_{j}}h_{j,k}^{\mathrm{corr}},0,\ldots,0)^{\mathsf{T}}=\sum_{k=0}^{m_{j}}\boldsymbol{h}_{j,k}^{\mathrm{corr}}$.
We recall that the family
$\big{(}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\boldsymbol{h}_{j,k}(\boldsymbol{\xi}^{\prime},0)\big{)}_{y\geq
0}$ is the norm-continuous semigroup corresponding to the (parameter-
dependent) initial value problem

$\displaystyle\frac{\partial}{\partial
y}\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},y,\lambda)$
$\displaystyle=\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}_{j,k},\quad
y>0,$ $\displaystyle
B_{\tilde{j}}^{0}(\boldsymbol{b})\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},0,\lambda)$
$\displaystyle=\delta_{j,\tilde{j}}\sum_{\tilde{k}=0}^{m_{\tilde{j}}}\delta_{k,\tilde{k}}\mathcal{P}_{\tilde{j},\tilde{k}}\frac{\mathcal{F}g_{\tilde{j}}(\boldsymbol{\xi}^{\prime})}{\rho^{\tilde{k}}},\quad\tilde{j}=1,\ldots,m.$

Since the function $\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}$ is the solution
to the boundary value problem

$\displaystyle\mathrm{D}_{y}\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},y,\lambda)$
$\displaystyle=\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},y,\lambda),$
$\displaystyle\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0,$
$\displaystyle
B_{j}^{0}(\boldsymbol{b})\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle=\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}\frac{\mathcal{F}g_{j}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}},$
$\displaystyle
j=1,\ldots,m,\,\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},$

we deduce that

$\displaystyle\mathcal{F}w^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},y,\lambda)$
$\displaystyle=\big{[}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}(\mathcal{F}\boldsymbol{v}^{\mathrm{corr}})(\boldsymbol{\xi}^{\prime},0)\big{]}_{1}=\big{[}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}M(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)\big{]}_{1}$
$\displaystyle=\big{[}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\big{(}\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}\frac{\mathcal{F}g_{j}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}}\big{)}_{j}\big{]}_{1}.$

By Cauchy’s theorem and homogeneity considerations

$\displaystyle h_{j,k}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},\tilde{y})$
$\displaystyle=\rho^{k+1-2m}\int_{\operatorname{\mathbb{R}}}\mathcal{B}_{j,k}(\boldsymbol{b},a)(\sigma+\mathcal{A}(\boldsymbol{b},a))^{-1}\mathrm{e}^{-\mathrm{i}\tilde{y}\rho
a}\,\mathrm{d}a$
$\displaystyle=\rho^{k+1-2m}\int_{\Gamma^{-}}\mathcal{B}_{j,k}(\boldsymbol{b},a)(\sigma+\mathcal{A}(\boldsymbol{b},a))^{-1}\mathrm{e}^{-\mathrm{i}\tilde{y}\rho
a}\,\mathrm{d}a,$

where $\Gamma^{-}$ is a closed curve in the open lower complex plane which
encircles $-\mathrm{i}P_{+}(\boldsymbol{b},\sigma)$, uniformly for all
$(\boldsymbol{b},\sigma)$.

Figure 4. A path $\Gamma^{-}$ surrounding
$-\mathrm{i}P_{+}(\boldsymbol{b},a)$.

We then obtain, after using a rotation as above and shifting the path of
integration as in [17], that

(13)
$\left|h_{j,k}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},\tilde{y})\right|\leq
C\left(\left|\lambda\right|+r^{2m}+s^{2m}\right)^{\nicefrac{{k+1-2m}}{{2m}}}\mathrm{e}^{-c\tilde{y}(\left|\lambda\right|+r^{2m}+s^{2m})^{\nicefrac{{1}}{{2m}}}},\quad\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0$

for sufficiently small values of
$\left|\operatorname{arg}(r^{2}+a^{2})\right|$. For any multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$, we, thus, obtain
the estimate

$\displaystyle\left|\mathrm{D}^{\boldsymbol{\alpha}}K^{\mathrm{corr,\lambda}}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})\right|$
$\displaystyle\leq\left|\mathrm{D}^{\boldsymbol{\alpha}}K^{\mathrm{corr,\lambda}}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})\right|$
$\displaystyle\leq
M\left|\lambda\right|^{\frac{n-2m+1+\left|\boldsymbol{\alpha}\right|}{2m}}\int_{0}^{\infty}\frac{s^{n-1}}{(1+s)^{2m-1-\left|\boldsymbol{\alpha}\right|}}\mathrm{e}^{-c(s+1)\left|\lambda\right|^{1/(2m)}(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})}\,\mathrm{d}s,$

cf. the calculations on [17, p. 80]. ∎

###  4.4. Solution operators

Using the kernel estimates from the last subsection, we can now derive
$\LL_{p}$-estimates for $w^{\mathrm{fs}}$, $w^{\mathrm{hs}}$ and
$w^{\mathrm{corr}}$. For this purpose, let us introduce the
$\LL_{p}$-realisation for the boundary value problem (4)–(5) as the closure
$A\&B=\overline{(A\&B)_{\mathrm{min}}}$ of the _minimal operator_
$(A\&B)_{\mathrm{min}}$ which is defined by

$\displaystyle(A\&B)_{\mathrm{min}}:\operatorname{\operatorname{dom}}((A\&B)_{\mathrm{min}})$
$\displaystyle=\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\subseteq\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
$\displaystyle\rightarrow\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\times\prod_{j=1}^{m}\left(\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\right),$
$\displaystyle(A\&B)_{\mathrm{min}}u$
$\displaystyle=\left(\begin{array}[]{c}\mathcal{A}(\mathrm{D})\\\
\mathcal{B}_{1}(\mathrm{D})\\\ \vdots\\\
\mathcal{B}_{m}(\mathrm{D})\end{array}\right)u.$

######  Remark 4.9.

Note that the differential operators $\mathcal{B}_{j}(\mathrm{D})$
corresponding to the boundary symbols act on the function
$u:\operatorname{\mathbb{R}}_{+}^{n+1}\rightarrow E$ and are evaluated on the
half-space $\operatorname{\mathbb{R}}_{+}^{n+1}$, and not only on its boundary
trace $\partial\operatorname{\mathbb{R}}_{+}^{n+1}$.
Also, it may not be immediately clear that the operator
$(A\&B)_{\mathrm{min}}$ is closable, thus let us comment on that. We may
introduce the higher-order reflexion
$\mathcal{E}u(\boldsymbol{x}^{\prime},y):=\sum_{i=0}^{2m}\alpha_{i}u(\boldsymbol{x}^{\prime},-\frac{y}{i+1})$
for $\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ and $y<0$, which
for suitable chosen scalar coefficients
$\alpha_{i}\in\operatorname{\mathbb{R}}$ defines a continuous extension of
functions $u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ to
functions
$\mathcal{E}u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}^{n+1};E)$. Then,
given any sequence $(u_{n})_{n\geq
1}\subseteq\operatorname{\operatorname{dom}}((A\&B)_{\mathrm{min}})$ such that
$u_{n}\rightarrow 0$ and $\mathcal{A}(\mathrm{D})u_{n}\rightarrow v$ in
$\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ for some function
$v\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, from the definition of
the extension operator it follows that $\mathcal{E}u_{n}\rightarrow 0$ and
$\mathcal{A}(\mathrm{D})u_{n}\rightarrow\tilde{v}$ in
$\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E)$ for the extension
$\tilde{v}=\mathcal{E}v\in\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E)$ of $v$.
We claim that $\tilde{v}=0$, which will imply that
$v=v|_{\operatorname{\mathbb{R}}_{+}^{n+1}}=0$ and, hence,
$(A\&B)_{\mathrm{min}}$ is closable. To this end, we use that
$\mathcal{A}(\boldsymbol{\xi})$ is parameter-elliptic, so that for every
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|\leq 2m$, the function
$\boldsymbol{\xi}^{\boldsymbol{\alpha}}(\mathcal{A}(\boldsymbol{\xi})+1)^{-1}$
is an $\mathcal{R}$-bounded Fourier multiplier. From
$(\mathcal{A}(\mathrm{D})+1)\mathcal{E}u_{n}=\mathcal{A}(\mathrm{D})\mathcal{E}u_{n}+\mathcal{E}u_{n}=\mathcal{E}(\mathcal{A}(\mathrm{D})u_{n}+u_{n})\rightarrow\tilde{v}$
it, thus, follows that also $\mathrm{D}^{\boldsymbol{\alpha}}\mathcal{E}u_{n}$
converges in $\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E)$ for each multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|\leq 2m$. Since
$\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}^{n+1};E)$ is a Banach space and
$\mathcal{E}u_{n}\rightarrow 0$, however, this can only be the case if
$\mathrm{D}^{\boldsymbol{\alpha}}(\mathcal{E}u_{n})\rightarrow\mathrm{D}^{\boldsymbol{\alpha}}(\mathcal{E}0)=0$,
which in turn implies that $\tilde{v}=0$. This shows that the operator
$(A\&B)_{\mathrm{min}}$ is indeed closable.

For each parameter $\lambda\in\Sigma_{\pi-\phi}$ and functions
$g_{j}\in\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
(for the moment defined on $\operatorname{\mathbb{R}}_{+}^{n+1}$, _not_ only
on $\partial\operatorname{\mathbb{R}}_{+}^{n+1}$) we consider the system of
differential equations

(14) $\lambda\boldsymbol{J}v+(A\&B)v=\left(\begin{array}[]{c}0\\\
\boldsymbol{g}\end{array}\right),$

where we write $\boldsymbol{g}=(g_{1},\ldots,g_{m})^{\mathsf{T}}$ and
$\boldsymbol{J}v=(v,0,\ldots,0)^{\mathsf{T}}\in E^{1+m}$ for $v\in E$, and get
the following result in which we consider two cases: On the one hand, the
situation as above or in [17, Proposition 6.8], where $\boldsymbol{g}$ is
given from the start, and on the other hand, the more subtle case where
$\boldsymbol{g}$ is only given on
$\partial\operatorname{\mathbb{R}}_{+}^{n+1}$, but we use a particular
extension to $\operatorname{\mathbb{R}}_{+}^{n+1}$ which allows us to assume
that $\partial_{y}g_{j,k}=-L_{\lambda}^{1/2m}g_{j,k}$ in
$\operatorname{\mathbb{R}}_{+}^{n+1}$, thus relating the partial derivative in
$y$-direction with a fractional differential operator acting on the
$\boldsymbol{x}^{\prime}$-components of the function $\boldsymbol{g}$.

######  Remark 4.10.

We will use that the pseudo-differential operator
$-L_{\lambda}^{1/2m}=-(\left|\lambda\right|+(-\Delta)^{m})^{1/2m}$, i.e. the
linear operator associated to the Fourier symbol
$-(\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m}$,
for fixed $\lambda\in\Sigma_{\pi-\phi}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$, generates a
bounded $\mathrm{C}_{0}$-semigroup on the Bochner–Lebesgue space
$\LL_{p}(\operatorname{\mathbb{R}}^{n};E)$ for any given $p\in[1,\infty)$. In
fact, for every $\mu>0$ and $\lambda\in\Sigma_{\pi-\phi}$, the resolvent
problem

$(\mu+L_{\lambda}^{1/2m})u=f$

can be solved in Fourier space as

$(\mu+(\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m})\hat{u}=\hat{f}\quad\Leftrightarrow\quad\hat{u}=\frac{1}{\mu+(\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m}}\hat{f}=:m_{\mu}(\boldsymbol{\xi}^{\prime})\hat{f},$

where $m_{\mu}(\boldsymbol{\xi}^{\prime})$ is scalar, and for boundedness of
the resolvent, it, thus, suffices to prove boundedness of
$m_{\mu}(\boldsymbol{\xi}^{\prime})$ and
$(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\alpha}}D_{\boldsymbol{\xi}^{\prime}}^{\boldsymbol{\alpha}}m_{\mu}(\boldsymbol{\xi}^{\prime})$
for all $\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n}$ of length
$\left|\boldsymbol{\alpha}\right|\leq\lfloor\frac{n}{2}\rfloor$, which can
easily be checked, and in fact gives
$\left\|(\mu(\mu+L_{\lambda}^{1/2m})^{-1})^{k}\right\|\leq C$ for all $\mu>0$.
Hence, by the Hille–Yosida Theorem $-L_{\lambda}^{1/2m}$ generates a bounded
$C_{0}$-semigroup on $\LL_{p}(\operatorname{\mathbb{R}}^{n};E)$, for every
$\lambda\in\Sigma_{\pi-\phi}$. For details, we refer to Proposition 7.1 in the
appendix.

We denote by $L_{\lambda}$ the operator
$\left|\lambda\right|+(-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}$ on
$\LL^{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, where
$\Delta_{\operatorname{\mathbb{R}}^{n}}$ is the closure of the minimal
realisation of the ($E$-valued) Laplace operator on
$\operatorname{\mathbb{R}}^{n}$, and extend the functions $g_{j,k}$ on the
boundary
$\partial\operatorname{\mathbb{R}}_{+}^{n+1}\cong\operatorname{\mathbb{R}}^{n}$
with use of the $C_{0}$-semigroup $(\mathrm{e}^{-L_{\lambda}^{1/2m}y})_{y\geq
0}$ on $\LL_{p}(\operatorname{\mathbb{R}}^{n};E)$, which is generated by
$-L_{\lambda}^{1/2m}$ to functions

(15)
$(\mathcal{E}_{\lambda}g_{j,k})(\boldsymbol{x}^{\prime},y)=\mathrm{e}^{-L_{\lambda}^{1/2m}y}g_{j,k}(\boldsymbol{x}^{\prime},y)\quad\text{for
}\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y\geq 0.$

This defines functions
$\mathcal{E}_{\lambda}g_{j}=\sum_{k=0}^{m_{j}}\mathcal{E}_{\lambda}g_{j,k}\in\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$.
This can be seen as follows: By assumption
$\mathcal{P}_{j,k^{\prime}}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))=\\{0\\}$
for $k\neq k^{\prime}$, and then by continuity of the projection operators
also
$\mathcal{P}_{j,k^{\prime}}(\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})})=\\{0\\}$.
Moreover, both the Fourier-transform and its inverse leave the spaces
$\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})}$ invariant,
e.g.
$\operatorname{\operatorname{ran}}(\mathcal{F}v)\subseteq\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})}$
whenever
$\operatorname{\operatorname{ran}}(v)\subseteq\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})}$
and the same is true for the scalar Fourier-symbol
$(\left|\lambda\right|+\left|\boldsymbol{\xi}\right|^{2m})^{1/2m}$ of the
operator $L_{\lambda}^{1/2m}$. Therefore, all spaces
$\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})}$ are
$L_{\lambda}^{1/2m}$-invariant, i.e.
$L_{\lambda}^{1/2m}(\operatorname{\operatorname{dom}}(L_{\lambda}^{1/2m})\cap\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})})\subseteq\overline{\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})}$,
and, then, these spaces are $(\mathrm{e}^{-yL_{\lambda}^{1/2m}})_{y\geq
0}$-invariant as well.

######  Proposition 4.11.

Let $p\in(1,\infty)$ and $\lambda\in\Sigma_{\pi-\phi}$ be given.

