text:
\section{Introduction}\label{sectionh1}

Establishing the existence of a bounded $H_\infty$-calculus is an important tool in the
modern analysis of nonlinear partial differential equations. The $H_\infty$-calculus
was introduced by McIntosh in \cite{McIn}. We refer the reader to Denk, Hieber, Pr\"uss
\cite{DHP} or Kunstmann, Weis \cite{KuWe} for recent and extensive surveys.

Let $A:\mathcal{D}(A)\subset Y\to Y$ be a closed and densely defined operator
in a Banach space $Y$. In the sequel, $\Lambda=\Lambda(\theta)$ will denote a sector
\begin{equation}\label{h2.B}
 \Lambda=\Lambda(\theta)=\{\lambda=re^{i\varphi}\mbox{\boldmath$\;|\;$\unboldmath}
 r\ge 0,\;\theta\le\varphi\le 2\pi-\theta\},\qquad 0<\theta<\pi,
\end{equation}
in the complex plane. By $H=H(\theta)$ we denote the space of all holomorphic functions
${\mathbb C}\setminus\Lambda\to{\mathbb C}$ for which
$|f(\lambda)|\le c(|\lambda|^\delta+|\lambda|^{-\delta})^{-1}$ for some $\delta>0$.
If $\Lambda\setminus\{0\}$ is contained in the resolvent set of $A$ and
$\|\lambda(\lambda-A)^{-1}\|_{\mathcal{L}(Y)}$ is uniformly bounded on
$\Lambda\setminus\{0\}$, then
\begin{equation}\label{h2.A}
  f(A):=\frac{1}{2\pi i}\int_{\partial\Lambda} f(\lambda)(\lambda-A)^{-1}\,d\lambda,
  \qquad f\in H,
\end{equation}
converges absolutely and defines an element in $\mathcal{L}(Y)$. By definition, the
operator $A$ has a bounded
$H_\infty$-calculus with respect to ${\mathbb C}\setminus\Lambda$, if
\begin{equation}\label{h2.C}
 \|f(A)\|_{\mathcal{L}(Y)}\le M\,\|f\|_{\infty} \qquad\forall\;f\in H
\end{equation}
with a constant $M$ not depending on $f\in H$ (here, $\|f\|_\infty$ denotes the supremum
norm of $f$). The name $H_\infty$-calculus originates from the fact that \eqref{h2.C} then
allows the definition of $f(A)$ in $\mathcal{L}(Y)$ for any bounded holomorphic function $f$ on
${\mathbb C}\setminus\Lambda$. In particular, the choice of  $f(\lambda)=\lambda^{it}$, $t\in{\mathbb R}$,
implies the boundedness of purely imaginary powers, $A^{it}\in\mathcal{L}(Y)$, and
$\|A^{it}\|\le Me^{|t|\theta}$ for all $t\in{\mathbb R}$.
Establishing \eqref{h2.C} requires a thorough understanding of the resolvent of $A$
that goes well beyond proving the boundedness of $\|\lambda(\lambda-A)^{-1}\|_{\mathcal{L}(Y)}$.

In view of its importance for nonlinear parabolic equations, there is a vast literature
concerned with the question of the existence of a bounded
$H_\infty$-calculus for operators $A$ in quite different contexts. For example,
Amann, Hieber, Simonett \cite{AHS} treat differential operators on ${\mathbb R}^n$ and
on compact manifolds with little regularity in the coefficients;
Duong in \cite{Duon} considers boundary
value problems on smooth manifolds, extending Seeley's work \cite{Seel2} on bounded
imaginary powers; Denk, Dore, Hieber, Pr\"uss, and Venni \cite{DDHPV} then investigate
boundary value problems of little regularity; in \cite{EsSe}, Escher and Seiler consider
the Dirichlet-Neumann operator for domains of low regularity; \cite{CSS1} shows boundedness
of the imaginary powers for differential operators on manifolds with conical singularities.

In the present paper we study realizations (i.e.\ closed extensions) $\underline{A}_T$ of a
$\mu$-th order cone differential operator $A$ subject to lower order differential
boundary conditions $T$ on a manifold ${\mathbb D}$ with conical singularities, where
the boundary value problem $\spalte{A}{T}$ is assumed to be Shapiro-Lopatinskii
elliptic. More precisely (for an explanation of the following notation see
Section \ref{sectionh4}), we consider $A$ as an unbounded operator in a weighted
$L_p$-space $\mathcal{H}^{0,\gamma}_p({\mathbb D},E)$ of sections of a vector bundle $E$, initially
defined on a space of smooth sections that vanish under the boundary condition $T$.
In general, $A$ will have a large number of
closed extensions with domain contained in $\mathcal{H}^{\mu,\gamma}_p({\mathbb D},E)_T$,
the space of sections of smoothness $\mu$ vanishing under $T$.
We let $\underline{A}_T$ denote one of these extensions.
All closed extensions can be described explicitly, cf.\ \cite{CSS2}.

In Theorem \ref{hinfty} we show that $\underline{A}_T$ admits a bounded $H_\infty$-calculus
provided the resolvent of $\underline{A}_T$ has the structure of an element of (a version of) the
parameter-dependent cone calculus for pseudodifferential boundary value problems.
In Section \ref{sectionh6} we next establish conditions on $\spalte{A}{T}$ which are
more easily verified and guarantee that the resolvent of the minimal extension $A_{T,\min}$
has the desired structure. In essence, these conditions ask for the invertibility of
all parameter-dependent principal symbols associated with $\spalte{\lambda-A}{T}$, for
then the cone calculus developed by Schulze (see e.g.\ \cite{KaSc} for a presentation)
allows the construction of a parametrix which yields the resolvent to $A_{T,\min}$.
Based on the results on resolvents of closed extensions of
cone differential operators in \cite{GKM2}, \cite{GKM1} by Gil, Krainer, and Mendoza,
in \cite{Krai} by Krainer,
and  in  \cite{ScSe2}, we expect to obtain
such natural conditions also for extensions different from the minimal one.
This analysis, however, is not the focus of the present paper.

In Section \ref{sectionh7} we
apply these results to the minimal extension of
the Laplacian on a manifold with straight conical singularity, subject to Dirichlet or
Neumann conditions. We find conditions on the dimension of ${\mathbb D}$ and on the weight
$\gamma$ such that $\Delta_{D,\min}$ and $\Delta_{N,\min}$ in
$\mathcal{H}^{0,\gamma}_p({\mathbb D})$ satisfy all the above assumptions. As a consequence, we derive
maximal regularity for the associated initial boundary value problem.

\forget{In their papers \cite{GKM2}, \cite{GKM1}, Gil, Krainer, and Mendoza study the resolvent
of cone differential operators in the boundaryless case and, in particular, find
conditions that imply the existence and decay property of the resolvent of an arbitrary
closed extension in a sector $\Lambda$. Krainer in \cite{Krai} generalizes these results
also to boundary value problems. }

\section{Differential operators on smooth manifolds with boundary}\label{sectionh3}
In this section we want to recall classical results of \cite{Seel1}, \cite{Seel2} and
\cite{Duon} on the resolvent of differential operators on manifolds with boundary and
their $H_\infty$-calculus.

In the following let $A$ be a differential operator of positive order $\mu\in{\mathbb N}$ on
a smooth manifold with boundary $X$, acting on sections into a vector bundle $E$,
 $$A:\;\mathcal{C}^\infty(X,E)\longrightarrow \mathcal{C}^\infty(X,E).$$
Moreover, let $T=(T_0,\ldots,T_{\mu-1})$ be a tuple of (normal) boundary conditions,
i.e.
 $$T_j=\gamma_0\circ B_j:\;\mathcal{C}^\infty(X,E)\longrightarrow \mathcal{C}^\infty(\partial X,F_j);$$
here, the $B_j$ are differential operators of order $\mu-j$ acting from sections
into $E$ to sections into some hermitian vector bundle $F_j$ and $\gamma_0$ is
the operator of restriction to the boundary of $X$. It is allowed that some of the
$F_j$ are zero dimensional, i.e. the corresponding boundary condition $T_j$ is void.

For a function space $\mathcal{F}$ on $X$ let us set
 $$\mathcal{F}_T=\{u \in\mathcal{F}\mbox{\boldmath$\;|\;$\unboldmath} Tu=0\},$$
provided application of $T$ to elements of $\mathcal{F}$ makes sense. Let us now consider $A$
as the unbounded operator
\begin{equation}\label{h3.A}
 A:\;\mathcal{C}^\infty(X,E)_T\subset L_p(X,E)\longrightarrow L_p(X,E)
\end{equation}
with (a fixed) $1<p<\infty$. If the boundary value problem $\spalte{A}{T}$
is (Shapiro-Lopatinskii) elliptic, the closure $A_T$ of this
operator can be shown to be defined by the action of $A$ on the domain
\begin{equation}\label{h3.B}
 \mathcal{D}(A_T)=H^\mu_p(X,E)_T.
\end{equation}

\subsection{Elements of Boutet de Monvel's calculus for boundary value problems}
\label{sectionh3.1}

In \cite{Bout} Boutet de Monvel introduced a pseudodifferential calculus on manifolds
with boundary which allowed the construction of parametrices to elliptic problems
$\spalte{A}{T}$. This calculus has a corresponding parameter-dependent version,
some of whose elements we describe now. For a short presentation see for example
\cite{CSS2}.

Let $\Sigma\subset{\mathbb C}$ be a closed sector with vertex in the origin. We recall the
definition of the space $B^{\nu,0}(X;\Sigma,E)$, $\nu\in{\mathbb R}$, of operator-families
on $X$,
\begin{equation}\label{h3.C}
 A(\eta)=P_+(\eta)+G(\eta),
\end{equation}
depending on $\eta\in\Sigma$ as parameter. The first term is a pseudodifferential
operator
 $$P_+(\eta)=r_+\, P(\eta)\, e_+;$$
here, $e_+:\mathcal{C}^\infty(X,E)\to\mathcal{C}^\infty(2X,2E)$ denotes the operator of extending sections on $X$
by zero to sections on the double $2X$ (or any other smooth closed manifold containing
$X$ as submanifold), while $r_+:\mathcal{C}^\infty(2X,2E)\to\mathcal{C}^\infty(X,E)$ is the operator of restriction.
$P(\eta)$ is a classical, parameter-dependent pseudodifferential operator of order
$\nu$ on $2X$. Furthermore it is required that $P(\eta)$ satisfies the
{\em transmission condition} with respect to $\partial X$. Since differential operators
as well as their parametrices always satisfy the transmission condition, we shall
skip the details.

The second term on the right-hand side of \eqref{h3.C} belongs to
$B^{\nu,0}_G(X;\Sigma,E)$, the space of (singular) {\em Green operators}.
Modulo {\em regularizing operators}, i.e.\ integral operators having a kernel in
$\mathcal{S}(\Sigma,\mathcal{C}^\infty(X\times X))$, each Green operator is localized in a
collar neighborhood of the boundary. By standard use of a partition of unity on $X$, this
localized part is determined by operators on the half space
$\overline{{\mathbb R}}{}^n_+=\{(x^\prime,x_n)\mbox{\boldmath$\;|\;$\unboldmath} x^\prime\in{\mathbb R}^{n-1},\,x_n\ge 0\}$
of the form
 $$\text{\rm op}^\prime({d}\,)(\eta):\mathcal{C}^\infty_{\text{\rm comp}}(\overline{{\mathbb R}}{}^n_+,{\mathbb C}^q)\longrightarrow
   \mathcal{C}^\infty(\overline{{\mathbb R}}{}^n_+,{\mathbb C}^q),\qquad q=\text{dim}\,E,$$
defined by
\begin{equation}\label{h3.D}
 \big(\text{\rm op}^\prime({d})(\eta)u\big)(x,t)=\int_{{\mathbb R}^{n-1}}e^{ix\xi}\widetilde{d}(x,\xi,\eta;s,t)
 (\mathcal{F}_{y\to\xi}u)(\xi,s)\,dsd\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi
\end{equation}
with a {\em symbol kernel}
\begin{equation}\label{h3.E}
 {d}(x,\xi,\eta;s,t)=\widetilde{d}(x,\xi,\eta;[\xi,\eta]s,[\xi,\eta]t),
\end{equation}
where $\widetilde{d}$ is a $(q\times q)$-matrix whose components are symbols
\begin{equation}\label{h3.F}
 \widetilde{d}_{ij}(x,\xi,\eta;s,t)\in S^{\nu+1}_{{\rm cl}}({\mathbb R}^{n-1}\times{\mathbb R}^{n-1}\times\Sigma,
 \mathcal{S}(\overline{{\mathbb R}}_+\times\overline{{\mathbb R}}_+)).
\end{equation}
In \eqref{h3.E} and the sequel, $[\xi,\eta]$ denotes a smoothed norm function, i.e.\
a smooth positive function in $(\xi,\eta)$ that coincides with $|(\xi,\eta)|$ outside the
unit ball.
Roughly speaking, $\text{\rm op}^\prime({d})$ acts as an integral operator with smooth
kernel in the direction normal to the boundary, while it is a pseudodifferential
operator of order $\nu$ in the tangential direction.