  1. (1)

Let
$g_{j}=\sum_{k=0}^{m_{j}}g_{j,k}\in\sum_{k=0}^{m_{j}}\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}^{n};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$
for $j=1,\ldots,m$ be given.
Then there is a unique solution
$v^{\mathrm{hs}}\in\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\cap\operatorname{\operatorname{dom}}(A\&B)$
of the initial-boundary value problem (14) for $\mathcal{E}_{\lambda}g_{j,k}$
as in (15) taking the role of
$g_{j,k}:\operatorname{\mathbb{R}}_{+}^{n+1}\rightarrow E$. It is given by
$v^{\mathrm{hs}}=\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}v^{\mathrm{hs}}_{j,k}$ for
functions

$v^{\mathrm{hs}}_{j,k}(\boldsymbol{x}^{\prime},y)=T^{\lambda}_{j,k}h^{\lambda}_{j,k}(\boldsymbol{x}^{\prime},y)$

where

$\displaystyle(T^{\lambda}_{j,k}h^{\lambda}_{j,k})(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}^{n}}T_{j,k}(\lambda,\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}},y+\tilde{y})h^{\lambda}_{j,k}(\boldsymbol{\tilde{x}}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\tilde{x}}^{\prime}\,\mathrm{d}\tilde{y}$
$\displaystyle=\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}^{n}}T_{j,k}(\lambda,\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}},y+\tilde{y})L_{\lambda}^{\frac{2m-k}{2m}}\mathcal{E}_{\lambda}g_{j,k}(\boldsymbol{\tilde{x}}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\tilde{x}}^{\prime}\,\mathrm{d}\tilde{y}$

and

$\displaystyle h^{\lambda}_{j,k}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=L_{\lambda}^{\frac{2m-k}{2m}}\mathrm{e}^{-L_{\lambda}^{1/2m}y}\mathcal{E}_{\lambda}g_{j,k}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\mathrm{e}^{-L_{\lambda}^{1/2m}y}L_{\lambda}^{\frac{2m-k}{2m}}\mathcal{E}_{\lambda}g_{j,k}(\boldsymbol{x}^{\prime},y)\quad\text{for
}\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}\text{ and }y>0.$

Here, the operator-valued kernel function
$T^{\lambda}_{j,k}(\boldsymbol{x}^{\prime},y)=T(\lambda,\boldsymbol{x}^{\prime},y)$
is defined by

$T^{\lambda}_{j,k}(\boldsymbol{x}^{\prime},y)=T_{j,k}(\lambda,\boldsymbol{x}^{\prime},y)=\big{(}\frac{\partial}{\partial
y}K_{j,k}^{\lambda}-K_{j,k}^{\lambda}L_{\lambda}^{1/2m}\big{)}(\boldsymbol{x}^{\prime},y)$

Moreover, there are constants $C>0$ (independent of the parameter
$\lambda\in\Sigma_{\pi-\phi}$ and the boundary data $g_{j}$) such that for
every $\lambda\in\Sigma_{\pi-\phi}$,
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ and $j=1,\ldots,m$
the following estimates are valid:

$\displaystyle\left\|\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}T^{\lambda}_{j,k}h^{\lambda}_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left\|((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}^{n};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq 2m,$
$\displaystyle\left\|\left|\lambda\right|^{\frac{k-2m}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}T^{\lambda}_{j,k}h^{\lambda}_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left\|g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}^{n};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq k\leq m_{j},$
$\displaystyle\left\|\left|\lambda\right|^{2-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\partial_{\lambda}T^{\lambda}_{j,k}h^{\lambda}_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left\|g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}^{n};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq k\leq m_{j}.$

  2. (2)

Given functions
$g_{j,k}\in\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$,
we set
$h^{\lambda}_{j,k}(\boldsymbol{x}^{\prime},y)=L_{\lambda}^{\frac{2m-k}{2m}}g_{j,k}$.
Then, the following estimates are valid:

$\displaystyle\left\|\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,I}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left\|((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle\left\|\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,II}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left\|((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k-1}{2m}}\partial_{y}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,I}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-k}{2m}}\left\|g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,II}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-k-1}{2m}}\left\|\frac{\partial}{\partial_{y}}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}.$

######  Remark 4.12.

A result analogous to part (2) can be found in [17, Proposition 6.8]. On the
other hand, in part (1) (and in [18]) one starts with functions defined on
$\operatorname{\mathbb{R}}^{n}$ and uses the particular extension
$\mathcal{E}_{\lambda}g_{j}$ to a function on
$\operatorname{\mathbb{R}}_{+}^{n+1}$. A convenient property of this
particular extension is the identity

$\frac{\partial}{\partial y}h^{\lambda}_{j,k}=\frac{\partial}{\partial
y}\mathrm{e}^{-L_{\lambda}^{1/2m}y}L_{\lambda}^{\frac{2m-k}{2m}}g_{j,k}=-L_{\lambda}^{1/2m}h^{\lambda}_{j,k}.$

###### Proof of Proposition 4.11.

We will focus on part (1). Part (2) follows with slight modifications, cf.
[17, Proposition 6.8]. Having the integral formulas for the solution of the
IBVP at hand, it remains to prove the asserted $\LL_{p}$-estimates. In fact,
from the previous subsection (i.e. for $f=0$, hence,
$\mathcal{F}w^{\mathrm{fs}}=\mathcal{F}w^{\mathrm{corr}}=0$), we already know
that

$w^{\mathrm{hs}}=[\boldsymbol{v}^{\mathrm{hs}}]_{1}\quad\text{for
}\boldsymbol{v}^{\mathrm{hs}}=\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}\boldsymbol{v}^{\mathrm{hs}}_{j,k}$

with

$\displaystyle\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\mathcal{F}^{-1}\big{(}\mathcal{F}\boldsymbol{V}_{\rho}^{2m}\cdot\mathcal{F}\boldsymbol{g}_{\rho}^{2m}\big{)}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=(2\pi)^{n}\big{(}\mathcal{F}^{-1}\boldsymbol{V}_{\rho}^{2m}\ast_{\boldsymbol{x}^{\prime}}\mathcal{F}^{-1}(\mathcal{F}\boldsymbol{g}_{\rho}^{2m})\big{)}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y)\mathcal{F}^{-1}\big{(}\sum_{k=0}^{m_{j}}\rho^{2m-k}\mathcal{F}g_{j,k}(\boldsymbol{\tilde{x}}^{\prime})\big{)}_{j=1}^{m}\,\mathrm{d}\boldsymbol{\tilde{x}^{\prime}}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\big{[}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})\mathcal{F}^{-1}(\sum_{k=0}^{m_{j}}\rho^{2m-k}\mathrm{e}^{-\rho\tilde{y}}\mathcal{F}g_{j,k}(\boldsymbol{\tilde{x}}^{\prime})\big{]}_{y=0}$
$\displaystyle=-\int_{\operatorname{\mathbb{R}}^{n}}\int_{0}^{\infty}\frac{\partial}{\partial\tilde{y}}\big{(}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})\mathcal{F}^{-1}\big{(}\sum_{k=0}^{m_{j}}\rho^{2m-k}\mathrm{e}^{-\rho\tilde{y}}\mathcal{F}g_{j,k}(\boldsymbol{\tilde{x}^{\prime}})\big{)}_{j=1}^{m}\,\mathrm{d}\tilde{y}\,\mathrm{d}\boldsymbol{\tilde{x}}^{\prime}$
$\displaystyle=-\int_{\operatorname{\mathbb{R}}^{n}}\int_{0}^{\infty}\big{(}\frac{\partial}{\partial\tilde{y}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})-\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{1/2m}\big{)}$
$\displaystyle\qquad\cdot\big{(}\sum_{k=0}^{m_{j}}((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{(2m-k)/2m}\mathrm{e}^{-L_{\lambda}^{1/2m}\tilde{y}}g_{j,k}(\boldsymbol{\tilde{x}}^{\prime})\big{)}_{j=1}^{m}$
$\displaystyle=-\big{(}\sum_{k=0}^{m_{j}}\int_{\operatorname{\mathbb{R}}^{n}}\int_{0}^{\infty}\big{(}\frac{\partial}{\partial
y}K^{\lambda}_{j,k}(\boldsymbol{x}-\boldsymbol{\tilde{x}},y+\tilde{y})-K^{\lambda}_{j,k}(\boldsymbol{x}-\boldsymbol{x}^{\prime},y+\tilde{y})L_{\lambda}^{1/2m}\big{)}$
$\displaystyle\qquad\cdot((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathrm{e}^{-L_{\lambda}^{1/2m}y}g_{j,k}(\boldsymbol{\tilde{x}})\mathrm{d}\boldsymbol{\tilde{x}}\big{)}_{j=1,\ldots,m}$
$\displaystyle=:\big{(}\sum_{k=0}^{m_{j}}S^{\lambda,\mathrm{I}}_{j,k}g_{j,k}+S^{\lambda,\mathrm{II}}_{j,k}g_{j,k}\big{)}_{j=1,\ldots,m}=:\big{(}\sum_{k=0}^{m_{j}}S^{\lambda}_{j,k}g_{j,k}\big{)}_{j=1,\ldots,m}\big{)}_{j=1,\ldots,m}$

where we write $\big{(}\mathrm{e}^{-L_{\lambda}^{1/2m}y}\big{)}_{y\geq 0}$ for
the strongly continuous semigroup generated by the pseudo-differential
operator $-L_{\lambda}^{1/2m}$, cf. Remark 4.10. Moreover, we exploited the
growth bounds for
$\boldsymbol{K}^{\lambda}(\boldsymbol{x}-\boldsymbol{\tilde{x}},y+\tilde{y})$
as $\tilde{y}\rightarrow\infty$ and the special form of the extension of
$g_{j,k}$ from $\operatorname{\mathbb{R}}^{n}$ to
$\operatorname{\mathbb{R}}_{+}^{n+1}$ by $\mathcal{E}_{\lambda}$. Analogously
to the proof of [17, Proposition 6.8], but for the additional parameter
$k=0,1,\ldots,m_{j}$, we thus may define

$\displaystyle v^{\mathrm{hs}}_{j,k}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=-\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}^{n}}T_{j,k}(\lambda,\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})h^{\lambda}_{j,k}(\boldsymbol{\tilde{x}}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\tilde{x}}^{\prime}\,\mathrm{d}\tilde{y}$
$\displaystyle:=-\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}^{n}}\big{[}\frac{\partial}{\partial\tilde{y}}K^{\lambda}_{j,k}-K^{\lambda}_{j,k}L_{\lambda}^{1/2m}\big{]}(\boldsymbol{x}^{\prime}-\boldsymbol{\tilde{x}}^{\prime},y+\tilde{y})h^{\lambda}_{j,k}(\boldsymbol{\tilde{x}}^{\prime},\tilde{y})\,\mathrm{d}\boldsymbol{\tilde{x}}^{\prime}\,\mathrm{d}\tilde{y}.$

As in the proof of [17, Proposition 6.8] and applying Proposition 4.7, we then
find that

$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}\big{(}\frac{\partial}{\partial\tilde{y}}K^{\lambda}_{j,k}(\cdot,\tilde{y})+K^{\lambda}_{j,k}(\cdot,\tilde{y})L_{\lambda}^{1/2m}\big{)}\right\|_{\LL_{1}(\operatorname{\mathbb{R}}^{n};\operatorname{\mathcal{B}}(E))}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{n-2m+\left|\boldsymbol{\alpha}\right|}{2m}}\int_{\operatorname{\mathbb{R}}^{n}}p_{2m,\left|\boldsymbol{\alpha}\right|}^{n+1}(c\left|\lambda\right|^{\nicefrac{{1}}{{2m}}}\tilde{y})\,\mathrm{d}\boldsymbol{x}^{\prime},\lambda$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{n-2m+\left|\boldsymbol{\alpha}\right|}{2m}}\int_{0}^{\infty}p_{2m,\left|\boldsymbol{\alpha}\right|}^{n+1}(c\left|\lambda\right|^{\nicefrac{{1}}{{2m}}}(r+\tilde{y}))r^{n-1}\,\mathrm{d}r$
$\displaystyle=n!\,C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-2m}{2m}}p_{2m+n,\left|\boldsymbol{\alpha}\right|}^{n+1}(c\left|\lambda\right|^{\nicefrac{{1}}{{2}}}\tilde{y}),\quad\tilde{y}>0.$

Then, by [17, Lemma 6.7],

$\left\|\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}T^{\lambda}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}^{n};E)}\leq
C\left\|((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}^{n};E)},$

so that the first estimate has been established. By slight modification of the
arguments, the second estimate can be proved as well. For the third estimate,
we may use that the integral operators $T^{\lambda}_{j,k}$ depend
holomorphically on the parameter $\lambda\in\Sigma_{\pi-\phi}$, so that the
third estimate is a direct consequence of the first one when applying the same
technique which has been used for the proof of Proposition 2.7.
The representation and regularity of the solution to (14) follow by
approximation just as at the end of the proof of [17, Proposition 6.8]: Given
$g_{j,k}\in\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$,
let
$g_{j,k}^{\nu}\in\mathrm{W}_{p}^{2m-k+1}(\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$
with $g_{j,k}^{\nu}\rightarrow
g_{j,k}\in\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$
as $\nu\rightarrow\infty$. By the estimates above,
$v^{\nu}_{j,k}:=T^{\lambda}_{j,k}g_{j,k}^{\nu}\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$
with $(\lambda J+A)v^{\nu}=\left(\begin{smallmatrix}0\\\
\boldsymbol{g}^{\nu}\end{smallmatrix}\right)$, i.e.
$v^{\nu}\in\operatorname{\operatorname{dom}}(A\&B)$, and further
$v^{\nu}_{j,k}\rightarrow v_{j,k}$ in
$\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$. As $A\&B$ is a
closed operator,
$v\in\operatorname{\operatorname{dom}}(A\&B)\cap\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
with $(\lambda\boldsymbol{J}+A\&B)v=\left(\begin{smallmatrix}0\\\
\boldsymbol{g}\end{smallmatrix}\right)$. ∎

Similarly, the results for the boundary value problem

(16) $\lambda Jv+(A\&B)v=\left(\begin{array}[]{c}f\\\
\boldsymbol{0}\end{array}\right)$

transfer from [17] to the situation considered here.

######  Proposition 4.13.

Let $p\in(1,\infty)$ and differential operators $\mathcal{A}(\mathrm{D})$ and
$\mathcal{B}_{j}(\mathrm{D})$, $j=1,\ldots,m$, be given as above. Let
$\lambda\in\Sigma_{\pi-\phi}$, and
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$. Then there is a unique
solution
$w\in\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\cap\operatorname{\operatorname{dom}}(A\&B)$
of the boundary value problem (16), and it is given by
$w=w^{\mathrm{fs}}+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}v^{\mathrm{corr}}_{j,k}$,
for functions $v^{\mathrm{corr}}_{j,k}$ as in Proposition 4.5 and
$w^{\mathrm{fs}}=P(\lambda+A_{\operatorname{\mathbb{R}}^{n+1}})^{-1}\mathcal{E}_{0}f$.
Moreover, there is a constant $C>0$ (independent of the data $f$) such that

$\left\|\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}w\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}\leq
C\left\|f\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}\quad\text{
for all }\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}\text{ such
that }\left|\boldsymbol{\alpha}\right|\leq 2m-1.$

###### Proof.

The proof can be transferred almost _expressis verbis_ from the proof of [17,
Proposition 6.9]. ∎

Summarising these results, we obtain the following theorem, which can be seen
as an analogue to [17, Theorem 6.10].

######  Theorem 4.14.

  1. (1)

Let $p\in(1,\infty)$ and let $E$ be an arbitrary Banach space. Let
$\mathcal{A}(\mathrm{D})$ be a parameter-elliptic operator of order $2m$ with
angle of ellipticity $\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$. Let
$\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$. For each $j=1,\ldots,m$ let
$\mathcal{B}_{j}(\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}b_{j,k,\boldsymbol{\beta}}\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k}$
be a boundary differential operator of order $m_{j}<2m$. Assume that the
Lopatinskii–Shapiro condition ${\bf(LS)}$ is valid. Let data
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and
$g_{j}\in\sum_{k=0}^{m_{j}}\mathrm{W}_{p}^{2m-k-\frac{1}{p}}(\operatorname{\mathbb{R}}^{n};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$,
$j=1,\ldots,m$, be given. Let $\lambda\in\Sigma_{\pi-\phi}$ and let $A\&B$ be
defined as above. Then the boundary value problem

$\lambda Ju+(A\&B)u=\left(\begin{array}[]{c}f\\\
\mathcal{E}_{\lambda}\boldsymbol{g}\end{array}\right)$

has a unique solution
$u\in\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\cap\operatorname{\operatorname{dom}}(A\&B)$
which, for
$R^{\lambda}_{0}:=P_{\operatorname{\mathbb{R}}_{+}^{n+1}}(\lambda+A_{\operatorname{\mathbb{R}}^{n+1}})^{-1}\mathcal{E}_{0}$,
is given by

$u=R^{\lambda}_{0}f+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}\left(R^{\lambda}_{j,k}f+S^{\lambda}_{j,k}\mathcal{E}_{\lambda}g_{j,k}\right)$

and there is a constant $C>0$ which is independent of the data
$(f,\boldsymbol{g})$ and the parameter $\lambda$ such that for every multi-
index $\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|\leq 2m-1$ and every
$\lambda\in\Sigma_{\pi-\phi}$ we have

$\displaystyle\left\|\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}u\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left(\left\|f\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}\left\|((-\Delta)^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathcal{E}_{\lambda}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}\right).$

  2. (2)

If we start with given functions
$g_{j,k}\in\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and
replace $\mathcal{E}_{\lambda}\boldsymbol{g}$ in the statement above by
$\boldsymbol{g}$, we obtain the same result, except for the modified estimates

(17)
$\displaystyle\left\|\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}u\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left(\left\|f\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}\left\|((-\Delta)^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}\right.$
$\displaystyle\qquad\left.+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}\left\|((-\Delta)^{m}+\left|\lambda\right|)^{\frac{2m-k-1}{2m}}\frac{\partial}{\partial
y}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}\right).$

###  4.5. $\mathcal{H}^{\infty}$-calculus

In this subsection, we transfer the results on the
$\mathcal{H}^{\infty}$-calculus as in [17, Section 7.1] to the situation
considered here. We start by combining Theorem 4.14 and [17, Lemma 7.1], and
immediately obtain the following theorem.

######  Theorem 4.15.

If, in the situation of Theorem 4.14, the Banach space $E$ is of class
$\mathcal{HT}$, then $u$ lies in
$\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and (17) holds
true for all $\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ with
$\left|\boldsymbol{\alpha}\right|=2m$ as well. Moreover, in this case

$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda}_{j,k}\mathcal{E}_{\lambda}g_{j}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-k}{2m}}\left\|\mathcal{E}_{\lambda}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq k\leq m_{j},$ if
$g_{j,k}$ is only given on
$\operatorname{\mathbb{R}}^{n}\cong\partial\operatorname{\mathbb{R}}_{+}^{n+1}$,
else
$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,\mathrm{I}}_{j,k}\mathcal{E}_{\lambda}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-k}{2m}}\left\|g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq k\leq m_{j},$
$\displaystyle\left\|\mathrm{D}^{\boldsymbol{\alpha}}S^{\lambda,\mathrm{II}}_{j,k}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)}$
$\displaystyle\leq
C\left|\lambda\right|^{\frac{\left|\boldsymbol{\alpha}\right|-k-1}{2m}}\left\|\tfrac{\partial}{\partial
y}g_{j,k}\right\|_{\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)},$
$\displaystyle 0\leq\left|\boldsymbol{\alpha}\right|\leq k+1\leq m_{j}+1$

and the $\LL_{p}$-realisation $A_{B}$ of the boundary value problem (16) is

(18) $\displaystyle A_{B}u$ $\displaystyle=\mathcal{A}(\mathrm{D})u,$ (19)
$\displaystyle\operatorname{\operatorname{dom}}(A_{B})$
$\displaystyle=\\{u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E):\,\mathcal{B}_{j}(\mathrm{D})u=0\quad\text{for
all }j=1,\ldots,m\\}.$

Together with the kernel estimates of Proposition 4.7, it follows that $A_{B}$
admits a _bounded $\mathcal{H}^{\infty}$-calculus_, see the following analogue
to [17, Theorem 7.4]:

######  Theorem 4.16.

Let $E$ be a Banach space of class $\mathcal{HT}$ and let
$A_{B}:\operatorname{\operatorname{dom}}(A_{B})\rightarrow\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
be defined as in Theorem 4.15. Then
$A_{B}\in\mathcal{H}^{\infty}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))$
with $\mathcal{H}^{\infty}$-angle
$\phi_{A_{B}}^{\infty}\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$.

The proof can be executed similarly to the proof of [17, Theorem 7.4], using
[17, Lemma 7.1] and an adjusted version of [17, Remark 7.2c].

###### Proof.

Let us fix any $\phi\in(\phi_{\mathcal{A}}^{\mathrm{ellipt}},\pi)$ and recall
that we may write the resolvents of the operator $A_{B}$ as

$(\lambda+A_{B})^{-1}=R^{\lambda}_{0}+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}R^{\lambda}_{j,k}\quad\text{for
all }\lambda\in\Sigma_{\pi-\phi}.$

The terms correspond to the resolvent for the full-space problem plus
corrective terms due to the homogeneous boundary conditions imposed at
$\partial\operatorname{\mathbb{R}}_{+}^{n+1}$.
Let $h\in\mathcal{H}^{\infty}(\Sigma_{\phi})$ be an arbitrary polynomially
decaying function, i.e. there is $C,s>0$ such that

$\left|h(z)\right|\leq C(1+\left|z\right|)^{-s},$

and denote by $\Gamma$ the contour given by
$(0,\infty]\mathrm{e}^{\mathrm{i}\theta}\cup(0,\infty)\mathrm{e}^{-\mathrm{i}\theta}$
for some $\theta\in(\phi_{\mathcal{A}}^{\mathrm{ellipt}},\phi)$. We may then
use the Dunford integral calculus to define

$h(A_{B}):=\frac{1}{2\pi\mathrm{i}}\int_{\Gamma}h(\lambda)(\lambda+A_{B})^{-1}\mathrm{d}\lambda.$

We want to show that
$A_{B}\in\mathcal{H}^{\infty}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))$
with $\mathcal{H}^{\infty}$-angle
$\phi_{A_{B}}^{\infty}\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$, thus it
remains to prove that for all such $\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$
and $h\in\mathcal{H}_{0}(\Sigma_{\phi})$ which polynomially decay to zero at
infinity, we have an estimate

$\left\|h(A_{B})\right\|_{\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))}\leq
C\left\|h\right\|_{\infty}$

for some constant $C>0$ which is independent of
$h\in\mathcal{H}_{0}(\Sigma_{\phi})$. For the full-space problem related term
$R^{\lambda}_{0}$ such an estimate has already been established in [17,
Theorem 5.5], relying on the condition that $E$ is a Banach space of class
$\mathcal{HT}$. The remaining terms $R^{\lambda}_{j,k}$ are the integral
operators associated with the integral kernels
$K^{\mathrm{corr},\lambda}_{j,k}$, hence,
$\int_{\Gamma}h(\lambda)R^{\lambda}_{j,k}\,\mathrm{d}\lambda$ is the integral
operator associated to the kernel
$\int_{\Gamma}h(\lambda)K^{\mathrm{corr},\lambda}_{j,k}(\cdot,\cdot,\cdot)\,\mathrm{d}\lambda$.
Since the estimate

$\displaystyle\left|\int_{\Gamma}h(\lambda)K^{\mathrm{corr},\lambda}_{j,k}(\boldsymbol{x}^{\prime},y,\tilde{y})\,\mathrm{d}\lambda\right|$
$\displaystyle\leq
C\left\|h\right\|_{\infty}\int_{0}^{\infty}r^{\frac{n+1}{2m}}p_{2m,0}^{n+1}(cr^{\frac{1}{2m}}(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})\,\mathrm{d}r$
$\displaystyle\leq
C\left\|h\right\|_{\infty}\frac{1}{(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})^{n+1}}\int_{0}^{\infty}\tau^{n}p_{2m,0}^{n+1}(c\tau)\,\mathrm{d}\tau$
$\displaystyle\leq
C\left\|h\right\|_{\infty}\frac{1}{(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})^{n+1}}$

is valid thanks to
$\int_{0}^{\infty}\tau^{n}p_{2m,0}^{n+1}(\tau)\mathrm{d}\tau<\infty$, we may
employ Proposition 2.18 and [17, Remark 7.2 c)] to conclude the assertion. ∎

######  Corollary 4.17.

Let $E$ be a Banach space of class $\mathcal{HT}$. Then

$\mathcal{R}(\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}(\lambda+A_{B})^{-1}:\,\lambda\in\Sigma_{\pi-\phi},\,\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}\text{
such that }0\leq\left|\boldsymbol{\alpha}\right|\leq 2m\\})<\infty.$

###### Proof.

By Theorem 4.16 and Remark 2.17,
$A_{B}\in\mathcal{H}^{\infty}(E)\subseteq\mathcal{BIP}(E)$, hence, for every
$\sigma\in(0,1)$

$\operatorname{\operatorname{dom}}(A_{B}^{\sigma})=\big{(}\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E),\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)\big{)}_{\sigma,p}=\mathrm{H}_{p}^{2m\sigma}(\operatorname{\mathbb{R}}_{+}^{n+1};E).$

Thus, for each multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|\leq 2m$, the linear operator
$\mathrm{D}^{\boldsymbol{\alpha}}A_{B}^{-\frac{k}{2m}}$ with
$k=\left|\boldsymbol{\alpha}\right|$ is a bounded operator on $E$. As
$\\{\lambda(\lambda+A_{B})^{-1}:\,\lambda\in\Sigma_{\pi-\phi}\\}$ is
$\mathcal{R}$-bounded, all the sets

$\\{\lambda^{-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\lambda(\lambda+A_{B})^{-1}:\,\lambda\in\Sigma_{\pi-\phi}\\}\quad\text{for
each }\left|\boldsymbol{\alpha}\right|\leq 2m$

are $\mathcal{R}$-bounded. Since, here, the set of admissible multi-indices
$\\{\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}:\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$ is finite, the family of operators

$\\{\lambda^{-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\lambda(\lambda+A_{B})^{-1}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$

is $\mathcal{R}$-bounded as well. ∎

###  4.6. $\mathcal{R}$-bounds for solution operators

By Corollary 4.17, the solution map for the boundary value problem (16) with
homogeneous boundary data $\boldsymbol{g}=\boldsymbol{0}$ admits
$\mathcal{R}$-bounds. Next, we consider the boundary value problem (16) for
$f=0$ and inhomogeneous boundary data $\boldsymbol{g}\neq 0$ and also conclude
$\mathcal{R}$-boundedness for the corresponding solution maps. By Proposition
4.11 the solution $v$ to
$(\lambda\boldsymbol{J}+A\&B)v=(0,\mathcal{E}_{\lambda}\boldsymbol{g})^{\mathsf{T}}$
may be expressed as

$v=\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}S^{\lambda}_{j,k}\mathcal{E}_{\lambda}g_{j,k}.$

Our goal is to prove that the boundary value problem

$\begin{cases}(\lambda+\mathcal{A}(\mathrm{D}))u=f&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\mathrm{D})u=g_{j,k}&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m\end{cases}$

has a (unique) solution in the class
$u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ if and only
if the given functions lie in the classes
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and
$g_{j,k}\in\mathrm{W}_{p}^{2m-k-1/p}(\partial\operatorname{\mathbb{R}}_{+}^{n+1};E)$
for $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$. Using the extension operator
$\mathcal{E}_{\lambda}$, the latter is equivalent to
$\mathcal{E}_{\lambda}g_{j,k}\in\mathrm{W}_{p}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
for all $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$, and since for
$\lambda\in\Sigma_{\pi-\phi}$, i.e. $\left|\lambda\right|>0$,
$((-\Delta)^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}$ acts as a top-linear
isomorphy between $\mathrm{W}_{p}^{k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
and $\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, we may reformulate the
problem as follows: Show that that

$\begin{cases}(\lambda+\mathcal{A}(\mathrm{D}))u=f&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathcal{B}_{j,k}(\mathrm{D})u|_{y=0}=\tilde{g}_{j,k}&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,1\leq j\leq m,\,0\leq k\leq
m_{j}\end{cases}$

where
$\tilde{g}_{j,k}:=((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathcal{E}_{\lambda}g_{j,k}|_{y=0}$,
has a (unique) solution
$u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+};E)$ if and only if
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and
$((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathcal{E}_{\lambda}g_{j,k}\in\LL_{p}(\operatorname{\mathbb{R}}_{+};E)$
for all $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$. Therefore, for
$\lambda\in\Sigma_{\pi-\phi}$, $j=1,\ldots,m$ and $k=0,\ldots,m_{j}$, and
using the operators $S_{j,k}^{\lambda,\mathrm{I}}$ and
$S_{j,k}^{\lambda,\mathrm{I}}$ from Proposition 4.13 we introduce operators

(20) $\displaystyle U^{\lambda}_{j,k}$
$\displaystyle=S^{\lambda,\mathrm{I}}_{j,k}((-\Delta)^{m}+\left|\lambda\right|)^{\frac{k-2m}{2m}}\mathcal{E}_{\lambda},$
(21) $\displaystyle V^{\lambda}_{j,k}$
$\displaystyle=S^{\lambda,\mathrm{II}}_{j,k}((-\Delta)^{m}+\left|\lambda\right|)^{\frac{k-2m}{2m}}\mathcal{E}_{\lambda}.$

The families consisting of these operators allow for the following
$\mathcal{R}$-bounds, cf. [17, Proposition 7.6].

######  Proposition 4.18.

For each $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$, the sets

$\displaystyle\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}U^{\lambda}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$
$\displaystyle\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)),$
$\displaystyle\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}V^{\lambda}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$
$\displaystyle\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)),$
$\displaystyle\\{\lambda^{2-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\partial_{\lambda}U^{\lambda}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$
$\displaystyle\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)),$
$\displaystyle\\{\lambda^{2-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\partial_{\lambda}V^{\lambda}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\}$
$\displaystyle\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))$

are $\mathcal{R}$-bounded.

###### Proof.

By Proposition 4.11, the operators
$\mathrm{D}^{\boldsymbol{\alpha}}U^{\lambda}_{j,k}$ and
$\mathrm{D}^{\boldsymbol{\alpha}}V^{\lambda}_{j,k}$ are integral operators on
$\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ with kernels
$\mathrm{D}^{\boldsymbol{\alpha}}K^{\lambda}_{U^{j,k}}$ and
$\mathrm{D}^{\boldsymbol{\alpha}}K^{\lambda}_{V_{j,k}}$, resp., where

$\displaystyle\mathcal{F}K^{\lambda}_{U^{j,k}}(\boldsymbol{\xi}^{\prime},y)$
$\displaystyle=\mathcal{F}\big{(}\frac{\partial}{\partial
y}K^{\lambda,\mathrm{I}}_{j,k}\big{)}(\boldsymbol{\xi}^{\prime},y)(\left|\boldsymbol{\xi}^{\prime}\right|^{2m}+\left|\lambda\right|)^{\frac{2m-k}{2m}}\mathcal{P}_{j,k}(\left|\boldsymbol{\xi}^{\prime}\right|^{2m}+\left|\lambda\right|)^{\frac{k-2m}{2m}}\mathcal{E}_{\lambda},$
$\displaystyle\mathcal{F}K^{\lambda}_{V^{j,k}}(\boldsymbol{\xi}^{\prime},y)$
$\displaystyle=\mathcal{F}K^{\lambda,\mathrm{II}}_{j,k}(\boldsymbol{\xi}^{\prime},y)\frac{\partial}{\partial
y}(\left|\boldsymbol{\xi}^{\prime}\right|^{2m}+\left|\lambda\right|)^{\frac{2m-k-1}{2m}}\mathcal{P}_{j,k}(\left|\boldsymbol{\xi}^{\prime}\right|^{2m}+\left|\lambda\right|)^{\frac{k+1-2m}{2m}}\mathcal{E}_{\lambda},$

for $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$. By [17, Proposition 4.1 and Lemma
7.1], it suffices to prove that there is $C>0$ such that for all
$\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ and $y,\tilde{y}>0$

$\displaystyle\mathcal{R}(\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial\tilde{y}}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y+\tilde{y}):\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\})$
$\displaystyle\leq\frac{C}{(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})^{n+1}},$
$\displaystyle\mathcal{R}(\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial\tilde{y}}K^{\lambda}_{V^{j,k}}(\boldsymbol{x}^{\prime},y+\tilde{y}):\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\})$
$\displaystyle\leq\frac{C}{(\left|\boldsymbol{x}^{\prime}\right|+y+\tilde{y})^{n+1}}.$

This can be proved as follows, analogously to [17, Proposition 7.7]. For each
$\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}$, $y>0$,
$\lambda\in\Sigma_{\pi-\phi}$ and
$\boldsymbol{\alpha},\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n}$,
$\gamma\in\operatorname{\mathbb{N}}_{0}$ such that
$\left|\boldsymbol{\alpha}\right|\leq 2m$ and
$\left|\boldsymbol{\beta}\right|+\gamma=\left|\boldsymbol{\alpha}\right|$, we
have the identity

$\displaystyle\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial
y}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}}\frac{\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}}{\rho^{2m}}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\beta}}(A_{0}(b,\sigma)\rho)^{\gamma}\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)M(b,\sigma)\rho^{2m-k}J_{\boldsymbol{\xi}^{\prime},\lambda}^{2m-k}\,\mathrm{d}\boldsymbol{\xi}^{\prime},$

where
$J_{\boldsymbol{\xi}^{\prime},\lambda}^{2m-k}=(\left|\boldsymbol{\xi}^{\prime}\right|^{2m}+\left|\lambda\right|)^{\frac{k-2m}{2m}}$.
As in the proof of Proposition 4.7, we choose a rotation
$\boldsymbol{Q}\in\operatorname{\mathbb{R}}^{n\times n}$ such that
$\boldsymbol{Q}\boldsymbol{x}^{\prime}=(\left|\boldsymbol{x}^{\prime}\right|,0,\ldots,0)^{\mathsf{T}}$
and write $\boldsymbol{Q}\boldsymbol{\xi}^{\prime}=(a,r\boldsymbol{\varphi})$
for some $a\in\operatorname{\mathbb{R}}$, $r>0$ and
$\boldsymbol{\varphi}\in\mathbb{S}^{n-2}$. Then