Both $B^{\nu,0}_G(X;\Sigma,E)$ and $B^{\nu,0}(X;\Sigma,E)$ are Fr\'{e}chet spaces in a
natural way, since the spaces of local symbols and symbol kernels as well as the
regularizing operators carry Fr\'{e}chet topologies.

\subsection{Parameter-ellipticity and the resolvent of $A_T$}\label{sectionh3.2}

Let $\Lambda=\Lambda(\theta)$ be a sector in ${\mathbb C}$ as introduced in \eqref{h2.B}.
We shall call $A_T$, cf.\ \eqref{h3.A} and \eqref{h3.B},
(parameter-){\em elliptic with respect to} $\Lambda$ if two conditions are
satisfied. First, the principal symbol of $A$ has no spectrum in $\Lambda$ for non-zero
covariables, i.e. in local coordinates,
\begin{equation}\label{h3.Z}
 \mbox{\rm det}\,(\lambda-\sigma^\mu_\psi(A)(x,\xi))\not=0\qquad\forall\;
 0\not=(\xi,\lambda)\in{\mathbb R}^n_{\xi}\times\Lambda.
\end{equation}
The second condition is the invertibility of the boundary symbol of
$\spalte{\lambda-A}{T}$; in local coordinates $(x^\prime,x_n)$ near the boundary of
$X$ this means that, for fixed $(x^\prime,\xi^\prime,\lambda)$,
 $$\begin{pmatrix}
    \lambda-\sigma^\mu_\psi(A)(x^\prime,0,\xi^\prime,D_{x_n})\\
    \left(\gamma_0\circ\sigma^{j}_\psi(B_j)
    (x^\prime,0,\xi^\prime,D_{x_n})\right)_{j=0,\ldots,\mu-1}
   \end{pmatrix}:\;
   \mathcal{S}(\overline{{\mathbb R}}_+,{\mathbb C}^q)\longrightarrow
   \begin{matrix}
    \mathcal{S}(\overline{{\mathbb R}}_+,{\mathbb C}^q)\\
    \oplus\\
    {\mathbb C}^{m_0}\oplus\ldots\oplus{\mathbb C}^{m_{\mu-1}}
   \end{matrix}$$
is an isomorphism whenever $(\xi^\prime,\lambda)\not=0$. Here,
$m_j=\mbox{\rm dim}\,F_j$ and $\gamma_0$ denotes evaluation at 0 of functions on
$\overline{{\mathbb R}}_+$. Now let
\begin{equation}\label{h3.G}
 \Sigma=\Sigma(\theta,\mu)=\left\{\eta=se^{i\alpha}\mbox{\boldmath$\;|\;$\unboldmath} s\ge 0,\;
 \mbox{$\frac{\theta}{\mu}\le\alpha\le\frac{2\pi-\theta}{\mu}$}\right\}.
\end{equation}
Then $\Sigma$ is a sector and $\Lambda=\{\eta^\mu\mbox{\boldmath$\;|\;$\unboldmath}\eta\in\Sigma\}$ for $\Lambda$
as in \eqref{h2.B}.

\begin{theorem}\label{h3.1}
Let $A_T$ be elliptic with respect to $\Lambda$. Then
 $$\lambda-A_T:\,H^\mu_p(X,E)_T\subset L_p(X,E)\longrightarrow L_p(X,E)$$
is invertible for large $\lambda\in\Lambda$ and there exists an operator-family
$P_+(\eta)+G(\eta)\in B^{-\mu,0}(X;\Sigma,E)$ such that, for large $\eta\in\Sigma$,
 $$(\eta^\mu-A_T)^{-1}=P_+(\eta)+G(\eta).$$
\end{theorem}

This theorem was essentially proven by Seeley \cite{Seel1} but without
using the `language' of Boutet de Monvel's calculus. For other proofs we
refer to \cite{Grub} and \cite{Grub2}.

\begin{corollary}\label{h3.2}
If $A_T$ is elliptic with respect to $\Lambda$ then there exists a constant $c_p\ge0$
such that for all large $\lambda\in\Lambda$
 $$\|(\lambda-A_T)^{-1}\|_{L_p(X,E)}\le c_p\,\frac{1}{|\lambda|}.$$
\end{corollary}

Using the above resolvent structure, \cite{Seel2} shows the existence of bounded
imaginary powers, while \cite{Duon} proves a bounded $H_\infty$-calculus:

\begin{theorem}\label{h3.3}
If $A_T$ is elliptic with respect to $\Lambda$ then there exists a constant $c\ge0$
such that $A_T+c$ has a bounded $H_\infty$-calculus with respect to ${\mathbb C}\setminus\Lambda$
$($simultaneously for all $1<p<\infty)$.
\end{theorem}

\section{Boundary value problems on conic manifolds}\label{sectionh4}
The main objective of this section is the description of boundary value problems on a
manifold with conical singularities and of a parameter-dependent Boutet de Monvel
calculus adapted to this situation. For simplicity we shall confine the description
to the case of one conical singularity.

\subsection{Cone differential operators}\label{sectionh4.0}

Let $\text{\rm int}\,{\mathbb D}$ be an $(n+1)$-dimensional riemannian manifold with boundary having a
cylindrical end, i.e.\ there exists a compact subset $C$ such that $\text{\rm int}\,{\mathbb D}\setminus C$ is
isometric to the product ${]0,1[}\times X$ for a smooth compact (not necessarily
connected) riemannian manifold $X$ with boundary.
We fix the coordinate in ${]0,1[}$ in such a way that every neighborhood
of $C$ in $\text{\rm int}\,{\mathbb D}$ has nonempty intersection with ${]\frac{1}{2},1[}\times X$.
We complete $\text{\rm int}\,{\mathbb D}$ with the help of the riemannian metric. If ${\mathbb D}$ is the resulting space,
then ${\mathbb D}\setminus C$ can be identified with ${[0,1[}\times X$. We next denote by
$\text{\rm int}\,{\mathbb B}$ the boundary of $\text{\rm int}\,{\mathbb D}$ and by ${\mathbb B}$ its completion in the metric
inherited from $\text{\rm int}\,{\mathbb D}$. Then ${\mathbb B}\setminus C$ can be identified with
${[0,1[}\times\partial X$ and ${\mathbb B}$ itself is a smooth manifold with boundary.

\forget{
To make the exposition simpler we shall assume in the following that ${\mathbb D}$ has only one
cylindrical end, denoted by ${[0,1[}\times X$. Here we use the canonical coordinates $(t,x)$,
$0\le t<1$, $x\in X$. The subset $\{0\}\times X$ of ${\mathbb D}$, where $t=0$, is  occasionally
called the {\em singularity} of ${\mathbb D}$.
}

In the sequel, a vector bundle over ${\mathbb D}$ will be a smooth hermitian vector bundle over
$\text{\rm int}\,{\mathbb D}$ such that $E|_{{]0,1[}\times X}$ is isometric to the pull-back under the
canonical projection ${]0,1[}\times X\to X$ of a hermitian bundle $E_0$ over $X$.

That we call ${\mathbb D}$ a {\em manifold with conical singularity} is due to the class of
differential operators we consider on it. A $\mu$-th order differential operator $A$ on
$\text{\rm int}\,{\mathbb D}$ with smooth coefficients, acting between sections of $E$, is called a
{\em cone differential operator} if
\begin{equation}\label{fuchs}
 A=t^{-\mu}\mathop{\mbox{\large$\sum$}}_{j=0}^\mu a_j(t)(-t\partial_t)^j,\qquad
 a_j\in\mathcal{C}^\infty([0,1[,\text{\rm Diff}^{\,\mu-j}(X;E_0)),
\end{equation}
on ${]0,1[}\times X$; here we use the canonical coordinates $(t,x)$ with
$0\le t<1$, $x\in X$, and $\text{\rm Diff}^{k}(X;E_0)$ denotes the space of $k$-th order
differential operators on $X$ with smooth coefficients, acting between sections of $E_0$.

\subsection{Function spaces}\label{sectionh4.1}

We first introduce some function spaces on ${\mathbb D}$ (the corresponding spaces on ${\mathbb B}$
are obtained analogously). Patching together two copies of
${\mathbb D}$, we obtain a smooth manifold with boundary $2{\mathbb D}$ containing ${\mathbb D}$.
Let $\omega$ be an arbitrary fixed cut-off function on ${\mathbb D}$ (extended by 1 to $2{\mathbb D}$).

\begin{definition}\label{h4.1}
Let $\gamma\in{\mathbb R}$. Then $\mathcal{C}^{\infty,\gamma}({\mathbb D})$ denotes the space of all smooth functions $u$
for which $(1-\omega)u\in\mathcal{C}^\infty(2{\mathbb D})$ and
\begin{equation}\label{h4.X}
 t\mapsto t^{\frac{n+1}{2}-\gamma}(\log t)^k (t\partial_t)^l (\omega u)(t,\cdot)
\end{equation}
is a bounded function on ${]0,1[}$ with values in $\mathcal{C}^\infty(X)$ for all $k,l\in{\mathbb N}_0$.
Moreover, we set
 $$\mathcal{C}^{\infty,\infty}({\mathbb D})=\mathop{\mbox{\Large$\cap$}}_{\gamma\in{\mathbb R}}\mathcal{C}^{\infty,\gamma}({\mathbb D}).$$
\end{definition}

\begin{definition}\label{h4.2}
For $s\in{\mathbb N}_0$, $\gamma\in{\mathbb R}$, and $1<p<\infty$, we let $\mathcal{H}^{s,\gamma}_p({\mathbb D})$
be the space of all functions $u$ such that $(1-\omega)u\in H^s_p(2{\mathbb D})$ and
 $$t^{\frac{n+1}{2}-\gamma}(t\partial_t)^l\partial^\alpha_x (\omega u)(t,x)\;\in\;
   L_p({]0,1[}\times X,\mbox{$\frac{dt}{t}dx$})\qquad\forall\;l+|\alpha|\le s,$$
where $dx$ refers to some metric on the manifold with boundary $X$.
\end{definition}

These definitions naturally extend to sections into $E$ and to real numbers $s$
(however, we shall only need the case of nonnegative integers). Then we write
$\mathcal{C}^{\infty,\gamma}({\mathbb D},E)$, $\mathcal{C}^{\infty,\infty}({\mathbb D},E)$, and $\mathcal{H}^{s,\gamma}_p({\mathbb D},E)$.