$\displaystyle\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial
y}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y)$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}\frac{\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}}{\rho^{2m-1}}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\beta}}$
$\displaystyle\quad\cdot(\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\rho)^{\gamma}\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\boldsymbol{M}(\boldsymbol{b},\sigma)\rho^{2m-k}J_{\boldsymbol{\xi}^{\prime},\lambda}^{2m-k}\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\mathbb{S}^{n-2}(\boldsymbol{\varphi})$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{\Gamma^{-}}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}\frac{\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}}{\rho^{2m-1}}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\beta}}$
$\displaystyle\quad\cdot(\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\rho)^{\gamma}\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\boldsymbol{M}(\boldsymbol{b},\sigma)\rho^{2m-k}J_{\boldsymbol{\xi}^{\prime},\lambda}^{2m-k}\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\mathbb{S}^{n-2}(\boldsymbol{\varphi}),$

where as in the proof of Proposition 4.7
$\gamma^{-}(s)=s+\mathrm{i}\varepsilon(r+\left|s\right|+\left|\lambda\right|^{\nicefrac{{1}}{{2}}m})$,
$s\in\operatorname{\mathbb{R}}$, is a parametrisation of the curve
$\Gamma^{-}$, and Cauchy’s integral theorem as well as holomorphy of
$\boldsymbol{A}_{0}$ and $\boldsymbol{M}$ have been employed. Setting
$c=\nicefrac{{c_{1}}}{{2}}$ for $c_{1}>0$ from (9), as in Proposition 4.7 we
obtain the identity

$\displaystyle\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial
y}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y)(\left|\boldsymbol{x}^{\prime}\right|+y)^{n+1}$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}\int_{0}^{\infty}F(r,\tau,\boldsymbol{\varphi},\lambda,\boldsymbol{x}^{\prime},y)r^{n-2}(\left|\boldsymbol{x}^{\prime}\right|+y)^{n}\mathrm{e}^{-\frac{1}{2}(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+cy)(r+\tau)}\,\mathrm{d}r\,\mathrm{d}\tau\,\mathrm{d}\mathbb{S}^{n-2}(\boldsymbol{\varphi}),$

where

$F(r,\tau,\boldsymbol{\varphi},\lambda,\boldsymbol{x}^{\prime},y)=G(r,\tau,\boldsymbol{\varphi},\lambda,y)H(\boldsymbol{x}^{\prime},y,r,\tau,\lambda)$

with

$\displaystyle G(r,\tau,\boldsymbol{\varphi},\lambda,y)$
$\displaystyle=\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\mathrm{e}^{cy(r+\tau+\left|\lambda\right|^{\nicefrac{{1}}{{2}}m})}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)^{\gamma}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\boldsymbol{M}(\boldsymbol{b},\sigma),$
$\displaystyle H(\boldsymbol{x}^{\prime},y,r,\tau,\lambda)$
$\displaystyle=(\left|\boldsymbol{x}^{\prime}\right|+y)\mathrm{e}^{-(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+cy)\left|\lambda\right|^{\nicefrac{{1}}{{2}}m}}\mathrm{e}^{-\frac{1}{2}(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+cy)(r+\tau)}{\boldsymbol{\xi}^{\prime}}^{\boldsymbol{\beta}}\rho^{\gamma}\frac{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}}{\rho^{2m-1}}J_{\boldsymbol{\xi}^{\prime},\lambda}^{2m-k}$

and where $\rho,\boldsymbol{b},\sigma$ depend on $r,\tau,\lambda$ and
$\left|\boldsymbol{\alpha}\right|=\left|\boldsymbol{\beta}\right|+\gamma$.
Setting
$M=\mathbb{S}^{n-2}\times\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}$,
this integral equals
$\int_{M}F(r,\tau,\boldsymbol{\varphi},\lambda,\boldsymbol{x}^{\prime},y)\,\mathrm{d}\mu(r,\tau,\boldsymbol{\varphi})$,
where

$\
\mathrm{d}\mu(r,\tau,\boldsymbol{\varphi})=r^{n-2}(\left|\boldsymbol{x}^{\prime}\right|+y)^{n}\mathrm{e}^{-\frac{1}{2}(\varepsilon\left|\boldsymbol{x}^{\prime}\right|+cy)(r+\tau)}\,\mathrm{d}r\,\mathrm{d}\tau\,\mathrm{d}\sigma(\boldsymbol{\varphi}).$

Here, $\mu$ is a finite measure on $M$ and such that
$\int_{M}\,\mathrm{d}\mu(r,\tau,\boldsymbol{\varphi})\leq C$ for some $C>0$
independent of $\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ and
$y>0$. By [17, Proposition 3.8] it follows that for
$\left|\boldsymbol{\beta}\right|+\gamma=\left|\boldsymbol{\alpha}\right|$

$\displaystyle(\left|\boldsymbol{x}^{\prime}\right|+y)^{n+1}\mathcal{R}(\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial
y}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y):\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\})$
$\displaystyle\leq\mathcal{R}(\\{(\left|\boldsymbol{x}^{\prime}\right|+y)^{n+1}\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\frac{\partial}{\partial
y}K^{\lambda}_{U^{j,k}}(\boldsymbol{x}^{\prime},y):\,\lambda\in\Sigma_{\pi-\phi},\,\left|\boldsymbol{\alpha}\right|\leq
2m,\,\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0\\})$
$\displaystyle=\mathcal{R}\big{(}\big{\\{}\int_{M}F(r,\tau,\boldsymbol{\varphi},\lambda,\boldsymbol{x}^{\prime},y)\,\mathrm{d}\mu(r,\tau,\boldsymbol{\varphi}):\,\lambda\geq
0,\,\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,\boldsymbol{\varphi}\in\mathbb{S}^{n-2}\big{\\}}\big{)}.$

The assertion for $V^{\lambda}_{j,k}$ follows similarly and those for the
$\lambda$-derivatives follow $\mathcal{R}$-boundedness for the operators
$U^{\lambda}_{j,k}$, $V^{\lambda}_{j,k}$ and the abstract result Proposition
2.7. ∎

###  4.7. Spatially dependent interior and boundary symbols

As situations with spatial dependent parameters are of utmost importance, in
particular for the localisation procedure by which results on the half-space
are transferred to general, but sufficiently regular domains, we consider
spatially dependent perturbations of the principle part next, i.e. the
interior symbol $\mathcal{A}$ is allowed to be of the form

$\mathcal{A}(\boldsymbol{x},D)=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}}$

with spatially dependent, but bounded and uniformly continuous coefficients
$a_{\boldsymbol{\alpha}}:\operatorname{\mathbb{R}}_{+}^{n+1}\rightarrow\operatorname{\mathcal{B}}(E)$,
and similarly for the boundary symbols

$\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}b_{j,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m$

where we demand regularity of the coefficients
$b_{j,\boldsymbol{\beta}}\in\mathrm{BUC}^{2m-k}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$
for each multi-index
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\beta}\right|=k$. Throughout this subsection, we assume
that some $\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ is given, the Banach
space $E$ is of class $\mathcal{HT}$ and the Lopatinskii–Shapiro condition is
satisfied at every boundary point
$\boldsymbol{x}\in\partial\operatorname{\mathbb{R}}_{+}^{n+1}$. For now, let
us fix $\boldsymbol{x}_{0}\in\operatorname{\mathbb{R}}_{+}^{n+1}$ and, as an
intermediate step, assume that, for some given small $\varepsilon>0$, the
coefficients $a_{\boldsymbol{\alpha}}$ and $b_{j,\boldsymbol{\beta}}$ have
uniformly small oscillation, in the sense that

$\displaystyle\sup_{\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}}\sum_{\left|\boldsymbol{\alpha}\right|=2m}\left\|a_{\boldsymbol{\alpha}}(\boldsymbol{x})-a_{\boldsymbol{\alpha}}(\boldsymbol{x}_{0})\right\|_{\operatorname{\mathcal{B}}(E)}$
$\displaystyle<\varepsilon,$
$\displaystyle\sup_{\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}}\sum_{\left|\boldsymbol{\beta}\right|=k}\left\|b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})-b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x}_{0})\right\|_{\operatorname{\mathcal{B}}(E)}$
$\displaystyle<\varepsilon\quad\text{for }j=1,\ldots,m,\,k=0,\ldots,m_{j}.$

We may then interpret $\mathcal{A}(\boldsymbol{x},D)$ and
$\mathcal{B}_{j}(\boldsymbol{x},D)$ as small perturbations from the constant
coefficient case:

$\displaystyle\mathcal{A}(\boldsymbol{x},\mathrm{D})$
$\displaystyle=\mathcal{A}(\boldsymbol{x}_{0},\mathrm{D})+\mathcal{A}^{\mathrm{sm}}(\boldsymbol{x},\mathrm{D})$
$\displaystyle\text{for }\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}$
$\displaystyle\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})$
$\displaystyle=\mathcal{B}_{j}(\boldsymbol{x}_{0},\mathrm{D})+\mathcal{B}_{j}^{\mathrm{sm}}(\boldsymbol{x},\mathrm{D})$
$\displaystyle\text{for
}\boldsymbol{x}\in\partial\operatorname{\mathbb{R}}_{+}^{n+1}.$

For each $\lambda\in\Sigma_{\pi-\phi}$ with
$\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ and a given right-hand side
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, the variable coefficient
problem

(22)
$\begin{cases}(\lambda+\mathcal{A}(\boldsymbol{x},\mathrm{D}))u=f&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})u=0&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m\end{cases}$

has a unique solution
$u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ if and only
if it is the unique solution
$u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ to the
problem

(23)
$\begin{cases}(\lambda+\mathcal{A}(\boldsymbol{x}_{0},\mathrm{D}))u=f-\mathcal{A}^{\mathrm{sm}}(\boldsymbol{x},D)u&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\boldsymbol{x}_{0})u(\boldsymbol{x}^{\prime},0)=-\mathcal{B}_{j}^{\mathrm{sm}}(\boldsymbol{x}^{\prime},\mathrm{D})u(\boldsymbol{x},0)&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\text{ for
}j=1,\ldots,m.\end{cases}$

A procedure as in [17, Section 7.3], based on the contraction mapping
principle, then delivers that, for sufficiently small $\varepsilon>0$ and
$\lambda\in\Sigma_{\pi-\phi}$ with sufficiently large modulus, the system of
equations (22) (or, equivalently, equation (23)) admits a unique solution
$u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ of the form

$u=[I+(\lambda+A_{B^{0}}^{0})^{-1}\mathcal{A}^{\mathrm{sm}}+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}S^{\lambda}_{j,k}\mathcal{B}_{j}^{\mathrm{sm}}]^{-1}(\lambda+A_{B^{0}}^{0})^{-1}f.$

Here, we set
$A_{B^{0}}^{0}:=\mathcal{A}(\boldsymbol{x}_{0},\mathrm{D})|_{\operatorname{ker}(\mathcal{B}(\boldsymbol{x}_{0},\mathrm{D}))}$
and then define

$\displaystyle A_{B}u$
$\displaystyle=\mathcal{A}(\boldsymbol{x},\mathrm{D})u,$
$\displaystyle\operatorname{\operatorname{dom}}(A_{B})$
$\displaystyle=\\{u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E):\,\mathcal{B}_{j}(\boldsymbol{x}_{0},D)u(\boldsymbol{x}^{\prime},0)=0,\,\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,j=1,\ldots,m\\}.$

Thus, for $\lambda\in\Sigma_{\pi-\phi}$ with sufficiently large modulus the
operator $\lambda+A_{B}$ is invertible and its inverse is implicitly given by
the relation

$(\lambda+A_{B})^{-1}=(\lambda+A_{B^{0}}^{0})^{-1}-(\lambda+A_{B^{0}}^{0})^{-1}\mathcal{A}^{\mathrm{sm}}(\cdot,\mathrm{D})(\lambda+A_{B})^{-1}-\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}S^{\lambda}_{j,k}\mathcal{P}_{j,k}\mathcal{B}_{j}^{\mathrm{sm}}(\lambda+A_{B})^{-1},$

hence, solving for $(\lambda+A_{B})^{-1}$ we find that

$(\lambda+A_{B})^{-1}=[I+(\lambda+A_{B^{0}}^{0})^{-1}\mathcal{A}^{\mathrm{sm}}(\cdot,\mathrm{D})+\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}S^{\lambda}_{j,k}\mathcal{P}_{j,k}\mathcal{B}^{\mathrm{sm}}_{j,k}(\cdot,\mathrm{D})]^{-1}(\lambda+A_{B^{0}}^{0})^{-1}.$

With this formula at hand, we may establish $\mathcal{R}$-bounds on the set of
operators
$\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}(\lambda+A_{B})^{-1}$
as follows.

######  Lemma 4.19.

There is $\lambda_{0}>0$ such that the set

$\\{\lambda^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}(\lambda+A_{B})^{-1}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0},\,0\leq\left|\boldsymbol{\alpha}\right|\leq
2m\\}\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))$

is $\mathcal{R}$-bounded.

###### Proof.

The proof is essentially the same as in [17, Section 7.3]. Instead of the
operators $\mathcal{B}_{j}(\mathrm{D})$, we consider each of the operators
$\mathcal{B}_{j,k}(\mathrm{D})=\mathcal{B}_{j}(\mathrm{D})\mathcal{P}_{j,k}$
($k=0,1,\ldots,m_{j}$) separately and use that
$\mathcal{B}_{j}(\mathrm{D})=\sum_{k=0}^{m_{j}}\mathcal{B}_{j,k}(\mathrm{D})$
is the sum of these operators. We leave out the details here. ∎

###  4.8. Variable and lower order coefficients in a half-space

To generalise the results of the previous subsection to more general spatially
dependent coefficients, i.e. to get rid of the small oscillation constraint,
and elliptic operators with lower order coefficients, we use the same
localisation procedure as in [17, Section 5.3]. In comparison with [17,
Section 7.4] we use a more subtle notion of the principal part, which allows
us to handle the mixed type boundary conditions considered here. Throughout
this subsection, we assume that there are projections $\mathcal{P}_{j,k}$
(independent of the spatial variable $\boldsymbol{x}$), for $j=1,\ldots,m$ and
$k=0,1,\ldots,m_{j}<2m$, such that
$\mathcal{P}_{j,k}\mathcal{P}_{j,k^{\prime}}=0$ for each $k\neq k^{\prime}$
and we allow that the interior symbol $\mathcal{A}$ and the boundary symbols
$\mathcal{B}_{j}$ have the spatial, but not temporal dependent form

$\mathcal{A}(\boldsymbol{x},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}},\quad\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m.$

The principle parts $\mathcal{A}_{\\#}$ of the interior symbol and
$\mathcal{B}_{j,\\#}$ of the boundary symbols are defined as

$\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}},\quad\mathcal{B}_{j,\\#}(\boldsymbol{x},\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m.$

Moreover, we demand the following regularity of the coefficients
$a_{\boldsymbol{\alpha}}$ and $b_{j,k,\boldsymbol{\beta}}$.

######  Assumption 4.20.

Assume that the following conditions are met by the interior and boundary
symbols and their coefficients.

  1. (1)

Smoothness assumptions:

    1. (a)

$a_{\boldsymbol{\alpha}}\in\mathrm{C}_{\mathrm{l}}(\overline{\operatorname{\mathbb{R}}_{+}^{n+1}};\operatorname{\mathcal{B}}(E))$
for all $\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|=2m$;

    2. (b)

$a_{\boldsymbol{\alpha}}\in[\LL_{\infty}+\LL_{r_{k}}](\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\mathcal{B}}(E))$
for all $\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|=k<2m$ where $r_{k}\geq p$ such that
$2m-k>\tfrac{n}{r_{k}}$;

    3. (c)

$b_{j,k,\boldsymbol{\beta}}\in\mathrm{C}^{2m-k}(\partial\operatorname{\mathbb{R}}_{+}^{n+1};\operatorname{\mathcal{B}}(E))$
and
$b_{j,k,\boldsymbol{\beta}}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})+\ldots+\operatorname{\operatorname{ran}}(\mathcal{P}_{j,m_{j}})$
for each $j=1,\ldots,m$, $0\leq k\leq m_{j}$ and
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\beta}\right|=k$.

  2. (2)

There is $\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$ such that

    1. (a)

Ellipticity of the interior symbol: The principal symbol
$\mathcal{A}_{\\#}(\boldsymbol{x},\boldsymbol{\xi})$ is parameter-elliptic
with angle of ellipticity $\phi\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ for
each $\boldsymbol{x}\in\overline{\operatorname{\mathbb{R}}_{+}^{n+1}}$.