\subsection{Elliptic boundary value problems}\label{sectionh4.2}

Let $A$ be a $\mu$-th order cone differential operator on ${\mathbb D}$ as described in the
introduction,
 $$A:\;\mathcal{C}^{\infty,\infty}({\mathbb D},E)\longrightarrow\mathcal{C}^{\infty,\infty}({\mathbb D},E),$$
and $T=(T_0,\ldots,T_{\mu-1})$ be a tuple of (normal) boundary conditions, i.e.
\begin{equation}\label{h4.A.6}
 T_j=\gamma_0\circ B_j:\;\mathcal{C}^{\infty,\infty}({\mathbb D},E)\longrightarrow \mathcal{C}^{\infty,\infty}({\mathbb B},F_j)
\end{equation}
with cone differential operators $B_j$ of order $j$, and $\gamma_0$ being the operator of
restriction to the boundary ${\mathbb B}$ of ${\mathbb D}$. Setting $F=F_0\oplus\ldots\oplus F_{\mu-1}$ we
thus have
\begin{equation}\label{h4.A.2}
 \begin{pmatrix}A\\T\end{pmatrix}:\;
 \mathcal{C}^{\infty,\infty}({\mathbb D},E)\longrightarrow
 \begin{matrix}\mathcal{C}^{\infty,\infty}({\mathbb D},E)\\\oplus\\ \mathcal{C}^{\infty,\infty}({\mathbb B},F)\end{matrix}.
\end{equation}
Let us shortly sketch what ${\mathbb D}$-ellipticity of $\spalte{A}{T}$ means;
for details we refer to
Section 3.2 of \cite{CSS2}. First, $A$ is elliptic on $\text{\rm int}\,{\mathbb D}$ in the standard sense,
i.e.\ the  principal symbol $\sigma^\mu_\psi(A)$ is everywhere
(i.e.\ for non-zero covariables) invertible.
In the splitting of coordinates $(t,x)\in{]0,1]}\times X$ near $t=0$ we have
 $$\sigma^\mu_\psi(A)(t,x,\tau,\xi)=t^{-\mu}\mathop{\mbox{\large$\sum$}}_{j=0}^\mu
   \sigma^{\mu-j}_\psi(a_j)(t,x,\xi)(-it\tau)^j.$$
Observe a characteristic ``degenerate'' structure: There is the singular factor $t^{-\mu}$
and the covariable $\tau$ appears only in the form $t\tau$. Removing this degeneracy and
freezing the coefficients in $t=0$ we obtain the so-called {\em rescaled symbol}
 $$\widetilde\sigma^\mu_\psi(A)(x,\tau,\xi)=\mathop{\mbox{\large$\sum$}}_{j=0}^\mu
   \sigma^{\mu-j}_\psi(a_j)(0,x,\xi)(-i\tau)^j.$$
We require this symbol to be invertible for all $x$ and $(\tau,\xi)\not=0$.
Since $\spalte{A}{T}$ is a usual boundary value problem on $\text{\rm int}\,{\mathbb D}$, we may associate
with it the standard principal boundary symbol. Ellipticity asks the invertibility of
this symbol. Since the boundary conditions are also given by cone differential operators,
the boundary symbol again has a degenerate structure. Removing the degeneracy and
freezing coefficients in $t=0$ yields the {\em rescaled boundary symbol} which is also
required to be invertible.

\subsection{Realizations of $A$ with respect to $T$}\label{sectionh4.3}

For a function space $\mathcal{F}$ on ${\mathbb D}$ we use again the notation
$\mathcal{F}_T=\{u\in\mathcal{F}\mbox{\boldmath$\;|\;$\unboldmath} Tu=0\}$. Let us now consider the unbounded operator
\begin{equation}\label{h4.A}
 A:\;\mathcal{C}^{\infty,\infty}({\mathbb D},E)_T\subset\mathcal{H}^{0,\gamma}_p({\mathbb D},E)\longrightarrow
 \mathcal{H}^{0,\gamma}_p({\mathbb D},E)
\end{equation}
with $\gamma\in{\mathbb R}$ and $1<p<\infty$ (it would be more precise to write
$A_{\gamma,p}$ but for convenience we exclude $\gamma$ and $p$ from the notation).
We shall assume that $\spalte{A}{T}$ is ${\mathbb D}$-elliptic in the sense of
Section \ref{sectionh4.2}.

We write $A_{T,\min}$ for the closure of $A$ and define $A_{T,\max}$ by the action of
$A$ on the space
 $$\mathcal{D}(A_{T,\max})=\{u\in\mathcal{H}^{\mu,\gamma}_p({\mathbb D},E)_T\mbox{\boldmath$\;|\;$\unboldmath}
   Au\in \mathcal{H}^{0,\gamma}_p({\mathbb D},E)\}.$$
By abuse of notation, $A_{T,\max}$ is not the true maximal extension of $A$, but it is
the largest one that still `feels' the boundary condition $T$. The closed extensions
between the minimal and the maximal are usually called the {\em realizations} of $A$
with respect to $T$. In \cite{CSS2}, Theorem 5.12, we have shown:

\begin{theorem}\label{h4.3.5}
There exists a finite dimensional space
$\mathcal{E}=\mathcal{E}_{A,T}\subset\mathcal{C}^{\infty,\gamma}({\mathbb D},E)$
of smooth functions on $\text{\rm int}\,{\mathbb D}$ such that
 $$\mathcal{D}(A_{T,\max})=\mathcal{D}(A_{T,\min})\oplus\mathcal{E}.$$
\end{theorem}

As a consequence, any realization $\underline{A}_T$ is determined by a subspace
$\underline{\mathcal{E}}$ of $\mathcal{E}$, i.e.
 $$\mathcal{D}(\underline{A}_T)=\mathcal{D}(A_{T,\min})\oplus\underline{\mathcal{E}}.$$
In \cite{CSS2} we also have characterized the domain of the closure, namely
 $$\mathcal{D}(A_{T,\min})=\Big\{u\in\mathop{\mbox{\Large$\cap$}}_{\varepsilon>0}\mathcal{H}^{\mu,\gamma+\mu-\varepsilon}_p({\mathbb D},E)_T\mbox{\boldmath$\;|\;$\unboldmath}
   Au\in\mathcal{H}^{0,\gamma}_p({\mathbb D},E)\Big\}.$$
In particular, $\mathcal{H}^{\mu,\gamma+\mu}_p({\mathbb D},E)_T\subset\mathcal{D}(A_{T,\min})
\subset\mathcal{H}^{\mu,\gamma+\mu-\varepsilon}_p({\mathbb D},E)_T$ for any $\varepsilon>0$. If in addition $A_T$ is
{\em conormally elliptic} with respect to the weight $\gamma+\mu$, see (E3) in
Section \ref{sectionh6} for an explanation, we even have
\begin{equation}\label{h4.A.5}
 \mathcal{D}(A_{T,\min})=\mathcal{H}^{\mu,\gamma+\mu}_p({\mathbb D},E)_T.
\end{equation}

\subsection{Parameter-dependent operators}\label{sectionh4.4}

A generalization of Boutet de Monvel's calculus to boundary value problems on
manifolds with conical singularities was introduced in \cite{ScSc1} and \cite{ScSc2}.
A corresponding parameter-dependent version can be found e.g.\ in \cite{KaSc}.
We give here a somewhat simplified presentation of some of the elements of this
calculus, focusing on the structures that are necessary for the description
of resolvents. For convenience, we shall assume $E={\mathbb C}$, since the general case is
a straightforward extension of this situation.

\subsubsection{Green operators}\label{sectionh4.4.1}

With $X^\wedge:={\mathbb R}_+\times X$ and for $\gamma\in{\mathbb R}$ let us set
\begin{align}
 \mathcal{H}^{0,\gamma}_p(X^\wedge)&=
 L_p(X^\wedge,t^{(\frac{n+1}{2}-\gamma)p}\mbox{$\frac{dt}{t}$}dx)=
   t^{\gamma-\frac{n+1}{2}} L_p(X^\wedge,\mbox{$\frac{dt}{t}$}dx),\label{h4.B.2}\\
 \mathcal{S}^\gamma_0(X^\wedge)&=\{u\in\mathcal{C}^\infty(X^\wedge)\mbox{\boldmath$\;|\;$\unboldmath}\omega u\in\mathcal{C}^{\infty,\gamma}({\mathbb D})
   \text{ and }(1-\omega)u\in\mathcal{S}({\mathbb R},\mathcal{C}^\infty(X))\}.\label{h4.B.4}
\end{align}
The latter is a Fr\'echet space in a natural way.

Let $\Sigma$ be a closed subsector of ${\mathbb C}$ with vertex in 0 and $\nu,\gamma\in{\mathbb R}$.
Then we denote by $R^{\nu,0}_G(X^\wedge;\Sigma,\gamma)$ the space of all
operator families $G(\eta)$, $\eta\in\Sigma$, such that
 $$\big(G(\eta)u\big)(t,x)=
   \big\langle k(\eta;t,x),\overline{u}\big\rangle_{\mathcal{H}^{0,0}_2(X^\wedge)}=
   \int_{X^\wedge}k(\eta;t,x,t^\prime,x^\prime)u(t^\prime,x^\prime)\,
   t^{\prime n}dt^\prime dx^\prime,$$
where the integral kernel $k$ has the form
\begin{equation}\label{h4.B}
 k(\eta;t,x,t^\prime,x^\prime)=\widetilde{k}(\eta;[\eta]t,x,[\eta]t^\prime,x^\prime)
\end{equation}
with
\begin{equation}\label{h4.C}
 \widetilde{k}(\eta;t,x,t^\prime,x^\prime)\in S^{\nu+n+1}\big(\Sigma,
 \mathcal{S}^{\gamma+\varepsilon}_0(X^\wedge_{(t,x)})\,{\widehat{\otimes}}_\pi\,
 \mathcal{S}^{-\gamma+\varepsilon}_0(X^\wedge_{(t^\prime,x^\prime)})\big)
\end{equation}
for some $\varepsilon>0$ (depending on $G$).

\subsubsection{Mellin pseudodifferential operators}\label{sectionh4.4.2}

For $\beta\in{\mathbb R}$ let $\Gamma_\beta=\{z\in{\mathbb C}\mbox{\boldmath$\;|\;$\unboldmath}\text{\rm Re}\, z=\beta\}$. We denote by
$M^{\nu,0}_\beta(X;\Sigma)$, $\nu\in{\mathbb R}$, the space of all functions
$h:\Gamma_{\beta}\times\Sigma\to B^{\nu,0}(X)$ such that
 $$h(\mbox{$\beta+i\tau$},\eta)\in B^{\nu,0}(X;{\mathbb R}_\tau\times\Sigma_\eta)$$
(cf.\ Section \ref{sectionh3.1} with parameter space ${\mathbb R}\times\Sigma$ instead of
$\Sigma$). A {\em Mellin symbol} is a function
$h=h(t,\beta+i\tau,\eta)\in\mathcal{C}^\infty({\mathbb R}_+,M^{\nu,0}_\beta(X;\Sigma))$; it induces the
Mellin pseudodifferential operator
$\text{\rm op}_M^{\frac{1}{2}-\beta}(h)(\eta):\;\mathcal{C}^\infty_{\text{\rm comp}}(X^\wedge)
\to\mathcal{C}^\infty(X^\wedge)$
by
 $$\big(\text{\rm op}_M^{\frac{1}{2}-\beta}(h)(\eta)u\big)(t)=
   \int_{-\infty}^\infty t^{-\beta-i\tau}h(t,\beta+i\tau,\eta)
   (\mathcal{M} u)(\beta+i\tau)\,d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\tau;$$
here we have identified $\mathcal{C}^\infty(X^\wedge)$ with $\mathcal{C}^\infty({\mathbb R}_+,\mathcal{C}^\infty(X))$,
and $\mathcal{M}$ denotes the Mellin transform, i.e.
$(\mathcal{M} u)(z)={\displaystyle\int_0^\infty} t^z u(t)\,\frac{dt}{t}$ for $z\in{\mathbb C}$.

\subsubsection{The full class and a norm estimate}\label{sectionh4.4.3}

We define ${C}^{\nu,0}({\mathbb D};\Sigma,\gamma)$ with $\nu\le 0$ and $\gamma\in{\mathbb R}$ as the set
of all operator-families $A(\eta)$ of the form
\begin{equation}\label{h4.D}
 A(\eta)=\sigma_0\left\{
           t^{-\nu}\text{\rm op}_M^{\gamma-\frac{n}{2}}(h)(\eta)+G(\eta)
           \right\}\sigma_1+
           (1-\sigma_3)P(\eta)(1-\sigma_4)+G_\infty(\eta),
\end{equation}
where $\sigma_j\in\mathcal{C}^\infty_{\text{\rm comp}}([0,1[)$ are cut-off functions and
\begin{itemize}
 \item[a)] $h(t,z,\eta)=\widetilde{h}(t,z,t\eta)$ with
  $\widetilde{h}\in \mathcal{C}^\infty(\overline{{\mathbb R}}_+,M^{\nu,0}_{\frac{n+1}{2}-\gamma}(X;\Sigma))$,
  cf.\ Section \ref{sectionh4.4.2},
 \item[b)] $P(\eta)\in B^{\nu,0}(2{\mathbb D};\Sigma)$, cf.\ Section \ref{sectionh3.1},
 \item[c)] $G(\eta)\in R^{\nu,0}_G(X^\wedge;\Sigma,\gamma)$,
  cf.\ Section \ref{sectionh4.4.1},
 \item[d)] $G_\infty\in\mathcal{S}(\Sigma,\mathcal{L}(\mathcal{H}^{s,\gamma}_p({\mathbb D}),
  \mathcal{H}^{s^\prime,\gamma}_p({\mathbb D})))$ for all $1<p<\infty$ and $s,s^\prime\in{\mathbb R}$
  with $s>-1+\frac{1}{p}$.
\end{itemize}
Initially, each such operator is defined on smooth functions, i.e.\
$A(\eta):\mathcal{C}^\infty_{\text{\rm comp}}(\text{\rm int}\,{\mathbb D})\to\mathcal{C}^\infty(\text{\rm int}\,{\mathbb D})$, where multiplications by $\sigma_1$ and
$\sigma_0$ are viewed as operators $\mathcal{C}^\infty_{\text{\rm comp}}(\text{\rm int}\,{\mathbb D})\to\mathcal{C}^\infty_{\text{\rm comp}}(X^\wedge)$ and
$\mathcal{C}^\infty_{\text{\rm comp}}(X^\wedge)\to\mathcal{C}^\infty_{\text{\rm comp}}(\text{\rm int}\,{\mathbb D})$, respectively. Similarly, $1-\sigma_3$ and
$1-\sigma_4$ act between smooth functions on ${\mathbb D}$ and $2{\mathbb D}$.

Let $A(\eta)\in{C}^{\nu,0}({\mathbb D};\Sigma,\gamma)$ as in \eqref{h4.D}.
We deduce from \eqref{h3.C} that there is a decomposition of $\widetilde{h}$ as
\begin{equation}\label{h4.D.5}
 \widetilde{h}(t,\mbox{$\frac{n+1}{2}$}-\gamma+i\tau,\eta)=
 \widetilde{p}_+(t,\tau,\eta)+\widetilde{g}(t,\tau,\eta)
\end{equation}
where $t\mapsto\widetilde{p}(t,\tau,\eta)$ is a smooth (up to $t=0$) family of
parameter-dependent pseudodifferential operators of order $\nu$ on $2X$ with parameter
space ${\mathbb R}\times\Sigma$, while $\widetilde{g}(t,\tau,\eta)$ belongs to
$\mathcal{C}^\infty(\overline{{\mathbb R}}_+,B^{\nu,0}_G(X;{\mathbb R}\times\Sigma))$. In local coordinates,
\begin{equation}\label{h4.E}
 \widetilde{p}_+(t,\tau,\eta)=\text{\rm op}_+(\widetilde{a})(t,\tau,\eta)\text{ with }
 \widetilde{a}(t,x,\tau,\xi,\eta)\in
 S^{\nu}_{\rm cl}(\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}_{(\tau,\xi)}\times\Sigma)
\end{equation}
and\footnote{By definition,
$S^{\nu}_{\rm cl}(\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}_{(\tau,\xi)}\times\Sigma)$ denotes the
space of all functions $b$ that are smooth on
$\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}_{(\tau,\xi)}\times\Sigma$ satisfy there uniform estimates
$|\partial_t^l\partial_x^\beta\partial^\alpha_{(\tau,\xi,\eta)}b(t,x,\tau,\xi,\eta)|\le
c\spk{\tau,\xi,\eta}^{\nu-|\alpha|}$ for any order of derivatives and that have
asymptotic expansions into components that are positively homogeneous in
$(\tau,\xi,\eta)$.}
$\widetilde{g}(t,\tau,\eta)$ is defined via a symbol kernel
\begin{equation}\label{h4.F}
 \widetilde{d}(t,x^\prime,\tau,\xi^\prime,\eta;u,v)=
 d(t,x^\prime,\tau,\xi^\prime,\eta;[\tau,\xi^\prime,\eta]u,[\tau,\xi^\prime,\eta]v)
\end{equation}
with
\begin{equation}\label{h4.G}
 d(t,x^\prime,\tau,\xi^\prime,\eta;u,v)\in
 S^{\nu+1}(\overline{{\mathbb R}}_+\times{\mathbb R}^{n-1}_{x^\prime}\times{\mathbb R}^{n}_{(\tau,\xi^\prime)}\times\Sigma;
 \mathcal{S}(\overline{{\mathbb R}}_+\times\overline{{\mathbb R}}_+)).
\end{equation}
The non-local, smoothing terms arising are as above in d).

\begin{theorem}\label{h4.2.4}
Let $A(\eta)\in{C}^{\nu,0}({\mathbb D};\Sigma,\gamma)$, $\nu\le0$. Then $A(\eta)$ extends
for each $\eta$ to a bounded operator in $\mathcal{H}^{0,\gamma}_p({\mathbb D})$ and there exists
a constant $c_p\ge0$ such that
 $$\|A(\eta)\|_{\mathcal{L}(\mathcal{H}^{0,\gamma}_p({\mathbb D}))}\le c_p\,\spk{\eta}^\nu\qquad
   \forall\;\eta\in\Sigma.$$
\end{theorem}
\begin{proof}
The proof is a combination of that for the boundaryless case (cf.\ Proposition 1 in
\cite{CSS1}) and the results of \cite{GrKo} on the $L_p$-continuity of operators from
Boutet's calculus. To give an idea, we shall derive the desired norm estimate for the
operator $\sigma_0\,t^{-\nu}\,\text{\rm op}_M^{\gamma-\frac{n}{2}}(g)\,\sigma_1$ with
$g(t,\tau,\eta)=\widetilde{g}(\tau,t\eta)$ as in \eqref{h4.D.5} (the $t$-independence of
$\widetilde{g}$ can always be reached by a tensor product argument). By conjugation with
$t$-powers, it is no restriction to assume that $\gamma=(n+1)(\frac{1}{2}-\frac{1}{p})$.
With this choice of $\gamma$, we have
$\mathcal{H}^{0,\gamma}_p(X^\wedge)=L_p(X^\wedge,t^ndtdx)$ and the operators
 $$\kappa_{\varrho}\in\mathcal{L}(\mathcal{H}^{0,\gamma}_p(X^\wedge)),\text{ defined by }
   (\kappa_{\varrho}u)(t,x)=\varrho^{\frac{n+1}{p}}u(\varrho t,x),$$
provide bijective isometries for each $\varrho>0$. Moreover, defining
$(Su)(r,x)=e^{(\frac{n}{2}-\gamma)r}u(e^{-r},x)$ for functions $u$ on
$X^\wedge$, we obtain an isometric isomorphism
 $$S:\mathcal{H}^{0,\gamma}_p(X^\wedge)\longrightarrow L_p({\mathbb R}\times X,drdx).$$
Thus we are done if we can prove that
 $$\spk{\eta}^{-\nu}\|S\,\kappa_{\spk{\eta}}^{-1}\,t^{-\nu}\,
   \text{\rm op}_M^{\gamma-\frac{n}{2}}(g)(\eta)\,\kappa_{\spk{\eta}}\,
   S^{-1}\|_{\mathcal{L}(L_p({\mathbb R}\times X,drdx))}\le c_p, $$
uniformly in $\eta$. Now note that conjugation of $\text{\rm op}_M^{\gamma-\frac{n}{2}}(g)(\eta)$
by $\kappa_{\spk{\eta}}$ amounts to replacing ${g}(t,z,\eta)$ by
${g}(\frac{t}{\spk{\eta}},z,\eta)$, and conjugation with $S$ transforms a Mellin
operator on $X^\wedge$ with symbol $h(t,\frac{n+1}{2}-\gamma+i\tau)$ into a
pseudodifferential operator on ${\mathbb R}\times X$ with symbol
$a(r,\varrho)=h(e^{-r},\frac{n+1}{2}-\gamma+i\varrho)$.
Passing to the local situation, cf.\ \eqref{h4.F} and \eqref{h4.G}, we have to show that
$\|\text{\rm op}^\prime(d_0)(\eta)\|_{\mathcal{L}(L_p(\overline{{\mathbb R}}^{n+1}_+))}\le c_p$
for the symbol kernel
 $$d_0(r,x^\prime,\tau,\xi^\prime,\eta;u,v)=e^{r\nu}
   \mbox{$\widetilde{d}(x^\prime,\tau,\xi^\prime,e^{-r}\frac{\eta}{\spk{\eta}};
   [\tau,\xi^\prime,e^{-r}\frac{\eta}{\spk{\eta}}]u,
   [\tau,\xi^\prime,e^{-r}\frac{\eta}{\spk{\eta}}]v)$},$$
see \eqref{h3.D} for the definition of $\text{\rm op}^\prime$. By a straightforward calculation,
 $$\|u^{k^\prime}D^k_u v^{l^\prime}D^l_v D^{m^\prime}_r D^m_\tau
   D^{\alpha^\prime}_{x^\prime} D^\alpha_{\xi^\prime}
   d_0(r,x^\prime,\tau,\xi^\prime,\eta;u,v)\|_{L^2({\mathbb R}_{+,u}\times{\mathbb R}_{+,v})}\le
   C\,\spk{\tau,\xi^\prime}^{-|\alpha|-m+k-k^\prime+l-l^\prime},$$
uniformly in $(r,x^\prime,\tau,\xi^\prime,\eta)$. Now Theorem 4.1.(5) in \cite{GrKo}
gives the continuity of $\text{\rm op}^\prime(d_0)(\eta)$ in $L_p(\overline{{\mathbb R}}^{n+1}_+)$, with
operator norm uniformly bounded in $\eta$.
\end{proof}

The local symbols $\widetilde{a}$ in \eqref{h4.E} are defined for
$(\tau,\xi,\eta)\in{\mathbb R}^{1+n}\times\Sigma$. The symbols that arise in the analysis
of resolvents extend -- holomorphically in $\eta$ -- to larger subsets of ${\mathbb C}$.
We shall need this property in particular for the principal part and define a
corresponding class:

\begin{definition}\label{h4.3}
For $c_0>0$ let $\Omega_{(\tau,\xi)}=\{\eta\in{\mathbb C}\mbox{\boldmath$\;|\;$\unboldmath}|\eta|\ge c_0\spk{\tau,\xi}\}$.
The set $C_{\rm hol}^{\nu,0}({\mathbb D};\Sigma,\gamma)$ consists of all operator-families
$A(\eta)\in{C}^{\nu,0}({\mathbb D};\Sigma,\gamma)$ such that the local symbols $\widetilde{a}$ from
\eqref{h4.E} admit a decomposition $\widetilde{a}=\widetilde{a}_0+\widetilde{a}_1$ with the following
properties:
\begin{itemize}
 \item[a)] $\widetilde{a}_0$ extends holomorphically to $\eta\in\Omega_{(\tau,\xi)}$
  $($for some $c_0)$ and satisfies the estimates
  \begin{equation}\label{h4.H}
   |\partial_t^l\partial_x^\beta\partial_\tau^k\partial_\xi^\alpha\partial_\eta^\gamma\,
   \widetilde{a}_0(t,x,\tau,\xi,\eta)|\le
   c\spk{\tau,\xi,\eta}^{\nu-k-|\alpha|-|\gamma|}
  \end{equation}
  uniformly in $(\tau,\xi)\in{\mathbb R}^{1+n}$ and $\eta\in\Omega_{(\tau,\xi)}\cup\Sigma$
  for any order of derivatives,
 \item[b)] $\widetilde{a}_1(t,x,\tau,\xi,\eta)\in
   S^{\nu-1}_{\rm cl}(\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}_{(\tau,\xi)}\times\Sigma_\eta)$.
\end{itemize}
For the pseudodifferential symbols of $P(\eta)$ in \eqref{h4.D} we require the
corresponding structure.
\end{definition}

\section{$H_\infty$-calculus of cone differential operators}\label{sectionh5}
The aim of this section is to prove the following theorem:

\begin{theorem}\label{hinfty}
If the resolvent set of the realization $\underline{A}_T$ contains
$\{\lambda\in\Lambda\mbox{\boldmath$\;|\;$\unboldmath} |\lambda|\ge R\}$ for some $R\ge0$ and there exists an
$A(\eta)\in C^{-\mu,0}({\mathbb D};\Sigma,\gamma)$ such that
 $$(\eta^\mu-\underline{A}_T)^{-1}=A(\eta)\qquad\forall\;|\eta|\ge R^{1/\mu},$$
then there exists a constant $c\ge0$ such
that $c+\underline{A}_T$ admits a bounded $H_\infty$-calculus with respect to ${\mathbb C}\setminus\Lambda$
$($simultaneously for all $1<p<\infty)$.
\end{theorem}

Before going into details, let us give an outline of the proof. In large parts it
follows the proof of Theorem 5.1 in \cite{CSS1}, where we showed the existence of
bounded imaginary powers for operators on conic manifolds without boundary.