    2. (b)

Lopatinskii–Shapiro condition: For each
$\boldsymbol{x}_{0}\in\overline{\operatorname{\mathbb{R}}_{+}^{n+1}}$,
$\lambda\in\overline{\Sigma_{\pi-\phi_{\mathcal{A}}^{\mathrm{ellipt}}}}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ with
$(\lambda,\boldsymbol{\xi}^{\prime})\neq(0,\boldsymbol{0})$, the initial value
problem for the principle symbols

$\displaystyle(\lambda+\mathcal{A}_{\\#}(\boldsymbol{x}_{0},\boldsymbol{\xi}^{\prime},\mathrm{D}_{n+1}))v(y)$
$\displaystyle=0,$ $\displaystyle y>0,$
$\displaystyle\mathcal{B}_{j,\\#}(\boldsymbol{x}_{0},\boldsymbol{\xi}^{\prime},\mathrm{D}_{n+1})v(0)$
$\displaystyle=h_{j},$ $\displaystyle j=1,\ldots,m$

has a unique solution $u\in\mathrm{C}_{0}(\operatorname{\mathbb{R}}_{+};E)$
for each $(h_{1},\ldots,h_{m})^{\mathsf{T}}\in E^{m}$.

A procedure similar to the approach in the preceeding subsection then gives
the following result.

######  Theorem 4.21.

Let $E$ be a Banach space of class $\mathcal{HT}$,
$n,m\in\operatorname{\mathbb{N}}$, $p\in(1,\infty)$ and assume that for some
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$ the boundary value problem
$(\mathcal{A},(\mathcal{B}_{j})_{j=1}^{m})$ satisfies Assumption 4.20. Let
$A_{B}$ be the $\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$-realisation of
the homogeneous boundary value problem with domain

$\operatorname{\operatorname{dom}}(A_{B})=\\{u\in\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}_{+}^{n+1};E):\,\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})u=0\,\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m\\}.$

Then, for each $\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$, there is
$\mu_{\phi}\geq 0$ such that $\mu_{\phi}+A_{B}$ is $\mathcal{R}$-sectorial
with $\phi_{\mu_{\phi}+A_{B}}^{\mathcal{R}}\leq\phi$. In particular, if
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}<\tfrac{\pi}{2}$, then the parabolic
initial-boundary value problem

$\displaystyle\partial_{t}u+A_{B}u+\mu_{\phi}u$ $\displaystyle=f,$
$\displaystyle t>0,$ $\displaystyle u(0)$ $\displaystyle=0$

has the property of maximal regularity in the class
$\LL_{q,p}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n+1};E))$
for each $q\in(1,\infty)$, i.e. for every given
$f\in\LL_{q,p}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n+1};E)$
there is a unique solution
$u\in\mathrm{W}_{q,p}^{(1,2)}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n+1};E)$
and

$\left\|u\right\|_{\mathrm{W}_{q,p}^{(1,2)}}\leq
C\left\|f\right\|_{\LL_{q,p}}\quad\text{for all
}f\in\LL_{q,p}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n+1};E).$

Since the technique of proof is very similar to parts which appear in the
proof for the general domain case, at this point we skip the details.

##  5\. Elliptic vector-valued PDE: The case of $\mathrm{C}^{2m}$-domains

###  5.1. Localisation techniques for domains

So far, we have considered the half-space
$\Omega=\operatorname{\mathbb{R}}_{+}^{n+1}$ as a model case. Via bending and
localisation techniques, we may now transfer the result to more general
domains which are sufficiently regular. Let $E$ be a Banach space of class
$\mathcal{HT}$ and $m,n,m_{1},\ldots,m_{m}\in\operatorname{\mathbb{N}}$ be
natural numbers such that $m_{j}<2m$ for $j=1,\ldots,m$. For each
$j=1,\ldots,m$ and $k=0,\ldots,m_{j}$, let $\mathcal{P}_{j,k}$ be a projection
such that $\mathcal{P}_{j,k}\mathcal{P}_{j,k^{\prime}}=0$ whenever $k\neq
k^{\prime}$. Let $\Omega\subseteq\operatorname{\mathbb{R}}^{n+1}$ be a domain
with compact $\mathrm{C}^{2m}$-boundary, i.e. we assume that
$\partial\Omega\subseteq\operatorname{\mathbb{R}}^{n+1}$ is compact, and for
each boundary point $\boldsymbol{x}_{0}\in\partial\Omega$ there exist local
coordinates of class $\mathrm{C}^{2m}$ which are obtained from the original
one by rotating and shifting, and such that the positive $x_{n+1}$-axis
corresponds to the direction of the inner normal vector
$-\boldsymbol{n}(\boldsymbol{x}_{0})$ to $\Omega$ at $\boldsymbol{x}_{0}$.
These local coordinates may be chosen such that they depend continuously on
boundary point $\boldsymbol{x}_{0}\in\partial\Omega$. We now consider the
differential operators with spatial dependent coefficients of the form

$\mathcal{A}(\boldsymbol{x},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}},\quad\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m,$

where the $\operatorname{\mathcal{B}}(E)$-valued coefficients satisfy the
following regularity and ellipticity conditions.

######  Assumption 5.1.

Let the following conditions be satisfied:

  1. (1)

Smoothness conditions:

    1. (a)

$a_{\boldsymbol{\alpha}}\in\mathrm{C}_{\mathrm{l}}(\overline{\Omega};\operatorname{\mathcal{B}}(E))$
for each multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|=2m$;

    2. (b)

$a_{\boldsymbol{\alpha}}\in[\LL_{\infty}+\LL_{r_{k}}](\Omega;\operatorname{\mathcal{B}}(E))$
for some $r_{k}\geq p$ such that $2m-k>\tfrac{n}{r_{k}}$, for each multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\alpha}\right|=k<2m$;

    3. (c)

$b_{j,k,\boldsymbol{\beta}}\in\mathrm{C}^{2m-k}(\partial\Omega;\operatorname{\mathcal{B}}(E))$
and
$b_{j,k,\boldsymbol{\beta}}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})+\ldots+\operatorname{\operatorname{ran}}(\mathcal{P}_{j,m_{j}})$
for each $j=1,\ldots,m$, $0\leq k\leq m_{j}$ and each multi-index
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\beta}\right|=k$.

  2. (2)

There is an angle $\phi_{\mathcal{A},\mathcal{B}}\in[0,\pi)$ such that the
following assertions hold true:

    1. (a)

ellipticity of the interior symbol: The principal symbol

$\mathcal{A}_{\\#}(\boldsymbol{x},\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\boldsymbol{\xi}^{\boldsymbol{\alpha}}$

is parameter elliptic with angle of ellipticity less than
$\phi_{\mathcal{A},\mathcal{B}}$, for each
$\boldsymbol{x}\in\overline{\Omega}\cup\\{\infty\\}$.

    2. (b)

Lopatinskii–Shapiro condition: At each boundary point
$\boldsymbol{x}\in\partial\Omega$, and for all
$\lambda\in\overline{\Sigma}_{\pi-\phi_{\mathcal{A},\mathcal{B}}}$ and
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$ such that
$\boldsymbol{\xi}\cdot\boldsymbol{n}(\boldsymbol{x}_{0})=0$, but
$(\lambda,\boldsymbol{\xi})\neq(0,\boldsymbol{0})$, the initial value problem

$\displaystyle\mathcal{A}_{\\#}(\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}\tfrac{\partial}{\partial
y})v(y)$ $\displaystyle=0$ $\displaystyle\text{for }y>0,$
$\displaystyle\mathcal{B}_{j}(\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}\tfrac{\partial}{\partial
y})v(0)$ $\displaystyle=h_{j}$ $\displaystyle\text{for }j=1,\ldots,m,$

has a unique solution $v\in\mathrm{C}_{0}(\operatorname{\mathbb{R}}_{+};E)$,
for each $(h_{1},\ldots,h_{m})^{\mathsf{T}}\in E^{m}$.

Under these assumptions, we may now prove maximal regularity of the parabolic
problem associated to the $\LL_{p}(\Omega;E)$-realisation of the boundary
value problem.

######  Theorem 5.2.

Let $E$ be a Banach space of class $\mathcal{HT}$,
$n,m\in\operatorname{\mathbb{N}}$ and $p\in(1,\infty)$. Let
$\Omega\subseteq\operatorname{\mathbb{R}}^{n+1}$ be a domain with compact
$\mathrm{C}^{2m}$-boundary. Suppose that for some angle
$\phi_{\mathcal{A},\mathcal{B}}\in[0,\pi)$ the boundary value problem
$(\mathcal{A}(\boldsymbol{x},D),(\mathcal{B}_{j}(\boldsymbol{x},D))_{j=1}^{m})$
satisfies Assumption 5.1. Let $A_{B}$ denote the
$\LL_{p}(\Omega;E)$-realisation of the boundary value problem with domain

$\operatorname{\operatorname{dom}}(A_{B})=\\{u\in\mathrm{W}_{p}^{2m}(\Omega;E):\,\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})u=0\,\text{on
}\partial\Omega,\,j=1,\ldots,m\\}.$

Then, for each $\phi>\phi_{\mathcal{A},\mathcal{B}}$, there is a constant
$\mu_{\phi}\geq 0$ such that $\mu_{\phi}+A_{B}$ is $\mathcal{R}$-sectorial
with $\phi^{\mathcal{R}}_{\mu_{\phi}+A_{B}}\leq\phi$. In particular, if
$\phi_{\mathcal{A},\mathcal{B}}<\tfrac{\pi}{2}$, the parabolic initial-
boundary value problem

$\displaystyle\partial_{t}u+A_{B}u+\mu_{\phi}u$ $\displaystyle=f,$
$\displaystyle t>0,$ $\displaystyle u(0)$ $\displaystyle=0$

has the property of maximal regularity in the class
$\LL_{q}(\operatorname{\mathbb{R}}_{+};\LL_{p}(\Omega;E))$ for each
$q\in(1,\infty)$.

###### Proof.

The localisation procedure as presented in [17, Section 8.2] carries over
almost literally; again we consider the operators
$\mathcal{B}_{j}(\mathrm{D})\mathcal{P}_{j,k}$ for each pair $(j,k)$
separately, but the technique stays the same.
As a first step, one needs to find suitable (local) diffeomorphisms which can
be used to _flatten_ the boundary. To this end, given any
$\boldsymbol{x}_{0}\in\partial\Omega$ there are local coordinates such that
the inner normal vector $-\boldsymbol{n}(\boldsymbol{x}_{0})$ to $\Omega$ at
$\boldsymbol{x}_{0}$ corresponds to the $n+1$-st direction, and then, since
$\Omega$ is a $\mathrm{C}^{2m}$-domain, one finds open sets
$U_{1}\subseteq\operatorname{\mathbb{R}}^{n}$ and
$U_{2}\subseteq\operatorname{\mathbb{R}}$ such that $\boldsymbol{x}_{0}\in
U:=U_{1}\times U_{2}\subseteq\operatorname{\mathbb{R}}^{n+1}$, and there is a
height function
$h\in\mathrm{C}^{2m}(\overline{U_{1}};\operatorname{\mathbb{R}})$ such that
$\Omega\cap U=\\{(\boldsymbol{x}^{\prime},y)\in
U:\,y>h(\boldsymbol{x}^{\prime})\\}$ and $\partial\Omega\cap
U=\\{(\boldsymbol{x}^{\prime},y)\in U:\,y=h(\boldsymbol{x}^{\prime})\\}$ and
$\Phi(\boldsymbol{x}^{\prime},y)=(\boldsymbol{x}^{\prime},y-h(\boldsymbol{x}^{\prime}))^{\mathsf{T}}$
defines a $\mathrm{C}^{2m}$-diffeomorphism between $U$ and
$\tilde{U}:=\Phi(U)$, which can be extended to a
$\mathrm{C}^{2m}$-diffeomorphism

$\tilde{\Phi}:\\{\boldsymbol{x}=(\boldsymbol{x}^{\prime},y)\in\operatorname{\mathbb{R}}^{n+1}:\,y>\tilde{h}(\boldsymbol{x}^{\prime})\\}\rightarrow\operatorname{\mathbb{R}}_{+}^{n+1}\quad\text{with
}\tilde{\Phi}|_{U}=\Phi.$

In the following, we often will not distinguish between $\Phi$ and its
extension $\tilde{\Phi}$, if no confusion may arise. Moreover, such a
diffeomorphism $\Phi$ can be chosen to be _$\mathrm{C}^{2m}$ -admissible_,
i.e. the normal direction to $\partial\Omega$ at
$\boldsymbol{x}_{0}\in\partial\Omega$ and the tangent space
$T_{\partial\Omega;\boldsymbol{x}_{0}}$ to $\partial\Omega$ at
$\boldsymbol{x}_{0}\in\partial\Omega$ are transformed into the normal
direction and the tangential space
$T_{\operatorname{\mathbb{R}}_{+}^{n+1};\Phi(\boldsymbol{x}_{0})}$ at
$\Phi(\boldsymbol{x}_{0})\in\partial\operatorname{\mathbb{R}}_{+}^{n+1}$ to
$\operatorname{\mathbb{R}}_{+}^{n+1}$. The latter property can be
characterised by a block-diagonal form of the Jacobi matrix
$\mathrm{D}\Phi(\boldsymbol{x}_{0})$, i.e.

$\mathrm{D}\Phi(\boldsymbol{x}_{0})=\left(\begin{array}[]{cc}\boldsymbol{H}(\boldsymbol{x}_{0})&0\\\
0&d(\boldsymbol{x}_{0})\end{array}\right),$

where $\boldsymbol{H}(\boldsymbol{x}_{0})\in\operatorname{\mathbb{R}}^{n\times
n}$ is invertible and $d(\boldsymbol{x}_{0})>0$ is a positive scalar. Here,
replacing $\Phi(\boldsymbol{x})$ by
$(\mathrm{D}\Phi(\boldsymbol{x}_{0}))^{-1}\Phi(\boldsymbol{x})$, we may
w.l.o.g. assume that
$(\mathrm{D}\Phi)(\boldsymbol{x}_{0})=\boldsymbol{I}_{n+1}$ is the identity
matrix. Using the admissible $\mathrm{C}^{2m}$-diffeomorphisms, we may now
_push forward_ any differential equation on $\Omega$ by defining

$\mathcal{G}^{\Phi}:u\mapsto\mathcal{G}^{\Phi}u:=v,\quad\mathcal{G}^{\Phi}u(\boldsymbol{y})=v(\boldsymbol{y}):=u(\Phi^{-1}(\boldsymbol{y})),\quad\boldsymbol{y}\in\tilde{U}\cap\operatorname{\mathbb{R}}_{+}^{n+1}.$

Given a multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}\setminus\\{\boldsymbol{0}\\}$,
by the Leibniz formula for derivatives we then obtain that

$\mathrm{D}^{\boldsymbol{\alpha}}(\mathcal{G}^{\Phi}u)(\boldsymbol{y})=\sum_{1\leq\left|\boldsymbol{\gamma}\right|\leq\left|\boldsymbol{\alpha}\right|}q_{\boldsymbol{\alpha},\boldsymbol{\gamma}}(\Phi;\boldsymbol{y})(\mathrm{D}^{\boldsymbol{\gamma}}u)(\Phi^{-1}(\boldsymbol{y})),$

where each $q_{\boldsymbol{\alpha},\boldsymbol{\gamma}}$ is a homogeneous
polynomial of degree $\left|\boldsymbol{\gamma}\right|$ in the derivatives
$\mathrm{D}^{\boldsymbol{\beta}}(\Phi^{-1})$ of order
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ such that
$1\leq\left|\boldsymbol{\beta}\right|\leq\left|\boldsymbol{\alpha}\right|-\left|\boldsymbol{\gamma}\right|-1$.
Therefore, in this way the differential operators
$\mathcal{A}(\boldsymbol{x},\boldsymbol{D})=\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}}$
and
$\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k}$
will be pushed forward to differential operators of the same order

$\displaystyle\mathcal{A}^{\Phi}(\boldsymbol{y},\mathrm{D})$
$\displaystyle=\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a^{\Phi}_{\boldsymbol{\alpha}}(\boldsymbol{y})\mathrm{D}^{\boldsymbol{\alpha}},$
$\displaystyle\mathcal{B}_{j}^{\Phi}(\boldsymbol{y},\mathrm{D})$
$\displaystyle=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b^{\Phi}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m.$

Alternatively, we may write
$\mathcal{A}^{\Phi}(\boldsymbol{y},\mathrm{D})=\mathcal{G}\mathcal{A}(\boldsymbol{x},\mathrm{D})\mathcal{G}^{-1}$
for $y\in\tilde{U}\cap\operatorname{\mathbb{R}}_{+}^{n+1}$ and
$\mathcal{B}^{\Phi}_{j}(\boldsymbol{y},\mathrm{D})=\mathcal{G}\mathcal{B}^{\Phi}_{j}(\boldsymbol{x},\mathrm{D})\mathcal{G}^{-1}$
for $y\in\partial\operatorname{\mathbb{R}}_{+}^{n+1}\cap\tilde{U}$. Since

$\displaystyle\mathcal{A}^{\Phi}(\boldsymbol{y},\mathrm{D})(\mathcal{G}u)(\boldsymbol{y})$
$\displaystyle=\big{(}\mathcal{G}\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}_{\boldsymbol{x}}^{\boldsymbol{\alpha}}\mathcal{G}^{-1}\mathcal{G}v\big{)}(\boldsymbol{y})=\mathcal{G}\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\Phi^{-1}(\boldsymbol{y}))\mathrm{D}_{\boldsymbol{x}}^{\boldsymbol{\alpha}}|_{\boldsymbol{x}=\Phi^{-1}(\boldsymbol{y})}u(\Phi^{-1}(\boldsymbol{y}))$
$\displaystyle=\sum_{\left|\boldsymbol{\alpha}\right|\leq
2m}a_{\boldsymbol{\alpha}}(\Phi^{-1}(\boldsymbol{y}))\big{(}\mathrm{D}\Phi(\Phi^{-1}y))^{\mathsf{T}}\mathrm{D}_{\boldsymbol{y}}\big{)}^{\boldsymbol{\alpha}}(\mathcal{G}u)(\boldsymbol{y}),$

the principal part of the interior differential operator is given by

$\mathcal{A}^{\Phi}_{\\#}(\boldsymbol{y},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}(\Phi^{-1}(\boldsymbol{y}))\big{(}\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\mathrm{D}\big{)}^{\boldsymbol{\alpha}}=\mathcal{A}_{\\#}(\Phi^{-1}(\boldsymbol{y}),\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\mathrm{D}).$

Similarly, for the boundary operators we obtain that

$\mathcal{B}^{\Phi}_{j}(\boldsymbol{y},\mathrm{D})=\mathcal{G}\big{(}\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b_{j,k,\boldsymbol{\beta}}(\boldsymbol{x})\mathrm{D}_{\boldsymbol{x}}^{\boldsymbol{\beta}}\mathcal{P}_{j,k}\big{)}\mathcal{G}^{-1}=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|\leq
k}b_{j,k,\boldsymbol{\beta}}(\Phi^{-1}(\boldsymbol{y}))\big{[}\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\mathrm{D}\big{]}^{\boldsymbol{\beta}}$

and from this their principle parts

$\mathcal{B}^{\Phi}_{j,\\#}(\boldsymbol{y},\boldsymbol{\xi})=\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}b_{j,k,\boldsymbol{\beta}}(\Phi^{-1}(\boldsymbol{y}))\big{[}\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\boldsymbol{\xi}\big{]}^{\boldsymbol{\beta}}=\mathcal{B}_{j,\\#}(\Phi^{-1}(\boldsymbol{y}),\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\boldsymbol{\xi}).$

With these relations at hand, we may check that the Lopatinskii–Shapiro
condition is unaffected under $\mathrm{C}^{2m}$-admissible transformations. By
this, we mean the following. The Lopatinskii–Shapiro condition for the
original problem reads: For every $\lambda\in\Sigma_{\pi-\phi}$, every point
$\boldsymbol{x}\in\partial\Omega$ and every
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}$ such that
$\boldsymbol{\xi}\cdot\boldsymbol{n}(\boldsymbol{x})=0$ for the outer normal
vector $\boldsymbol{n}(\boldsymbol{x}_{0})$ at $\boldsymbol{x}_{0}$, and every
$\boldsymbol{w}=(w_{1},\ldots,w_{m})^{\mathsf{T}}\in E^{m}$, the initial
boundary value problem

$\begin{cases}(\lambda+\mathcal{A}_{\\#}(\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}\frac{\partial}{\partial\eta})v(\eta)=0,&\eta>0,\\\
\mathcal{B}_{j,\\#}(\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}\frac{\partial}{\partial\eta})v(0)=w_{j},&j=1,\ldots,m\end{cases}$

has a unique solution
$v=v_{\boldsymbol{x},\lambda,\boldsymbol{\xi}}(\boldsymbol{w})$ in the class
$v\in\mathrm{C}_{0}(\operatorname{\mathbb{R}}_{+};E)$. The Lopatinskii–Shapiro
condition for the transformed problem involves the initial value problem

$\begin{cases}(\lambda+\mathcal{A}^{\Phi}_{\\#}(\boldsymbol{y},\boldsymbol{\xi}^{\prime},-\mathrm{i}\frac{\partial}{\partial\sigma})v^{\Phi}(\sigma)=0,&\sigma>0,\\\
\mathcal{B}^{\Phi}_{j,\\#}(\boldsymbol{y},\boldsymbol{\xi}^{\prime},-\mathrm{i}\frac{\partial}{\partial\sigma})v^{\Phi}(\sigma)=w^{\Phi}_{j},&j=1,\ldots,m.\end{cases}$

From the previous considerations, this means that

$\begin{cases}(\lambda+\mathcal{A}_{\\#}(\Phi^{-1}(\boldsymbol{y}),\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}(\boldsymbol{\xi}^{\prime},\frac{\partial}{\partial\sigma})^{\mathsf{T}})v^{\Phi}(\sigma)=0,&\sigma>0,\\\
\mathcal{B}_{j,\\#}(\Phi^{-1}(\boldsymbol{y}),\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}(\boldsymbol{\xi}^{\prime},\frac{\partial}{\partial\sigma})^{\mathsf{T}})v^{\Phi}(\sigma)=w^{\Phi}_{j},&j=1,\ldots,m,\end{cases}$

where we may write
$\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}(\boldsymbol{\xi}^{\prime},-\mathrm{i}\frac{\partial}{\partial\sigma})^{\mathsf{T}}=\kappa(\boldsymbol{y})\big{(}\tilde{\boldsymbol{\xi}}(\Phi^{-1}(\boldsymbol{y}),\boldsymbol{\xi}^{\prime})+\mathrm{i}\boldsymbol{n}(\Phi^{-1}(\boldsymbol{y}))\big{)}$
for some $\kappa(\boldsymbol{y})>0$ and
$\tilde{\boldsymbol{\xi}}\in\operatorname{\mathbb{R}}^{n}$. Since $\Phi$ is an
admissible $\mathrm{C}^{2m}$-diffeomorphism it holds that
$\tilde{\boldsymbol{\xi}}\cdot\boldsymbol{n}=0$. Therefore, the solutions to
the original (at $\boldsymbol{x}$) and the transformed (at
$\boldsymbol{y}=\Phi(\boldsymbol{x})$) initial value problem are related via

$w^{\Phi}_{\boldsymbol{y},\lambda,\boldsymbol{\xi}^{\prime}}(\sigma)=w_{\Phi^{-1}(\boldsymbol{y}),\lambda,\tilde{\boldsymbol{\xi}}}(\frac{\sigma}{\kappa}),\quad\text{that
is}\quad
w_{\boldsymbol{x},\lambda,\boldsymbol{\xi}}(\eta)=w^{\Phi}_{\Phi(\boldsymbol{y}),\lambda,\boldsymbol{\xi}^{\prime}}(\kappa\eta),$

where $\kappa>0$ is the norm of
$\mathrm{D}\Phi(\Phi^{-1}(\boldsymbol{y}))^{\mathsf{T}}\boldsymbol{n}=\mathrm{D}\Phi(\boldsymbol{x})^{\mathsf{T}}\boldsymbol{n}$.
This shows, in particular, that the validity of the Lopatinskii–Shapiro
condition is not affected by admissible $\mathrm{C}^{2m}$-diffeomorphisms.

Now, we may find such an admissible $\mathrm{C}^{2m}$-diffeomorphism
$\Phi_{\boldsymbol{x}_{0}}:U_{\boldsymbol{x}_{0}}\rightarrow\tilde{U}_{\boldsymbol{x}_{0}}$
for each boundary point $\boldsymbol{x}_{0}\in\partial\Omega$, and moreover,
since $U_{\boldsymbol{x}_{0}}$ is an open neighbourhood of
$\boldsymbol{x}_{0}$, we find radii $r(\boldsymbol{x}_{0})>0$ such that
$\Phi_{\boldsymbol{x}_{0}}^{-1}\big{(}B_{2r(\boldsymbol{x}_{0})}(\boldsymbol{y}_{0})\big{)}\subseteq
U_{\boldsymbol{x}_{0}}$ for each $\boldsymbol{y}_{0}=\Phi(\boldsymbol{x}_{0})$
and, for some $\varepsilon_{0}>0$ which is given a priori, the coefficients
for the principle parts of the pushed-forward operators have locally small
oscillations in the sense that

$\displaystyle\sum_{\left|\boldsymbol{\alpha}\right|}\left\|a^{\Phi}_{\boldsymbol{\alpha}}(\boldsymbol{y})-a^{\Phi}_{\boldsymbol{\alpha}}(\boldsymbol{y}_{0})\right\|$
$\displaystyle<\varepsilon_{0},$
$\displaystyle\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}\left\|b^{\Phi}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y})-b^{\Phi}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y}_{0})\right\|$
$\displaystyle<\varepsilon_{0}\quad\text{for all }\boldsymbol{y}\in
B_{2r(\boldsymbol{x}_{0})}(\boldsymbol{y}_{0}).$

Then, trivially

$\bigcup_{\boldsymbol{x}_{0}\in\partial\Omega}\Phi_{\boldsymbol{x}_{0}}^{-1}\big{(}B_{r(\boldsymbol{x}_{0})}(\boldsymbol{x}_{0})\big{)}\supset\partial\Omega$

is an open cover for the compact boundary $\partial\Omega$, so that we may
choose a finite subcover $\bigcup_{\nu=1}^{N}U_{\nu}$ with
$U_{\nu}=\Phi_{x_{\nu}}^{-1}(B_{r(\boldsymbol{x}_{\nu})}(\boldsymbol{y}_{\nu}))$
for some $\\{\boldsymbol{x}_{1},\ldots,x_{N}\\}\subseteq\partial\Omega$ and
$\boldsymbol{y}_{\nu}:=\Phi(\boldsymbol{x}_{\nu})$, $\nu=1,\ldots,N$.
For each of these $\nu=1,\ldots,N$, we may then extend the top-order
coefficients $a^{\Phi}_{\boldsymbol{\alpha}}(\boldsymbol{y})$ on
$B_{r(\boldsymbol{x}_{\nu})}(\boldsymbol{y}_{\nu})\cap\operatorname{\mathbb{R}}_{+}^{n+1}$
to the half-space $\operatorname{\mathbb{R}}_{+}^{n+1}$ by reflection at the
surface $\partial B_{r(\boldsymbol{x}_{\nu})}(\boldsymbol{y}_{\nu})$, viz.

$\displaystyle
a^{\nu}_{\boldsymbol{\alpha}}(\boldsymbol{y}):=\begin{cases}a^{\Phi_{\boldsymbol{x}_{\nu}}}_{\boldsymbol{\alpha}}(\boldsymbol{y}),&\boldsymbol{y}\in\overline{B_{r(\boldsymbol{x}_{\nu})}(\boldsymbol{y}_{\nu})}\cap\overline{\operatorname{\mathbb{R}}_{+}^{n+1}},\\\
a^{\Phi_{\boldsymbol{x}_{\nu}}}_{\boldsymbol{\alpha}}(\boldsymbol{y}_{\nu}+\frac{r(\boldsymbol{x}_{\nu})^{2}}{\left|\boldsymbol{y}-\boldsymbol{y}_{\nu}\right|^{2}}(\boldsymbol{y}-\boldsymbol{y}_{\nu})),&\boldsymbol{y}\in\overline{\operatorname{\mathbb{R}}_{+}^{n+1}}\setminus\overline{B_{r(\boldsymbol{x}_{\nu})}(\boldsymbol{y}_{\nu})}\end{cases}\quad\text{for
}\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}\text{ with
}\left|\boldsymbol{\alpha}\right|=2m,$

and also extend the coefficients of the boundary symbols for all
$j=1,\ldots,m$ and $k=0,\ldots,m_{j}$ according to

$\displaystyle b^{\nu}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y})$
$\displaystyle=b^{\Phi_{\boldsymbol{x}_{\nu}}}(\boldsymbol{y}_{\nu}+\chi(\frac{\boldsymbol{y}-\boldsymbol{y}_{\nu}}{r(\boldsymbol{x}_{\nu})})(\boldsymbol{y}-\boldsymbol{y}_{\nu})),\quad\boldsymbol{y}\in\operatorname{\mathbb{R}}_{+}^{n+1}$

where $\chi\in\mathrm{C}_{0}^{\infty}(\operatorname{\mathbb{R}}^{n})$ has
compact support which is contained in $B_{2}(\boldsymbol{0})$, and is
identically $1$ on $\overline{B_{1}(\boldsymbol{0})}$. Note that these
definitions lead to top order coefficients on
$\overline{\operatorname{\mathbb{R}}_{+}^{n+1}}$ which still have small
oscillations, i.e.

$\sum_{\left|\boldsymbol{\alpha}\right|=2m}\left\|a^{\nu}_{\boldsymbol{\alpha}}(\boldsymbol{y})-a^{\nu}_{\boldsymbol{\alpha}}(\boldsymbol{y}_{\nu})\right\|,\quad\sum_{k=0}^{m_{j}}\sum_{\left|\boldsymbol{\beta}\right|=k}\left\|b^{\nu}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y})-b^{\nu}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y}_{\nu})\right\|<\varepsilon_{0},\quad
j=1,\ldots,m,\,\boldsymbol{y}\in\overline{\operatorname{\mathbb{R}}_{+}^{n+1}}.$

For any given $\lambda\in\Sigma_{\pi-\phi}$ and $f\in\LL_{p}(\Omega;E)$, let
us consider the boundary value problem

$\begin{cases}\lambda u+\mathcal{A}(\boldsymbol{x},\mathrm{D})u=f,&\text{in
}\Omega,\\\ \mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})u=0,&\text{on
}\partial\Omega,\,j=1,\ldots,m.\end{cases}$

We choose a partition of unity
$(\varphi_{\nu})_{\nu=1,\ldots,M}\subseteq\mathrm{C}^{\infty}(\operatorname{\mathbb{R}}^{n})$
with $M\geq N$ and such that $0\leq\varphi_{\nu}\leq 1$ for all
$\nu=1,\ldots,M$, and $\operatorname{supp}\varphi_{\nu}\subseteq U_{\nu}$ for
$\nu=1,\ldots,N$ whereas
$\operatorname{supp}\varphi_{\nu}\subseteq\operatorname{\mathbb{R}}^{n}\setminus\partial\Omega$
for $\nu=N+1,\ldots,M$. Then a function $u\in\mathrm{W}_{p}^{2m}(\Omega;E)$
solves the boundary value problem (5.1) if and only if for $\nu=1,\ldots,N$

$\begin{cases}\lambda(\varphi_{\nu}u)+\varphi_{\nu}\mathcal{A}(\boldsymbol{x},\mathrm{D})u=\varphi_{\nu}f&\text{in
}\Omega\cap U_{\nu},\\\
\varphi_{\nu}\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D})u=0&\text{on
}\partial\Omega\cap U_{\nu},\,j=1,\ldots,m\end{cases}$

and for $\nu=N+1,\ldots,M$

$\lambda(\varphi_{\nu}u)+\varphi_{\nu}\mathcal{A}(\boldsymbol{x},\mathrm{D})u=\varphi_{\nu}f\quad\text{on
}\Omega.$

Note that in the latter case no boundary conditions are involved since
$\varphi_{\nu}\equiv 0$ in a neighbourhood of $\partial\Omega$. For
$\nu=N+1,\ldots,M$, we may write the equation equivalently as

$\lambda(\varphi_{\nu}u)+\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D})(\varphi_{\nu}u)=\varphi_{\nu}f+\big{[}\mathcal{A}_{\\#},\varphi_{\nu}\big{]}u-\varphi_{\nu}(\mathcal{A}(\boldsymbol{x},\mathrm{D})-\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D}))u=:\varphi_{\nu}f+\mathcal{C}_{\nu}(\boldsymbol{x},\mathrm{D})u,$

where the latter terms on the right-hand side include the commutator
$[\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D}),\varphi_{\nu}]$ between the
principle part $\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D})$ of the elliptic
differential operator $\mathcal{A}(\boldsymbol{x},\mathrm{D})$ and the
multiplication operator for multiplication with $\varphi_{\nu}$, and
$\mathcal{A}(\boldsymbol{x},\mathrm{D})-\mathcal{A}_{\\#}(\boldsymbol{x},\mathrm{D})$
are the lower-order terms of $\mathcal{A}(\boldsymbol{x},\mathrm{D})$. We may,
therefore, for $\nu=N+1,\ldots,M$, denote by $A_{\nu}$ the
$\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$-realisation of the operator

$\mathcal{A}^{\nu}(\boldsymbol{x},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a^{\nu}_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\alpha}$

and set $R^{\lambda}_{\nu}:=(\lambda+A_{\nu})^{-1}$ for the resolvent
operator, if it exists. Then, $u$ has to respect the identity

$\varphi_{\nu}u=R^{\lambda}_{\nu}(\varphi_{\nu}f)+R^{\lambda}_{\nu}\mathcal{C}_{\nu}(\boldsymbol{x},\mathrm{D})u,\quad\nu=N+1,\ldots,M.$

On the other hand, for $\nu=1,\ldots,N$, we use the local coordinates and the
$\mathrm{C}^{2m}$-diffeomorphisms introduced above and note that the pull-back
operators $\mathcal{G}_{\nu}$ corresponding to the diffeomorphisms
$\Phi_{\nu}=\Phi_{\boldsymbol{x}_{\nu}}$, $\nu=1,\ldots,N$, act as continuous
operators between the Sobolev spaces $\mathrm{W}_{p}^{l}(U_{k};E)$ and
$\mathrm{W}_{p}^{l}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, viz.