By replacing from the very beginning the differential
operator $A$ by $c+A$ we can assume that the resolvent set contains all of $\Lambda$ and
 $$A(\eta):=(\eta^\mu-\underline{A}_T)^{-1}\in C^{-\mu,0}({\mathbb D};\Sigma,\gamma).$$
We then show that $\underline{A}_T$ itself has a bounded $H_\infty$-calculus.
Let us remark that, for $f\in H$,
 $$f(\underline{A}_T)=\int_{\partial\Lambda}f(\lambda)(\lambda-\underline{A}_T)^{-1}\,d\lambda=
   \int_{\partial\Sigma}f(\eta^\mu)A(\eta)\eta^{\mu-1}\,d\eta.$$
Inserting the explicit formula for $A(\eta)$, cf.\ \eqref{h4.D}, we obtain four
integrals over the boundary of $\Sigma$ which have to be estimated from above by
$M\|f\|_\infty$ with a constant $M$ independent of $f\in H$, cf.\eqref{h2.C}.

Obviously, the integral associated with $G_\infty(\eta)$ can be estimated as desired,
since $\eta^{\mu-1}G_\infty(\eta)$ is reapidly decreasing in $\eta$, hence integrable.

For the integral associated with $(1-\sigma)P(\eta)(1-\sigma_1)$ we can apply
Theorem \ref{h3.3}, since away from the singularity the Sobolev spaces and
operator classes coincide with the usual ones on the smooth manifold with
boundary $2{\mathbb D}$.

In Propositions \ref{lower}, \ref{principal}, and \ref{singg} we shall treat the term
\begin{equation}\label{mellinterm}
 \sigma_0\int_{\partial\Sigma} f(\eta^\mu) \, t^\mu\opm{\gamma-\frac{n}{2}}(h)(\eta)
 \,\eta^{\mu-1}\,d\eta\,\sigma_1=
 \opm{\gamma-\frac{n}{2}}\Big(\sigma_0\,t^\mu\int_{\partial\Sigma} f(\eta^\mu)\,h(\eta)
 \,\eta^{\mu-1}\,d\eta\Big)\sigma_1.
\end{equation}
First, however, we shall study the term induced by $G(\eta)$.
To this end note that  multiplication by a cut-off function
$\sigma\in\mathcal{C}^\infty_{\text{\rm comp}}([0,1[)$ induces continuous
operators $\mathcal{H}^{0,\gamma}_p({\mathbb D})\to\mathcal{H}^{0,\gamma}_p(X^\wedge)$ and
$\mathcal{H}^{0,\gamma}_p(X^\wedge)\to\mathcal{H}^{0,\gamma}_p({\mathbb D})$, cf.\ \eqref{h4.B.2}.
Thus it suffices to show the following:

\begin{proposition}\label{greenpart}
Let $G(\eta)\in R^{-\mu,0}_G(X^\wedge;\Sigma,\gamma)$ and
$G_f=\sigma_0 \displaystyle\int_{\partial\Sigma}
f(\eta^\mu) G(\eta)\eta^{\mu-1}\,d\eta \, \sigma_1$ for
$f\in H$. Then $G_f\in\mathcal{L}(\mathcal{H}^{0,\gamma}_p(X^\wedge))$, and there
exists a constant $M_p\ge0$ such that
\begin{equation}\label{normbound}
 \|G_f\|_{\mathcal{L}(\mathcal{H}^{0,\gamma}_p(X^\wedge))}\le
 M_p\| f \|_{\infty}\qquad\forall\; f\in H.
\end{equation}
\end{proposition}
\begin{proof}
By conjugation with $t^\gamma$ we can assume that $\gamma=0$.
Due to the symmetry of $\partial\Sigma$ and the fact that $\eta^{\mu-1}G(\eta)$ is
integrable on compact parts of $\partial\Sigma$, we may confine ourselves to the
integration over the set
  $$\mathcal{C}=\{\varrho e^{i\alpha}\mbox{\boldmath$\;|\;$\unboldmath} 1\le\varrho<\infty\}.$$
According to Section \ref{sectionh4.4.1}, $G(\eta)$ is
an integral operator (with respect to the scalar product in
$\mathcal{H}^{0,0}_2(X^\wedge)$). Suppressing, for notational simplicity,
the dependence on the variables $x$, $x^\prime$, the kernel of
$G(\eta)$, $|\eta|\ge1$ is $k(\eta,t,s)=\widetilde{k}(\eta,|\eta|\,t,|\eta|\,s)$,
where, for some $\varepsilon>0$,
  $$\widetilde{k}(\eta,t,s)\in S^{-\mu+n+1}(\Sigma) \,{\widehat{\otimes}}_\pi\,
    \mathcal{S}^\varepsilon_0(X^\wedge)\,{\widehat{\otimes}}_\pi\, \mathcal{S}^\varepsilon_0(X^\wedge).$$
Then $G_f$ the an integral operator with kernel
\begin{equation}\label{kernelgz}
   k_f(t,s)=\sigma_0(t)\,\sigma_1(s)\int_\mathcal{C}
   f(\eta^\mu)k(\eta,t,s)\,\eta^{\mu-1} \, d\eta.
\end{equation}
Writing
$\widetilde{k}(\eta,t,s)=({\chi}(t)+(1-{\chi})(t))\widetilde{k}(\eta,t,s)({\chi}(s)+(1-{\chi})(s))$
with the characteristic function ${\chi}$ of $[0,1]$, the proposition will be
true, if we can show that in each one of the four cases
 \begin{align}
  k_f(t,s)&=\sigma_0(t)\,\sigma_1(s)\int_{\mathcal{C}} f(\eta^\mu)
            {\chi}( |\eta|\,t )k(\eta,t,s)
            {\chi}( |\eta| \, s )\,\eta^{\mu-1} \, d\eta
            \label{kernel1}\\
  k_f(t,s)&=\sigma_0(t)\,\sigma_1(s)\int_{\mathcal{C}} f(\eta^\mu)
            {\chi}( |\eta|\,t ) k(\eta,t,s)
            (1-{\chi})( |\eta| \, s )\,\eta^{\mu-1} \, d\eta
            \label{kernel2a}\\
  k_f(t,s)&=\sigma_0(t)\,\sigma_1(s)\int_{\mathcal{C}} f(\eta^\mu)
            (1-{\chi})( |\eta|\,t)k(\eta,t,s)
            {\chi}( |\eta| \, s )\,\eta^{\mu-1} \, d\eta
            \label{kernel2b}\\
  k_f(t,s)&=\sigma_0(t)\,\sigma_1(s)\int_{\mathcal{C}} f(\eta^\mu)
            (1-{\chi})( |\eta| \, t)k(\eta,t,s)
            (1-{\chi})( |\eta| \, s )\,\eta^{\mu-1} \, d\eta
            \label{kernel3}
 \end{align}
the associated integral operators satisfies \eqref{normbound}.
To begin with case \eqref{kernel1} we use the fact
that, for some fixed $\varepsilon > 0$,
  $$|{k}(\eta,t,s)|\le c|\eta|^{-\mu+2\varepsilon}
    t^{-\frac{n+1}{2}+\varepsilon}s^{-\frac{n+1}{2}+\varepsilon},$$
uniformly in $\eta\in{\mathcal{C}}$ and $t,s>0$, cf.\ \eqref{h4.B.4} and \eqref{h4.X}. Then
\begin{align*}
  |k_f(t,s)|&\le
    c\,\| f \|_{\infty}\,\sigma_0(t)\,\sigma_1(s)\,
    t^{-\frac{n+1}{2}+\varepsilon}s^{-\frac{n+1}{2}+\varepsilon}
    \int_1^\infty \varrho^{-1+2\varepsilon}
    {\chi}(\varrho t){\chi}(\varrho s)\,d\varrho\\
  &\le\| f \|_{\infty}\, \frac{c}{2\varepsilon}\,
    \min\left(\frac{1}{t},\frac{1}{s}\right)^{2\varepsilon}\sigma_0(t)\,\sigma_1(s)\,
    t^{-\frac{n+1}{2}+\varepsilon}s^{-\frac{n+1}{2}+\varepsilon}.
\end{align*}
Since the factor $\frac{c}{2\varepsilon}\sigma_0(t)\,\sigma_1(s)$ can be estimated from above
by a constant, it remains to consider the kernel
$t^{-\frac{n+1}{2}+\varepsilon}s^{-\frac{n+1}{2}+\varepsilon}\min(\frac{1}{t},\frac{1}{s})^{2\varepsilon}$.
Because this kernel is symmetric in $s$ and $t$, indeed it suffices to treat
\begin{equation}\label{kern1.4}
  k(t,s)=\begin{cases}
          t^{-\frac{n+1}{2}-\varepsilon}s^{-\frac{n+1}{2}+\varepsilon}&:s\le t\\
          0&:s>t.
         \end{cases}
\end{equation}
Recalling the Hardy inequality
  $$\int_0^\infty\Big(\int_0^t g(s)\,ds\Big)^pt^{-1-r}\,dt\le
    \mbox{$\left(\frac{p}{r}\right)^p$}\int_0^\infty g(t)^pt^{p-1-r}\,dt,$$
which holds for any non-negative function $g$ on ${\mathbb R}_+$ and $r>0$
(cf.\ \cite{StWe}, Lemma 3.14, page 196), and denoting by $G$ the integral operator
with kernel \eqref{kern1.4}, we have
\begin{align*}
  \|Gu\|^p_{\mathcal{H}^{0,0}_p(X^\wedge)}&\le
     \int_0^\infty\Big(\int_0^\infty
     k(t,s)|u(s)|\,s^nds\Big)^pt^{\frac{n+1}{2}p-1}\,dt
     =\int_0^\infty\Big(\int_0^t s^{\frac{n-1}{2}+\varepsilon}|u(s)|\,ds\Big)^p t^{-1-p\varepsilon}\,dt\\
  &\le\mbox{$\left(\frac{p}{p\varepsilon}\right)^p$}\int_0^\infty
     |u(t)|^p\,t^{\frac{n+1}{2}p-1}dt=
     \mbox{$\left(\frac{1}{\varepsilon}\right)^p$}\|u\|^p_{\mathcal{H}^{0,0}_p(X^\wedge)},
\end{align*}
which completes the proof for the case \eqref{kernel1}. The proofs for the
cases \eqref{kernel2a}, \eqref{kernel2b} and \eqref{kernel3} can be obtained similarly,
cf. \cite{CSS1}.
\end{proof}

Let us now turn our attention to the analysis of (\ref{mellinterm}). We make use of the
structure of $\widetilde{h}$ as described in \eqref{h4.E}, \eqref{h4.F}, \eqref{h4.G}, and
Definition \ref{h4.3}. Then we have to handle two terms involving
$\widetilde{a}_0$ and $\widetilde{a}_1$, and a term coming from the symbol kernel $\widetilde{d}$.