$\mathcal{G}_{\nu}:\quad u\mapsto
u^{\Phi_{\nu}},\,u^{\Phi_{\nu}}(\boldsymbol{y})=u(\Phi_{\nu}^{-1}(\boldsymbol{y})),\quad\mathrm{W}_{p}^{l}(U_{\nu};E)\mapsto\mathrm{W}_{p}^{l}(\operatorname{\mathbb{R}}_{+}^{n+1};E),\quad\nu=1,\ldots,N,\,l=0,1,\ldots,2m.$

We set

$\displaystyle\mathcal{A}^{\nu}(\boldsymbol{y},\mathrm{D})$
$\displaystyle:=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a^{\nu}_{\boldsymbol{\alpha}}(\boldsymbol{y})\mathrm{D}^{\boldsymbol{\alpha}},$
$\displaystyle\mathcal{B}^{\nu}_{j}(\boldsymbol{y},\mathrm{D})$
$\displaystyle:=\sum_{k=0}^{m_{j}}b^{\nu}_{j,k,\boldsymbol{\beta}}(\boldsymbol{y})\mathrm{D}^{\boldsymbol{\beta}}\mathcal{P}_{j,k},\quad
j=1,\ldots,m,$

and rewrite the corresponding boundary value problems for $\varphi_{\nu}u$,
$\nu=1,\ldots,N$, as

$\begin{cases}\lambda\mathcal{G}_{\nu}(\varphi_{\nu}u)+\mathcal{A}^{\nu}(\boldsymbol{y},\mathrm{D})\mathcal{G}_{\nu}(\varphi_{\nu}u)&=\mathcal{G}_{\nu}(\varphi_{\nu}f)+\mathcal{G}_{\nu}\big{(}\big{[}\mathcal{A}(\boldsymbol{x},\mathrm{D}),\varphi_{\nu}\big{]}-\mathcal{G}_{k}^{-1}(\mathcal{A}^{\Phi_{\nu}}(\boldsymbol{y},\mathrm{D})-\mathcal{A}^{\nu}(\boldsymbol{y},\mathrm{D}))\mathcal{G}_{k}\varphi_{\nu}\big{)}u,\\\
\mathcal{B}^{\nu}_{j}(\boldsymbol{y},\mathrm{D})\mathcal{G}_{\nu}(\varphi_{\nu}u)&=\mathcal{G}_{\nu}\big{(}\big{[}\mathcal{B}_{j}(\boldsymbol{x},\mathrm{D}),\varphi_{\nu}\big{]}-\mathcal{G}_{\nu}^{-1}(\mathcal{B}^{\Phi_{\nu}}_{j}(\boldsymbol{y},\mathrm{D})-\mathcal{B}^{\nu}_{j}(\boldsymbol{y},\mathrm{D}))\mathcal{G}_{\nu}\varphi_{\nu}\big{)}u.\end{cases}$

Since we imposed the condition that
$b_{j,k,\boldsymbol{\beta}}(\operatorname{\operatorname{ran}}\mathcal{P}_{j,k})\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})+\ldots+\operatorname{\operatorname{ran}}(\mathcal{P}_{m_{j},k})$,
on the right-hand side only lower order differential operators appear, i.e.
the right-hand side includes only derivatives up to order $2m-1$ resp. up to
order $k-1$ on every subspace
$\operatorname{\operatorname{ran}}\mathcal{P}_{j,k}$, $k=1,\ldots,m_{j}$.
Hence, these terms can be considered as a small perturbation to the operators
$\mathcal{A}^{\nu}(\boldsymbol{y},\mathrm{D})$ and
$\mathcal{B}^{\nu}_{j}(\boldsymbol{y},\mathrm{D})$, $j=1,\ldots,m$,
respectively. The solution then will satisfy the implicit relation

$\mathcal{G}_{\nu}(\varphi_{\nu}u)=\tilde{R}^{\lambda,\nu}_{0}\big{(}\mathcal{G}_{\nu}(\varphi\nu
u)+\mathcal{G}_{\nu}\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D})u\big{)}+\sum_{j=0}^{m}\sum_{k=0}^{m_{j}}\tilde{S}^{\lambda,\nu}_{j,k}\mathcal{G}_{\nu}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})u,$

where we write

$\displaystyle\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D})u$
$\displaystyle:=\big{[}\mathcal{A}(\boldsymbol{x},\mathrm{D}),\varphi_{\nu}\big{]}u-\mathcal{G}_{\nu}^{-1}(\mathcal{A}^{\Phi_{\nu}}(\boldsymbol{y},\mathrm{D})-\mathcal{A}^{\nu}(\boldsymbol{y},\mathrm{D}))\mathcal{G}_{\nu}u,$
$\displaystyle\mathcal{D}^{\nu}_{j,k}u$
$\displaystyle:=\big{[}\mathcal{B}_{j,k}(\boldsymbol{x},\mathrm{D}),\varphi_{\nu}\big{]}-\mathcal{G}_{\nu}^{-1}(\mathcal{B}_{j,k}^{\Phi_{\nu}}(\boldsymbol{y},\mathrm{D})-\mathcal{B}^{\nu}_{j,k}(\boldsymbol{y},\mathrm{D}))\mathcal{G}_{\nu}u.$

From the solution and maximal regularity theory on the half-space we know that
we may write

$\displaystyle\tilde{S}^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})$
$\displaystyle=\big{(}\tilde{S}^{\lambda,\nu,\mathrm{I}}_{j,k}+\tilde{S}^{\lambda,\nu,\mathrm{II}}_{j,k}\big{)}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})$
$\displaystyle=\tilde{T}^{\lambda,\nu}_{j,k}(\left|\lambda\right|+(-\Delta)^{2m})^{(2m-k)/2m}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})$
$\displaystyle\quad+\tilde{R}^{\lambda,\nu}_{j,k}(\left|\lambda\right|+(-\Delta)^{2m})^{(2m-k-1)/2m}\partial_{n+1}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})$

where the operators $\tilde{T}^{\lambda,\nu}_{j,k}$ and
$R^{\lambda,\nu}_{j,k}$ satisfy $\mathcal{R}$-bounds

$\displaystyle\mathcal{R}(\\{\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\tilde{T}^{\lambda,\nu}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\})$ $\displaystyle<\infty,$
$\displaystyle\mathcal{R}(\\{\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\tilde{R}^{\lambda,\nu}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0},\,\left|\boldsymbol{\alpha}\right|\leq
2m\\})$ $\displaystyle<\infty,$

for each fixed $j=1,\ldots,m$, $k=0,1,\ldots,m_{j}$ and $\nu=1,\ldots,N$.
However, the set of admissible indices $(j,k,\nu)$ is finite, which implies
that

$\displaystyle\mathcal{R}(\\{\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\tilde{T}^{\lambda,\nu}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0},\,\left|\boldsymbol{\alpha}\right|\leq
2m,\,1\leq j\leq m,\,0\leq k\leq m_{j},\,1\leq\nu\leq N\\})$
$\displaystyle<\infty,$
$\displaystyle\mathcal{R}(\\{\left|\lambda\right|^{1-\frac{\left|\boldsymbol{\alpha}\right|}{2m}}\mathrm{D}^{\boldsymbol{\alpha}}\tilde{R}^{\lambda,\nu}_{j,k}:\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0},\,\left|\boldsymbol{\alpha}\right|\leq
2m,\,1\leq j\leq m,\,0\leq k\leq m_{j},\,1\leq\nu\leq N\\})$
$\displaystyle<\infty.$

Introducing

$T^{\lambda,\nu}_{j,k}:=\mathcal{G}_{\nu}^{-1}\tilde{T}^{\lambda,\nu}_{j,k}\mathcal{G}_{\nu}\quad\text{and}\quad
R^{\lambda,\nu}_{j,k}:=\mathcal{G}_{\nu}^{-1}\tilde{R}^{\lambda,\nu}_{j,k}\mathcal{G}_{\nu}$

and

$\displaystyle\mathcal{D}^{\nu,I}_{j,k}(\boldsymbol{x},\mathrm{D})$
$\displaystyle:=\mathcal{G}_{\nu}^{-1}\big{(}(-\Delta)^{2m}+\left|\lambda\right|\big{)}^{(2m-k)/2m}\mathcal{G}_{\nu}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D}),$
$\displaystyle\mathcal{D}^{\nu,II}_{j,k}(\boldsymbol{x},\mathrm{D})$
$\displaystyle:=\mathcal{G}_{\nu}^{-1}\big{(}(-\Delta)+\left|\lambda\right|\big{)}^{(2m-k-1)/2m}\partial_{n+1}\mathcal{G}_{\nu}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})$

by pull-back we get (for $\nu=1,\ldots,N$) the identities

$\displaystyle\varphi_{\nu}u$
$\displaystyle=R^{\lambda,\nu}_{0}\big{(}\varphi\nu
u+\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D})u\big{)}+\sum_{j=0}^{m}\sum_{k=0}^{m_{j}}\tilde{S}^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu}_{j,k}(\boldsymbol{x},\mathrm{D})u$
$\displaystyle=R^{\lambda,\nu}_{0}\big{(}\varphi\nu
u+\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D})u\big{)}+\sum_{j=0}^{m}\sum_{k=0}^{m_{j}}T^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu,I}_{j,k}(\boldsymbol{x},\mathrm{D})u+R^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu,II}_{j,k}(\boldsymbol{x},\mathrm{D})u.$

Summing over $\nu=1,\ldots,M$, we arrive at

$\displaystyle(\lambda+A_{B})^{-1}f$
$\displaystyle=u=\sum_{\nu=1}^{M}\phi_{\nu}u$
$\displaystyle=\sum_{\nu=1}^{M}R^{\lambda,\nu}(\phi_{\nu}f+\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D})u)+\sum_{\nu=1}^{N}\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}(T^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu,I}_{j,k}(\boldsymbol{x},\mathrm{D})+R^{\lambda,\nu}_{j,k}\mathcal{D}^{\nu,II}_{j,k}(\boldsymbol{x},\mathrm{D}))u$
$\displaystyle=R(\lambda)f+S(\lambda)u,$

where we set

$R(\lambda)=\sum_{\nu=1}^{M}R^{\lambda,\nu}_{0}\phi_{\nu},\quad
S(\lambda)=\sum_{\nu=1}^{N}\sum_{j=1}^{m}\sum_{k=0}^{m_{j}}(T^{\lambda,\nu}_{j,k}\mathcal{D}^{\lambda,\nu}_{j,k}+R^{\lambda,\nu}\mathcal{D}^{\lambda,\nu}_{j,k}).$

Thus, provided $\left\|S(\lambda)\right\|<1$, we may solve for the resolvent
operator by employing the Neumann series

$(\lambda+A_{B})^{-1}=(I-S(\lambda))^{-1}R(\lambda)f=\sum_{l=0}^{\infty}S(\lambda)^{l}R(\lambda)f.$

Here, each summand $S(\lambda)^{l}$, $l\geq 0$, is a sum of products of the
form

$U_{\lambda}^{0}\prod_{\eta=1}^{l}V_{\lambda}^{\eta}U_{\lambda}^{\eta},$

where
$U_{\lambda}^{\eta}\in\\{R^{\lambda,\nu},R^{\lambda,\nu}_{j,k},T^{\lambda,\nu}_{j,k}:\,j=1,\ldots,m,\,k=0,\ldots,m_{j}\\}$
is one of the resolvent or solution operators for the BVP, and
$V_{\lambda}^{\eta}\in\\{\mathcal{C}^{\nu}(\boldsymbol{x},\mathrm{D}),\mathcal{D}^{\nu,I}_{j,k},\mathcal{D}^{\nu,II}_{j,k}:\,j=1,\ldots,m,\,k=0,\ldots,m_{j}\\}$
are all lower-order operators, such that any product
$U_{\lambda}^{\eta}V_{\lambda}^{\eta}$ is, in fact, a compact operator on $X$.
Finally, we may employ Remark 2.3 to conclude that for sufficiently small
$\varepsilon>0$ and sufficiently large $\lambda_{0}\geq 0$, the family of
operators given by such Neumann series
$\\{\sum_{l=0}^{\infty}S(\lambda)^{l}R(\lambda):\,\lambda\in\Sigma_{\pi-\phi},\,\left|\lambda\right|\geq\lambda_{0}\\}$
not only is well-defined, but actually $\mathcal{R}$-bounded. By Theorem 2.12,
this implies that the abstract Cauchy problem

$\begin{cases}\frac{\mathrm{d}}{\mathrm{d}t}u(t)+A_{B}u(t)=f(t),&t\geq 0,\\\
u(0)=0\end{cases}$

has $\LL_{q}$-maximal regularity for all $q\in(1,\infty)$. ∎

##  6\. Inhomogeneous initial data and anisotropic Sobolev spaces

In [18], the authors extended their results in [17] to the case of
inhomogeneous initial data and anisotropic Sobolev spaces, i.e. distinct
integrability parameters $p,q\in(1,\infty)$ for the time and spatial variable,
resp., by identifying the optimal spaces for the data to obtain optimal
$\LL_{p}$-$\LL_{q}$-estimates. These results again heavily rely on the results
and techniques as in [17], especially vector-valued multiplier theorems on
Banach spaces with property $\mathcal{HT}$ (i.e. UMD-spaces), Gagliardo-
Nirenberg type estimates as well as optimal embedding results for Sobolev,
Besov and Triebel-Lizorkin spaces. The methods used in [18] translate more or
less directly to the situation considered here: Obviously, the optimal
regularity spaces have to be adjusted since the boundary operators have
different orders $k$ in the subspaces
$\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})$ corresponding to the
projections $\mathcal{P}_{j,k}$. We introduce the following assumptions for
this section.

  1. (D)

Assumptions on the data:

    1. (i)

$f\in\LL_{p}(J\times\Omega;E)$, where $J=[0,T]$ for some $T>0$;

    2. (ii)

$g_{j,k}\in\mathrm{W}_{p}^{(1,2m)\cdot\kappa_{k}}(J\times\partial\Omega;\operatorname{\operatorname{ran}}\mathcal{P}_{j,k})$,
where $\kappa_{k}=\frac{2m-k-\nicefrac{{1}}{{p}}}{2m}$, and we then set
$g_{j}=\sum_{k=0}^{m_{j}}g_{j,k}$;

    3. (iii)

$u_{0}\in\mathrm{B}_{p,p}^{2m(1-\nicefrac{{1}}{{p}})}(\Omega;E)$;

    4. (iv)

if $\kappa_{k}>\nicefrac{{1}}{{p}}$, then
$\mathcal{B}_{j}(0,\boldsymbol{x})\mathcal{P}_{j,k}u_{0}(\boldsymbol{x})=g_{j,k}(0,\boldsymbol{x})$
for $\boldsymbol{x}\in\partial\Omega$.

  2. (SD)

Regularity assumptions on the interior symbol:
There are $r_{l},s_{l}\geq p$ with
$\frac{1}{s_{l}}+\frac{n}{2mr_{l}}<1-\frac{l}{2m}$ such that

$\displaystyle a_{\boldsymbol{\alpha}}$
$\displaystyle\in\LL_{s_{l}}(J;[\LL_{r_{l}}+\LL_{\infty}](\Omega;\operatorname{\mathcal{B}}(E))),$
$\displaystyle\left|\boldsymbol{\alpha}\right|=l<2m,$ $\displaystyle
a_{\boldsymbol{\alpha}}$
$\displaystyle\in\mathrm{C}_{\mathrm{l}}(J\times\overline{\Omega};\operatorname{\mathcal{B}}(E)),$
$\displaystyle\left|\boldsymbol{\alpha}\right|=2m.$

  3. (SB)

Regularity assumptions on the boundary symbols:
There are $s_{j,k,l},r_{j,k,l}\geq p$ such that
$\frac{1}{s_{j,k,l}}+\frac{n-1}{2mr_{j,k,l}}<\kappa_{k}+\frac{l-k}{2m}$ and

$b_{j,k,\boldsymbol{\beta}}\in\mathrm{W}_{s_{j,k,l},r_{j,k,l}}^{(1,2m)\cdot\kappa_{k}}(J\times\partial\Omega;\operatorname{\mathcal{B}}(E)),\quad\text{for
each }\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}\text{ of length
}\left|\boldsymbol{\beta}\right|=l\leq k\leq m_{j}.$

Moreover,
$b_{j,k,\boldsymbol{\beta}}(t,\boldsymbol{x})(\operatorname{\operatorname{ran}}\mathcal{P}_{j,k})\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})+\ldots+\operatorname{\operatorname{ran}}(\mathcal{P}_{j,m_{j}})$
for a.e. $(t,\boldsymbol{x})\in J\times\partial\Omega$ and $j=1,\ldots,m$,
$k=0,1,\ldots,m_{j}$ and
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ such that
$\left|\boldsymbol{\beta}\right|\leq k$.

  4. (E)

Ellipticity of the interior symbol: For all $t\in J$,
$\boldsymbol{x}\in\overline{\Omega}$ and
$\boldsymbol{\xi}\in\mathbb{S}^{n-1}$, the spectrum of the principle part of
the interior symbol lies in the open right-half plane,

$\sigma(\mathcal{A}_{\\#}(t,\boldsymbol{x},\boldsymbol{\xi}))\subseteq\operatorname{\mathbb{C}}_{0}^{+},$

i.e. $\mathcal{A}(t,\boldsymbol{x},\mathrm{D})$ is _normally elliptic_. If
$\Omega$ is unbounded, the same condition is imposed at
$\boldsymbol{x}=\infty$ for
$\mathcal{A}_{\\#}(t,\infty,\boldsymbol{\xi}):=\lim_{\left|\boldsymbol{x}\right|\rightarrow\infty}\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}(t,\boldsymbol{x},\boldsymbol{\xi})\boldsymbol{\xi}^{\boldsymbol{\alpha}}$.

  5. (LS)

Lopatinskii-Shapiro condition: For all $t\in J$,
$\boldsymbol{x}\in\partial\Omega$ and all
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}$ with
$\boldsymbol{\xi}\cdot\boldsymbol{n}(\boldsymbol{x})=0$, and all
$\lambda\in\overline{\operatorname{\mathbb{C}}_{0}^{+}}$ such that
$(\lambda,\boldsymbol{\xi})\neq(0,\boldsymbol{0})$, the initial value problem

$\displaystyle\lambda
v(y)+\mathcal{A}_{\\#}(t,\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}(\boldsymbol{x})\tfrac{\partial}{\partial
y})v(y)$ $\displaystyle=0,$ $\displaystyle y>0,$
$\displaystyle\mathcal{B}_{j,\\#}(t,\boldsymbol{x},\boldsymbol{\xi}+\mathrm{i}\boldsymbol{n}(\boldsymbol{x})\frac{\partial}{\partial
y})v(0)$ $\displaystyle=h_{j},$ $\displaystyle j=1,\ldots,m,$

has a unique solution in the class
$v\in\mathrm{C}_{0}(\operatorname{\mathbb{R}}_{+};E)$.

The theorem on $\LL_{p}$-$\LL_{p}$-optimal regularity then reads as follows.

######  Theorem 6.1 ($\LL_{p}$–$\LL_{p}$-optimal regularity).

Let $E$ be a Banach space of class $\mathcal{HT}$ and
$\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ be a domain with compact
boundary of class $\partial\Omega\in\mathrm{C}^{2m}$. Let $p\in(1,\infty)$ and
suppose that assumptions (E), (LS), (SD) and (SB) hold true. Then the
inhomogeneous parabolic initial-boundary value problem

$\displaystyle\partial_{t}u+\mathcal{A}(t,\boldsymbol{x},\mathrm{D})u$
$\displaystyle=f(t,\boldsymbol{x}),$ $\displaystyle t\in
J,\,\boldsymbol{x}\in\Omega,$ (iIBVP)
$\displaystyle\mathcal{B}_{j}(t,\boldsymbol{x},\mathrm{D})u$
$\displaystyle=g_{j}(t,\boldsymbol{x}),$ $\displaystyle t\in
J,\,\boldsymbol{x}\in\partial\Omega,\,j=1,\ldots,m,$ $\displaystyle
u(0,\boldsymbol{x})$ $\displaystyle=u_{0}(\boldsymbol{x}),$
$\displaystyle\boldsymbol{x}\in\Omega$

has a unique solution in the class

$u\in\mathrm{W}_{p}^{(1,2)}(J\times\Omega;E)$

if and only if the data $f$, $\boldsymbol{g}$ and $u_{0}$ are subject to
condition (D).

We refrain from giving a proof, but only refer to the proof of [18, Theorem
2.1] which can be modified in a similar manner as the proof of Theorem 6.2
below.
For $\LL_{p}$–$\LL_{q}$-estimates with $p\neq q$, where $p\in(1,\infty)$
corresponds to the integrability parameter w.r.t. time $t\geq 0$ and
$q\in(1,\infty)$ reflects integrability w.r.t. the spatial variable
$\boldsymbol{x}\in\Omega$, a similar result holds true, but the smoothness
conditions on the coefficients and on the data have to be slightly modified,
cf. the corresponding result [18, Theorem 2.3] by Denk, Hieber and Prüss.

  1. (D1)

Assumptions on the data in case the $p\neq q$:

  2. (i)

$f\in\LL_{p}(J;\LL_{q}(\Omega;E))$,

  3. (ii)

$g_{j,k}\in\mathrm{F}_{p,q}^{\kappa_{k}}(J;\LL_{q}(\partial\Omega;\operatorname{\operatorname{ran}}\mathcal{P}_{j,k}))\cap\LL_{p}(J;\mathrm{B}_{q,q}^{2m\kappa_{k}}(\partial\Omega;\operatorname{\operatorname{ran}}\mathcal{P}_{j,k}))$
with $\kappa_{k}=\frac{2m-k-1/q}{2m}$, and we then set
$g_{j}=\sum_{k=0}^{m_{j}}g_{j,k}$,

  4. (iii)

$u_{0}\in\mathrm{B}_{q,p}^{2m(1-\nicefrac{{1}}{{p}})}(\Omega;E)$,

  5. (iv)

if $\kappa_{k}>1/q$, then
$\mathcal{B}_{j}(0,\boldsymbol{x})\mathcal{P}_{j,k}u_{0}(\boldsymbol{x})=g_{j,k}(0,\boldsymbol{x})$
for $\boldsymbol{x}\in\partial\Omega$, for each $j=1,\ldots,m$ and
$k=0,1,\ldots,m_{j}$.

  6. (SD1)

There are $s_{l}\geq p$ and $r_{l}\geq q$ with
$\frac{1}{s_{l}}+\frac{n}{2mr_{l}}<1-\frac{l}{2m}$ such that

$\displaystyle a_{\boldsymbol{\alpha}}$
$\displaystyle\in\LL_{s_{l}}(J;(\LL_{r_{l}}+\LL_{\infty})(\Omega;\operatorname{\mathcal{B}}(E)),$
$\displaystyle\left|\boldsymbol{\alpha}\right|=l<2m,$ $\displaystyle
a_{\boldsymbol{\alpha}}$
$\displaystyle\in\mathrm{C}_{\mathrm{l}}(J\times\overline{\Omega};\operatorname{\mathcal{B}}(E)),$
$\displaystyle\left|\boldsymbol{\alpha}\right|=2m.$

  7. (SB1)

There are $s_{j,k,l}\geq p$ and $r_{j,k,l}\geq q$ with
$\frac{1}{s_{j,k,l}}+\frac{n-1}{2mr_{j,k,l}}<\kappa_{k}+\frac{k-l}{2m}$ such
that

$b_{j,k,\boldsymbol{\beta}}\in\mathrm{W}_{s_{j,k,l},r_{j,k,l}}^{(1,2m)\cdot\kappa_{k}}(J\times\partial\Omega;\operatorname{\mathcal{B}}(E)))\quad\text{for
each }\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}\text{ of length
}\left|\boldsymbol{\beta}\right|=l\leq k\leq m_{j}.$

Moreover,
$b_{j,k,\boldsymbol{\beta}}(t,\boldsymbol{x})(\operatorname{\operatorname{ran}}\mathcal{P}_{j,k})\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})+\ldots+\operatorname{\operatorname{ran}}(\mathcal{P}_{j,m_{j}})$
for a.e. $(t,\boldsymbol{x})\in J\times\partial\Omega$ and $j=1,\ldots,m$,
$k=0,1,\ldots,m_{j}$ and
$\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n}$ such that
$\left|\boldsymbol{\beta}\right|\leq k$.

######  Theorem 6.2 ($\LL_{p}$–$\LL_{q}$-optimal regularity).

Let $E$ be a Banach space of class $\mathcal{HT}$ and
$\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ be a domain with compact
boundary of class $\partial\Omega\in\mathrm{C}^{2m}$. Let $p,q\in(1,\infty)$
and suppose that assumptions (E), (LS), (D1), (SD1) and (SB1) hold true. Then
the problem (iIBVP) has a unique solution in the class

$u\in\mathrm{W}_{p,q}^{(1,2)}(J\times\Omega;E)$

if and only if the data $f$, $\boldsymbol{g}$ and $u_{0}$ are subject to
conditions (D1).

Proof. Let us sketch the strategy which has been employed in [18] and comment
on adjustments that are necessary for the proof to carry over to the situation
considered here. As for the case of homogeneous initial data, i.e.
$\boldsymbol{u}_{0}=\boldsymbol{0}$, the basic strategy is to consider step by
step the full-space, the half-space and finally via localisation the general
domain problem. As the full-space problem obviously does not involve any
boundary conditions, by [18, Proposition 6.1], we know that the full-space
problem with constant coefficients admits a unique solution of iIBVP in the
class
$\mathrm{W}^{(1,2m)}_{p,q}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}^{n};E)$
if and only if
$f\in\LL_{p,q}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}^{n};E)$
and
$u_{0}\in\mathrm{B}_{q,p}^{2m(1-\frac{1}{p})}(\operatorname{\mathbb{R}}^{n};E)$.
For the next step, the initial-boundary value problem on the half-space, [18,
Proposition 6.4] and its proof can easily be modified as follows:

######  Proposition 6.3.

Assume that the BVP associated to the operators
$(\mathcal{A}(\mathrm{D}),\mathcal{B}_{1}(\mathrm{D}),\ldots,\mathcal{B}_{m}(\mathrm{D}))$
satisfy conditions (E) and (LS) and let $\mu>0$. Then the parabolic initial-
boundary value problem

$\displaystyle(\mu+\partial_{t}+\mathcal{A}(\mathrm{D}))u$ $\displaystyle=f$
$\displaystyle\text{in
}\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n},$ (24)
$\displaystyle\mathcal{B}_{j}(\mathrm{D})u$ $\displaystyle=g_{j},$
$\displaystyle\text{in
}\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}^{n-1},$ (25)
$\displaystyle u(t,\cdot)$ $\displaystyle=u_{0}$ $\displaystyle\text{in
}\operatorname{\mathbb{R}}_{+}^{n}$

has a (unique) solution in the class
$\mathrm{W}_{p,q}^{(1,2m)}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n};E)$
if and only if
$f\in\LL_{p,q}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n};E)$,
$u_{0}\in\mathrm{B}_{q,p}^{2m(1-\frac{1}{p})}(\operatorname{\mathbb{R}}_{+}^{n};E)$,
$g_{j,k}=\mathcal{P}_{j,k}g_{j}\in\mathrm{F}_{p,q}^{1-\frac{1}{p}-\frac{k}{2m}}(\operatorname{\mathbb{R}}_{+};\LL_{q}(\operatorname{\mathbb{R}}^{n-1};E))\cap\LL_{p}(\operatorname{\mathbb{R}}_{+};\mathrm{B}_{q,q}^{2m(1-\frac{1}{p})-k}(\operatorname{\mathbb{R}}^{n-1};E))$
and satisfy the compatibility conditions

$\mathcal{B}_{jk}(\mathrm{D})u_{0}|_{y=0}=g_{j,k}|_{t=0}\quad\text{whenever
}(1-\frac{1}{q})-\frac{k}{2m}>\frac{1}{q}.$

Proof. Necessity of the regularity of the data: The necessity part can again
be splitted into necessary spatial regularity and necessary of the time
regularity. For the spatial regularity, we may follow the lines of the proof
for [18, Proposition 6.4] where by means of [18, Proposition 6.1] and taking
boundary resp. time trace of a presupposed solution
$u\in\mathrm{W}_{p,q}^{(1,2m)}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n};E)$,
we may conclude that
$u_{0}\in\mathrm{B}_{q,p}^{2m(1-\frac{1}{p})}(\operatorname{\mathbb{R}}_{+}^{n};E)$
and
$u|_{y=0}\in\LL_{p}(\operatorname{\mathbb{R}}_{+};\mathrm{B}_{q,q}^{2m-\frac{1}{q}}(\partial\operatorname{\mathbb{R}}_{+}^{n};E))$,
so that each
$g_{j,k}=\mathcal{B}_{jk}(\mathrm{D})u_{0}|_{y=0}\in\LL_{p}(\operatorname{\mathbb{R}}_{+};\mathrm{B}_{q,q}^{2m-k-\frac{1}{q}}(\operatorname{\mathbb{R}}^{n-1};E))$,
$j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$.
The necessity of the time regularity can be observed based on the important
fact that since $E$ is a Banach space of class $\mathcal{HT}$ and
$p\in(1,\infty)$ lies in the reflexive range, then the auxiliary space
$\tilde{E}=\LL_{q}(\operatorname{\mathbb{R}}^{n-1};E)$ is of class
$\mathcal{HT}$ as well, see Lemma 2.1. Moreover, since $q\in(1,\infty)$ we may
identify $\LL_{q}(\operatorname{\mathbb{R}}_{+}^{n};E)$ with
$\LL_{q}(\operatorname{\mathbb{R}}_{+};\tilde{E})$ and
$\mathrm{W}_{q}^{2m}(\operatorname{\mathbb{R}}_{+}^{n};E)$ with a subspace of
$\mathrm{W}_{q}^{2m}(\operatorname{\mathbb{R}}_{+};\tilde{E})$ via the
relation $u(\boldsymbol{x}^{\prime},y)=(u(\cdot,y))(\boldsymbol{x}^{\prime})$.
In this sense,
$u\in\mathrm{W}_{p,q}^{(1,2m)}(\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}^{n};E)$
implies that
$u\in\mathrm{W}_{p}^{1}(\operatorname{\mathbb{R}}_{+};\tilde{E})$. Considering
the (resolvent-commutative) operators $A=\partial_{y}$ with
$\operatorname{\operatorname{dom}}(A)=\LL_{p}(\operatorname{\mathbb{R}}_{+};\mathrm{W}_{q}^{1}(\operatorname{\mathbb{R}}_{+};\tilde{E}))$
and $B=(\partial_{t})^{1/2m}$ on
$\operatorname{\operatorname{dom}}(B)=\mathring{\mathrm{W}}_{p}^{1/2m}(\operatorname{\mathbb{R}}_{+};\LL_{q}(\operatorname{\mathbb{R}}_{+};\tilde{E}))$,
after reducing the problem to the special case $u_{0}=0$, as in [18] it
follows for the functions
$u_{\boldsymbol{\alpha}}:=B^{2m-\left|\boldsymbol{\alpha}\right|-1}\mathrm{D}^{\boldsymbol{\alpha}}u$
that
$u_{\boldsymbol{\alpha}}|_{y=0}\in\mathring{F}_{p,q}^{\frac{1}{2m}(1-\frac{1}{q})}(\operatorname{\mathbb{R}}_{+};\tilde{E})$,
thus