Let us recall that any Mellin symbol
$h\in M\!S^{\nu}({\mathbb R}_+\times{\mathbb R}^n\times\Gamma_{\beta}\times{\mathbb R}^n)$, i.e.\ any
smooth function satisfying, for any order of derivatives,
 $$|(t\partial_{t})^k\partial^{\alpha^\prime}_{x}\partial^l_{\tau}\partial^\alpha_{\xi}
   h(t,x,\beta+i\tau,\xi)|\le c  \spk{\tau,\xi}^{\nu-l-|\alpha|}, $$
induces continuous operators from
$\mathcal{H}^{s,\frac{n+1}{2}-\beta}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n)$ to
$\mathcal{H}^{s-\nu,\frac{n+1}{2}-\beta}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n)$.

\begin{proposition}\label{lower}
Let $\widetilde{a}_1\in S^{-\mu-1}(\overline{{\mathbb R}}_+\times{\mathbb R}^n\times{\mathbb R}^{1+n}\times\Sigma)$
be compactly supported in $(t,x)$ and
 $$h_f(t,x,\mbox{$\frac{n+1}{2}$}-\gamma+i\tau,\xi)=
   t^\mu\int_{\partial\Sigma} f(\eta^\mu)\,
   \widetilde{a}_1(t,x,\tau,\xi,t \eta)\,\eta^{\mu-1}\,d\eta$$
with $f\in H$. This defines a symbol
$h_f\in M\!S^{-1}({\mathbb R}_+\times{\mathbb R}^n\times\Gamma_{\frac{n+1}{2}-\gamma}\times{\mathbb R}^n)$
and the symbol estimates of $\| f \|^{-1}_{\infty}\,h_f$ are uniform in $0\not=f\in H$.
Consequently,
  $$\|\text{\rm op}_M^{\gamma-\frac{n}{2}}\text{\rm op}_+(h_f)\|_{\mathcal{L}
     (\mathcal{H}^{0,\gamma}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n))}\le
     M_p\, \| f \|_{\infty}\qquad \forall\;f\in H$$
for a suitable constant $M_p\ge0$.
\end{proposition}
\begin{proof}
Without loss of generality, we consider the case $\gamma = \frac{n+1}{2}$.
We have to show that
\begin{equation}\label{ms0estimate}
 | \partial^l_{\tau} (t\partial_{t})^k \partial^\alpha_{\xi} \partial^\beta_{x}
 h_f(t,x,i\tau,\xi) | \,
 \langle\tau,\xi \rangle^{1+l+|\alpha|}  \, \| f \|^{-1}_{\infty}
\end{equation}
is uniformly bounded for $t>0$, $x \in {\mathbb R}^n$, $\tau\in{\mathbb R}$ and $0\not=f\in H$.
Observing that $t\partial_t t^\mu=\mu t^\mu$ and
  $$t\partial_t\left(\widetilde{a}_1
    (t,x,i\tau,\xi,t\eta)\right)=
    (t\partial_t\widetilde{a}_1)(t,x,i\tau,\xi,t\eta)+
    (\eta\partial_\eta\widetilde{a}_1)(t,x,i\tau,\xi,t\eta),$$
we see that the totally characteristic derivative gives rise to terms
of the same type as $\widetilde{a}_1$.
Since the derivatives with respect to $x$, $\tau$ and $\xi$ can be
taken under the integral sign, we may assume from the beginning that
$\widetilde{a}_1\in S^{-\mu-1-j}(\overline{{\mathbb R}}_+\times{\mathbb R}^n\times{\mathbb R}^{1+n}\times\Sigma)$
and show that
  $$|h_f(t,x,i\tau,\xi)|\le c  \, \langle\tau, \xi \rangle^{-1-j} \,
    \| f \|_{\infty}$$
uniformly in $t>0$, $x \in {\mathbb R}^n$ and $f\in H$. By hypothesis, we have
  $$|h_f(t,x,i\tau,\xi)|\le c \, \| f \|_{\infty} \,
    t^\mu\int_0^{+\infty}\spk{\tau,\xi,t\varrho}^{-\mu-1-j}\,
    \varrho^{\mu-1} \, d\varrho,$$
The transformation $\varrho=t^{-1}\langle \tau,\xi\rangle\sigma$ together with the identity
$\spk{\tau,\xi,\spk{\tau,\xi}\sigma}=\spk{\tau,\xi}\spk{\sigma}$ yields
\begin{align*}
 |h_f(t,x,i\tau,\xi)|&\le
    c \, \| f \|_{\infty} \int_{0}^{+\infty}
    \spk{\tau,\xi}^{-\mu-1-j}\spk{\sigma}^{-\mu-1-j}\spk{\tau,\xi}^{\mu-1}\sigma^{\mu-1}
    \spk{\tau,\xi}\,d\sigma\\
 &\le c \, \| f \|_{\infty} \spk{\tau,\xi}^{-1-j}
    \int_{0}^{+\infty}\spk{\sigma}^{-2}\,d\sigma.
\end{align*}
This finishes the proof.
\end{proof}

\begin{proposition}\label{principal}
Let $\widetilde{a}_0\in S^{-\mu}(\overline{{\mathbb R}}_+\times{\mathbb R}^n\times{\mathbb R}^{1+n}\times\Sigma)$ be as described
in Definition {\rm\ref{h4.3}} with compact $(t,x)$-support. Define
 $$h_f(t,x,\mbox{$\frac{n+1}{2}$}-\gamma+i\tau,\xi)=
   t^\mu\int_{\partial\Sigma} f(\eta^\mu)\,
   \widetilde{a}_0(t,x,\tau,\xi,t \eta)\,\eta^{\mu-1}\,d\eta$$
with $f\in H$. Then
$h_f\in MS^{0}({\mathbb R}_+\times{\mathbb R}^n\times\Gamma_{\frac{n+1}{2}-\gamma}\times{\mathbb R}^n)$,
and the symbol estimates of $\| f \|^{-1}_{\infty}\,h_f$ are uniform in $0\not=f\in H$.
Therefore, for a suitable constant $M_p\ge0$,
  $$\|\text{\rm op}_M^{\gamma-\frac{n}{2}}\text{\rm op}_+(h_f)\|_{\mathcal{L}
     (\mathcal{H}^{0,\gamma}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n))}\le
     M_p\, \| f \|_{\infty}\qquad \forall\;f\in H.$$
\end{proposition}
\begin{proof}
It is sufficient to show the symbol estimate for $h_f$; the estimates for derivatives
of $h_f$ are obtained similarly, arguing as in the proof of Proposition \ref{lower}.
The change of variables $\eta\mapsto t^{-1}\eta$ yields
\begin{equation}\label{h5.Z}
 h_f(t,x,\mbox{$\frac{n+1}{2}$}-\gamma+i\tau,\xi)=
 \int_{\partial\Sigma} f(t^{-\mu}\eta^\mu)\,
 \widetilde{a}_0(t,x,\tau,\xi,\eta)\,\eta^{\mu-1}\,d\eta.
\end{equation}
Let us now denote by $\mathcal{C}_{(\tau,\xi)}$ the natural parametrization of the boundary of
$\{\eta\in{\mathbb C}\setminus\Sigma\mbox{\boldmath$\;|\;$\unboldmath} |\eta|<c_0\spk{\tau,\xi}\}$, where $c_0$ is associated
with $\widetilde{a}_0$ as in Definition \ref{h4.3}.
As the integrand in \eqref{h5.Z}, for fixed $(t,x,\tau,\xi)$, is holomorphic in $\eta$
outside $\Sigma\cup\{|\eta|<c_0\spk{\tau,\xi}\}$ and decays there as
$|\eta|^{-1-\varepsilon}$ for some $\varepsilon>0$ (recall the decay property of functions in $H$),
we may replace the integration over $\partial\Sigma$ in \eqref{h5.Z} by integration
over $\mathcal{C}_{(\tau,\xi)}$  and obtain
 $$|h_f(t,x,\mbox{$\frac{n+1}{2}$}-\gamma+i\tau,\xi)|\le c\,\|f\|_\infty\,
   \text{\rm length}(\mathcal{C}_{(\tau,\xi)})\,\spk{\tau,\xi}^{-\mu}\,\spk{\tau,\xi}^{\mu-1}
   \le c\,\|f\|_\infty,$$
since the length of $\mathcal{C}_{(\tau,\xi)}$ is less than $(2+2\pi)c_0\spk{\tau,\xi}$.
\end{proof}

\begin{proposition}\label{singg}
Let ${d}$ be a symbol kernel as in \eqref{h4.F}, \eqref{h4.G}, with
compact support in $(t,x^\prime)$. Define, for $f\in H$, the symbol kernel
 $$h_f(t,x^\prime,\mbox{$\frac{n+1}{2}-\gamma+i\tau$},\xi^\prime;u,v) =
   t^\mu\int_{\partial\Sigma} f(\eta^\mu) \,
   {d}(t,x^\prime,\tau,\xi^\prime;u,v)\,\eta^{\mu-1}\,d\eta.$$
Then there exists a constant $M_p\ge0$ such that
\begin{equation}\label{h5.W}
 \|\opm{\gamma-\frac{n}{2}}\text{\rm op}^\prime
 (h_f)\|_{\mathcal{L}(\mathcal{H}^{0,\gamma}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n))}
 \le M_p\, \| f \|_{\infty}\qquad \forall\;f\in H.
\end{equation}
\end{proposition}
\begin{proof}
Without loss of generality, $\gamma=\frac{n+1}{2}-\frac{1}{p}$. For this choice
$\mathcal{H}^{0,\gamma}_p({\mathbb R}_+\times\overline{{\mathbb R}}_+^n)=L_p({\mathbb R}_+\times{\mathbb R}^n_+)$.
We are now going to show that
\begin{equation}\label{h5.Y}
 |\partial^k_\tau \partial^\alpha_{\xi^\prime} (t\partial_t)^l \partial^\beta_{x^\prime}
 h_f(t,x^\prime,\mbox{$\frac{1}{p}$}+i\tau,\xi^\prime;u,v)|
 \le c\,\|f\|_{\infty} \, \frac{\spk{\tau,\xi^\prime}^{-k-|\alpha|}}{u+v}
\end{equation}
for any $k,l\in{\mathbb N}_0$ and $\alpha,\beta\in{\mathbb N}_0^{n-1}$.
It is enough to consider the case $k=l=0$ and $\alpha=\beta=0$, since the terms for
higher order derivatives are of the same kind, cf.\ the proof of Proposition \ref{lower}.

Inserting the explicit form of ${d}$, cf.\ \eqref{h4.F}, we obtain
\begin{equation*}
 h_f(t,x^\prime,\mbox{$\frac{1}{p}$}+i\tau,\xi^\prime;u,v)=
 t \int_{\partial\Sigma} f(\eta^\mu) \,
 \widetilde{d}(x^\prime,\xi^\prime,t\eta;[\xi^\prime,\tau,t\eta] u,
 [\xi^\prime,\tau,t\eta] v)\,(t\eta)^{\mu-1}\,d\eta.
\end{equation*}
Since $[\cdot]\sim\spk{\cdot}$ and $d$ is rapidly decreasing in $(u,v)$, hence in $u+v$,
we can estimate
\begin{align*}
  |h_f(t,x^\prime,\mbox{$\frac{1}{p}$}+i\tau,\xi^\prime;u,v)|
& \le
  c \|f\|_{\infty}\int_0^\infty (\varrho t)^{\mu-1} \,
  \spk{(u+v)\spk{\xi^\prime,\tau,\varrho t}}^{-2} \,
  \spk{\xi^\prime,\tau,\varrho t}^{1-\mu} \, t \, d\varrho\\
& = c \|f\|_{\infty} \int_0^\infty \sigma^{\mu-1} \,
  \spk{(u+v)\spk{\tau,\xi^\prime}\spk{\sigma}}^{-2} \, \spk{\sigma}^{1-\mu}\,
  \spk{\tau,\xi^\prime}\,d\sigma;
\end{align*}
for the last identity we made use of the change of variables
$\varrho=t^{-1}\spk{\tau,\xi^\prime}\sigma$. The change of variables
$r=(u+v)\spk{\tau,\xi^\prime}(1+\sigma)$ then yields
\begin{align*}
  |h_f(t,x^\prime,\mbox{$\frac{1}{p}$}+i\tau,\xi^\prime;u,v)|
& \le c \|f\|_{\infty} \int_0^\infty \sigma^{\mu-1} \,
  \spk{(u+v)\spk{\tau,\xi^\prime}(1+\sigma)}^{-2} \, \spk{\sigma}^{1-\mu}\,
  \spk{\tau,\xi^\prime}\,d\sigma\\
& \le c \|f\|_{\infty} \frac{1}{u+v}\int_0^\infty \spk{r}^{-2}\,dr,
\end{align*}
i.e. \eqref{h5.Y}. Using the continuity of the Hilbert transform,
 $$u\mapsto Hu=\int_0^\infty\frac{u(s)}{\cdot+s}\,ds:
   L_p({\mathbb R}_+)\longrightarrow L_p({\mathbb R}_+),$$
and the $L_p({\mathbb R}^n)$-continuity of standard zero order pseudodifferential operators,
assertion \eqref{h5.W} is obtained as follows: Let us write for short $y=(t,x^\prime)$
and $\text{\rm op}_{y}=\text{\rm op}_{M,t}^{\gamma-\frac{n}{2}}\text{\rm op}_{x^\prime}$. Then,
for $\varphi=\varphi(u,y)\in\mathcal{C}^\infty_{\text{\rm comp}}({\mathbb R}_+\times{\mathbb R}_+^n)$,
 $$\big(\text{\rm op}_M^{\gamma-\frac{n}{2}}\text{\rm op}^\prime(h_f)\varphi\big)(u,y)=
   \int_0^\infty\big(\text{\rm op}_{y}(h_f(u,v))\varphi\big)(v,y)\,dv,$$
and therefore
\begin{align*}
 \|\text{\rm op}^\prime(h_f)\varphi\|_{L_p({\mathbb R}_+\times{\mathbb R}^{n}_+)}^p&\le
   \int_0^\infty\int\Big(\int_0^\infty|\big(\text{\rm op}_y(h_f(u,v))\varphi\big)(u,y)|dv\Big)^p\,
   dydu\\
 &\le \int_0^\infty\Big(\int_0^\infty\Big(
   \int|\big(\text{\rm op}_y(h_f(u,v))\varphi\big)(u.y)|^p\,dy\Big)^{1/p}dv\Big)^p du.
\end{align*}
The second estimate is due to Minkowski's inequality for integrals. Thus
\begin{align*}
 \|\text{\rm op}^\prime(h_f)\varphi\|_{L_p({\mathbb R}_+\times{\mathbb R}^{n}_+)}^p&\le
   c\|f\|_\infty^p\int_0^\infty\Big(\int_0^\infty\frac{1}{u+v}
   \|\varphi(v,y)\|_{L_p({\mathbb R}_+\times{\mathbb R}^n_{+,y})}\,dv\Big)^p\,du\\
 &\le c\|f\|_\infty^p \int_0^\infty \|\varphi(u,y)\|_{L_p({\mathbb R}_+\times{\mathbb R}^n_{+,y})}^p\,du
   =c\|f\|_\infty^p\|\varphi\|_{L_p({\mathbb R}_+\times{\mathbb R}^n_+)}^p.
\end{align*}
This finishes the proof.
\end{proof}

\section{Parameter-ellipticity of the minimal extension}\label{sectionh6}
In Theorem \ref{hinfty} we showed the existence of a bounded $H_\infty$-calculus for
a closed extension $\underline{A}_T$, assuming that the resolvent exists in the sector
$\Lambda$ and has a suitable structure. An obvious problem is now to find conditions on $A$
and $T$ which are more easily checked and which ensure all the required assumptions
on $\underline{A}_T$. In this section we shall give such conditions for the case
$\underline{A}_T=A_{T,\min}$. In fact, these conditions are obtained by combining the concept of
parameter-ellipticity in Schulze's cone calculus and the observations from
Section 3.2 of \cite{CSS2}.

As described in Section \ref{sectionh4.2}, ${\mathbb D}$-ellipticity of $\spalte{A}{T}$ is
characterized by the invertibility of the principal symbol $\sigma_\psi^\mu(A)$, the
rescaled symbol $\widetilde\sigma_\psi^\mu(A)$, the boundary symbol
$\sigma_\partial^\mu\spalte{A}{T}$, and the rescaled boundary symbol
$\widetilde\sigma_\partial^\mu\spalte{A}{T}$. We shall now pose
stronger conditions, which we call {\em parameter-ellipticity} with respect to the sector
$\Lambda$. The first condition is an analog of condition \eqref{h3.Z}:
\begin{itemize}
 \item[{\bf (E1)}] {\em Both $\sigma_\psi^\mu(A)$ and $\widetilde\sigma_\psi^\mu(A)$
  pointwise do not have spectrum in $\Lambda$ $($for non-zero covariables$)$}.
\end{itemize}
For $\spalte{A}{T}$ as in \eqref{h4.A.2} we consider the boundary symbol and
rescaled boundary symbol
\forget{
For notational convenience we use in the following the same letters $E$ and $F$
also for the restrictions of those bundles to subsets of ${\mathbb D}$, e.g.\ we write
$E$ instead of $E|_{\mathbb B}$. Let $\pi_{\mathbb B}:T^*\text{\rm int}\,{\mathbb B}\to\text{\rm int}\,{\mathbb B}$ and
$\pi_{\partial{\mathbb B}}:T^*\partial{\mathbb B}\to\partial{\mathbb B}$ denote the canonical projections.
Then}
\begin{equation*}
 \sigma_\partial^\mu\begin{pmatrix}A\\ T\end{pmatrix}:
   \mathcal{S}(\overline{{\mathbb R}}_+,E^\prime)\longrightarrow
   \begin{pmatrix}
    \mathcal{S}(\overline{{\mathbb R}}_+,E^\prime)\\ \oplus\\ F^\prime
   \end{pmatrix},\qquad
 \widetilde\sigma_\partial^\mu\begin{pmatrix}A\\ T\end{pmatrix}:
   \mathcal{S}(\overline{{\mathbb R}}_+,E^{\prime\prime})\longrightarrow
   \begin{pmatrix}
    \mathcal{S}(\overline{{\mathbb R}}_+,E^{\prime\prime})\\ \oplus\\ F^{\prime\prime}
   \end{pmatrix},
\end{equation*}
where $E^\prime,\,E^{\prime\prime}$ and $F^\prime,\,F^{\prime\prime}$ are the
corresponding pull-backs of $E$ and $F$ to $T^*\text{\rm int}\,{\mathbb B}$ and $T^*\partial{\mathbb B}$, respectively.
We require that
\begin{itemize}
 \item[{\bf (E2)}] {\em Both
  $\spalte{\lambda}{0}-\sigma_\partial^\mu\spalte{A}{T}$ and
  $\spalte{\lambda}{0}-\widetilde\sigma_\partial^\mu\spalte{A}{T}$ are pointwise
  invertible on $(T^*\text{\rm int}\,{\mathbb B}\times\Lambda)\setminus\{0\}$ and
  $(T^*\partial{\mathbb B}\times\Lambda)\setminus\{0\}$, respectively}.
\end{itemize}

Mainly to ensure the identity \eqref{h4.A.5}, we pose a condition on the
so-called {\em conormal symbol}
$\sigma_M^\mu\begin{pmatrix}A\\ T\end{pmatrix}(z)=
\begin{pmatrix}\sigma_M^\mu(A)(z)\\ \sigma_M^\mu(T)(z)\end{pmatrix}$.
This is a holomorphic family in $z\in{\mathbb C}$ of boundary value problems on $X$ obtained
in the following way: Using the representation of $A$ as in \eqref{fuchs},
 $$\sigma_M^\mu(A)(z):=\mathop{\mbox{\large$\sum$}}_{j=0}^\mu a_j(0)z^j$$
is a holomorphic family of differential operators on $X$. Similarly, using the notation
from \eqref{h4.A.6}, one defines
$\sigma_M^\mu(T)=(\sigma_M^0(T_0),\ldots,\sigma_M^{\mu-1}(T_{\mu-1}))$ by
 $$\sigma_M^j(T_j)(z)=\gamma_0\circ\sigma_M^j(B_j)(z).$$
\forget{
where $\gamma_0$ denotes restriction to the boundary of $X$.
}

It can be shown that $\sigma_M^\mu\spalte{A}{T}$ is meromorphically
invertible in case $\spalte{A}{T}$ is ${\mathbb D}$-elliptic (in the sense of
Section \ref{sectionh4.2}). Then the condition is that
\begin{itemize}
 \item[{\bf (E3)}]
  {\em
  $\sigma_M^\mu\begin{pmatrix}{A}\\{T}\end{pmatrix}(z):\mathcal{C}^\infty(X,E)\longrightarrow
  \begin{matrix}\mathcal{C}^\infty(X,E)\\ \oplus\\ \mathcal{C}^\infty(\partial X,F)\end{matrix}$ is
  invertible for each $z$ with $\text{\rm Re}\, z=\frac{n+1}{2}-\gamma-\mu$}.
\end{itemize}

\begin{remark}\label{h6.1}
The invertibility of the conormal symbol from {\rm(E3)} is equivalent to that of
 $$\sigma_M^\mu\begin{pmatrix}A\\ T\end{pmatrix}(z):H^s_p(X,E)\longrightarrow
   \begin{matrix}
    H^{s-\mu}_p(X,E)\\ \oplus\\
    \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
    B_{pp}^{s-j-1/p}(\partial X,F_j)
   \end{matrix}$$
for $s\ge 0$ and $1<p<\infty$. In fact, according to a result of
Grubb \cite[Theorem 1.12]{Grub}, invertibility for one choice of $s$ and $p$ implies
that the inverse also is an element in Boutet de Monvel's calculus and thus infers
the invertibility for every other choice. Equivalence with the invertibility on spaces
of smooth functions then follows from the fact that the kernel and the cokernel of an
elliptic operator consist of smooth functions.
\end{remark}

For $A$ as in \eqref{fuchs}, we define the so-called {\em model cone operator} $\widehat{A}$
on $X^\wedge={\mathbb R}_+\times X$ as
 $$\widehat{A}=t^{-\mu}\mathop{\mbox{\large$\sum$}}_{j=0}^\mu a_j(0)(-t\partial_t)^j.$$
Similarly, we define $\widehat{T}=(\widehat{T}_0,\ldots,\widehat{T}_{\mu-1})$ by
$\widehat{T}_j=\gamma_0\circ\widehat{B}_j$, cf.\ \eqref{h4.A.6}, where now $\gamma_0$ denotes the
restriction to $\partial X^\wedge={\mathbb R}_+\times\partial X$. For the analysis of $\widehat{A}$,
one uses a special scale of Sobolev spaces $\mathcal{K}^{s,\gamma}_p(X^\wedge)$
on $X^\wedge$ with $s\in{\mathbb R}$ and $1<p<\infty$, namely
 $$\mathcal{K}^{s,\gamma}_p(X^\wedge)=\{u\in H^s_{p,{\rm loc}}(X^\wedge)\mbox{\boldmath$\;|\;$\unboldmath}
   \omega u\in\mathcal{H}^{s,\gamma}_p({\mathbb D})\text{ and } (1-\omega)u\in
   H^s_{p,{\rm cone}}(X^\wedge)\};$$
here $\omega$ is a cut-off function located near $t=0$, and the subscript `{\rm cone}'
indicates that we do not consider $X^\wedge$ with its product structure, but as an
$SG$-manifold, cf.\ Section 4.2 in \cite{ScSc2} for more details.
If $E^\wedge$ is the pull-back of $E|_X$ to $X^\wedge$ under the canonical projection
$X^\wedge\to X$, this definition also extends to sections, i.e.\
we may define $\mathcal{K}^{s,\gamma}_p(X^\wedge,E^\wedge)$.

Analogously to Section \ref{sectionh4.3}, we then consider $\widehat{A}$ as an unbounded
operator
\begin{equation}\label{h6.C}
 \widehat{A}:\mathcal{S}^\infty(X^\wedge,E^\wedge)_{\widehat{T}}\subset
 \mathcal{K}^{0,\gamma}_p(X^\wedge,E^\wedge)\longrightarrow
 \mathcal{K}^{0,\gamma}_p(X^\wedge,E^\wedge),
\end{equation}
where $\mathcal{S}^\infty(X^\wedge,E^\wedge)$ are the smooth sections of $E^\wedge$ that
vanish rapidly for $t\to\infty$ and vanish to infinite order in $t=0$.
The main ingredients for the analysis of the closed extensions of $\widehat{A}$ are:
\begin{itemize}
 \item[i)] $\widehat{T}$ has a right-inverse that belongs to the cone calculus for boundary
  value problems for the infinite cone (cf.\ for example \cite{KaSc}).
 \item[ii)] For a fixed $0\not=\lambda_0\in\Lambda$,
  $\spalte{\lambda_0-\widehat{A}}{\widehat{T}}$ is an elliptic element
  in the cone calculus and one can construct a parametrix
  $\begin{pmatrix}\widehat{R}&\widehat{K}\end{pmatrix}$ inverting it modulo finite rank
  operators and such that $\widehat{T}\widehat{R}=0$.
\end{itemize}
For the analysis of closed extensions of $A$ on ${\mathbb D}$ both the corresponding right-inverse
as well as the parametrix were constructed in Lemma 3.4 and Propositions 3.3, 3.7 of
\cite{CSS2} relying on results of \cite{Grub} for boundary value problems on smooth
manifolds. Both constructions extend to $X^\wedge$.

\begin{theorem}\label{h6.2}
Under conditions {\rm(E1)} to {\rm(E3)} the following statements hold:
\begin{itemize}
 \item[a)] $\mathcal{S}^\infty(X^\wedge,E^\wedge)_{\widehat{T}}$ is a dense subspace of
  $\mathcal{K}^{s,\gamma}_p(X^\wedge,E^\wedge)_{\widehat{T}}$ for any $1<p<\infty$ and
  $s,\gamma\in{\mathbb R}$ with $s>\mu-1+\frac{1}{p}$.
 \item[b)] The domain of the closure of $\widehat{A}$ from \eqref{h6.C}, which we denote by
  $\widehat{A}_{\widehat{T},\min}$,  coincides with
  $\mathcal{K}^{\mu,\gamma+\mu}_p(X^\wedge,E^\wedge)_{\widehat{T}}$.
\end{itemize}
\end{theorem}
\begin{proof}
a) Using i) above, we obtain a projection
$P:\mathcal{K}^{s,\gamma}_p(X^\wedge,E^\wedge)\to\mathcal{K}^{s,\gamma}_p(X^\wedge,E^\wedge)_{\widehat{T}}$
within the cone calculus on the infinite cone $X^\wedge$
(cf.\ Section 2.2.3 in \cite{KaSc}).
Then one argues as in the proof of Corollary 3.10 in \cite{CSS2}.

b) The continuity of $\widehat{A}$ together with a) implies that
$\mathcal{K}^{\mu,\gamma+\mu}_p(X^\wedge,E^\wedge)_{\widehat{T}}\subset\mathcal{D}(\widehat{A}_{\widehat{T},\min})$.
The reverse inclusion follows as in the proof of Proposition 4.2 in \cite{CSS2} with
the special parametrix from ii).
\end{proof}

Our next -- and final -- requirement is that
\begin{itemize}
 \item[{\bf(E4)}] {\em $\widehat{A}_{\widehat{T},\min}$ does not have spectrum in
  $\Lambda\setminus\{0\}$}.
\end{itemize}
As $\widehat{A}$ and $\widehat{T}$ are invariant under dilations, the spectrum of
$\widehat{A}_{\widehat{T},\min}$ is automatically a conical subset of ${\mathbb C}$. The next
proposition shows that the spectrum of $\widehat{A}_{\widehat{T},\min}$ does not depend on
the choice of $1<p<\infty$:

\begin{proposition}\label{h6.3}
Assume that conditions {\rm(E1)} to {\rm(E3)} hold and fix a $\lambda_0\not=0$.
Let us denote, for the moment, by $A_p$ the minimal extension of $\lambda_0-\widehat{A}$
subject to $\widehat{T}$ in $\mathcal{K}^{0,\gamma}_p(X^\wedge,E^\wedge)$.
Suppose that for some $1<p_0<\infty$ the operator $A_{p_0}$ is invertible.
Then $A_p$ is invertible for all $1<p<\infty$.
\end{proposition}
\begin{proof}
By Theorem \ref{h6.2}, the domain of $A_p$ is
$\mathcal{K}^{\mu,\gamma+\mu}_p(X^\wedge,E^\wedge)_{\widehat{T}}$.
Then the invertibility of $A_p$ is equivalent to the invertibility of
 $$\widehat\mathcal{A}:=
   \begin{pmatrix}{\lambda_0-\widehat A}\\{\widehat T}\end{pmatrix}:
   \mathcal{K}^{\mu,\gamma+\mu}_p(X^\wedge,E^\wedge)\longrightarrow
   \begin{matrix}
     \mathcal{K}^{0,\gamma}_p(X^\wedge,E^\wedge)\\ \oplus\\
     \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
     \mathcal{B}_{pp}^{\mu-j-1/p,\gamma+\mu-j-1/2}(\partial X^\wedge,F_j^\wedge)
   \end{matrix},$$
see e.g.\ Corollary 7.2 in \cite{CSS2}. Since $\widehat\mathcal{A}$ is an
elliptic element in the cone calculus on $X^\wedge$,
we find a parametrix $\mathcal{B}$ to $\widehat\mathcal{A}$ such that
\begin{eqnarray}\label{linksrechts}
\widehat\mathcal{A}\mathcal{B}=I+\mathcal{R}_1\quad \text{and}\quad\mathcal{B}\widehat\mathcal{A}=I+\mathcal{R}_2,
\end{eqnarray}
where $\mathcal{R}_1$ and $\mathcal{R}_2$ are operators of order $-\infty$ and types $0$ and $\mu$,
respectively. They have the following mapping properties:
\begin{equation*}
\mathcal{R}_1:
\begin{matrix}
 \mathcal{K}^{s,\gamma}_p(X^\wedge,E^\wedge)\\ \oplus\\
 \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
 \mathcal{B}^{s+\mu-j-1/p,\gamma+\mu-j-1/2}_{pp}(\partial X^\wedge,F_j^\wedge)
\end{matrix}
\longrightarrow
\begin{matrix}
 \mathcal{S}^{\gamma}(X^\wedge,E^\wedge)\\ \oplus\\
 \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
 \mathcal{S}^{\gamma+\mu-j-1/2}(\partial X^\wedge,F_j^\wedge)
\end{matrix}
\end{equation*}
for any $1<p<\infty$ and $s>-1+\frac{1}{p}$, and
 $$\mathcal{R}_2:\mathcal{K}^{s,\gamma+\mu}_p(X^\wedge,E^\wedge)\longrightarrow
   \mathcal{S}^{\gamma+\mu}(X^\wedge,E^\wedge),
   \qquad 1<p<\infty,\;s>\mu-1+\mbox{$\frac{1}{p}$}.$$
In fact, in Section 2.1.6 of \cite {KaSc} these mapping properties are shown for the
case $p=2$; for the extension to arbitrary $p$ we use \cite{GrKo}. In case $p=p_0$,
multiplying \eqref{linksrechts} by $\widehat\mathcal{A}^{-1}$ yields that
\begin{equation}\label{inv}
 \widehat\mathcal{A}^{-1}=\mathcal{B}+\mathcal{B}\mathcal{R}_1-\mathcal{R}_2\widehat\mathcal{A}^{-1}\mathcal{R}_1.
\end{equation}
For each $1<p<\infty$ the right hand side extends to a bounded map
 $$\begin{matrix}
   \mathcal{K}^{0,\gamma}_p(X^\wedge,E^\wedge)\\ \oplus\\
   \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
   \mathcal{B}^{\mu-j-1/p,\gamma+\mu-j-1/2}_{pp}(\partial X^\wedge,F_j^\wedge)
   \end{matrix}
   \longrightarrow \mathcal{K}^{\mu,\gamma+\mu}_p(X^\wedge,E^\wedge)$$
and moreover restricts to a continuous map
 $$\begin{matrix}
   \mathcal{S}^{\gamma}_0(X^\wedge,E^\wedge)\\ \oplus\\
   \mathop{\mbox{\Large$\oplus$}}\limits_{j=0}^{\mu-1}
   \mathcal{S}^{\gamma+\mu-j-1/2}_0(\partial X^\wedge,F_j^\wedge)
   \end{matrix}
   \longrightarrow\mathcal{S}^{\gamma+\mu}_0(X^\wedge,E^\wedge).$$
By density, the right hand side of \eqref{inv} therefore furnishes an inverse to
$\widehat\mathcal{A}$ for arbitrary $p$.
\end{proof}

\begin{theorem}\label{thm:himin}
Let $\spalte{A}{T}$ satisfy the conditions \text{\rm (E1)} to
\text{\rm (E4)}. Then
 $$A_{T,\min}:\mathcal{H}^{\mu,\gamma+\mu}_p({\mathbb D},E)_T\subset
   \mathcal{H}^{0,\gamma}_p({\mathbb D},E)\longrightarrow\mathcal{H}^{0,\gamma}_p({\mathbb D},E)$$
fulfills the assumptions of Theorem {\rm\ref{hinfty}}. Hence there exists a
$c\ge0$ such that $c+A_{T,\min}$ has a bounded $H_\infty$-calculus with respect to
${\mathbb C}\setminus\Lambda$.
\end{theorem}
\begin{proof}
Let us choose parameter-dependent order reductions in the cone algebra on ${\mathbb B}$,
 $$R_j(\eta)\in C^{\mu-j}({\mathbb B},\Sigma;\gamma+\mu-j,\gamma,\theta;F_j,F_j),
   \qquad j=0,\ldots,\mu-1,$$
where the $F_j$ are the bundles from \eqref{h4.A.6} and $\theta\in{\mathbb N}$ is arbitrary.
Let $R(\eta)=\text{\rm diag}(R_0(\eta),\ldots,R_{\mu-1}(\eta))$.
Now the conditions (E1) to (E4) are chosen in such a way that
 $$\mathcal{A}(\eta):=\begin{pmatrix}1&0\\0&R(\eta)\end{pmatrix}
   \begin{pmatrix}\eta^\mu-A\\T\end{pmatrix}=
   \begin{pmatrix}\eta^\mu-A\\R(\eta)T\end{pmatrix}\in
   C^{\mu,\mu}({\mathbb D},\Sigma;\gamma+\mu,\gamma,\theta;E;E,F)$$
with $F:=F_0\oplus\ldots\oplus F_{\mu-1}$ is a parameter-elliptic element the
cone calculus for boundary value problems. It follows that there exists a parametrix
$\mathcal{B}(\eta)=\begin{pmatrix}B(\eta)&K(\eta)\end{pmatrix}$ and that, for $p=2$,
\begin{equation}\label{h6.A}
 \mathcal{A}(\eta):\mathcal{H}^{\mu,\gamma+\mu}_p({\mathbb D},E)\longrightarrow
   \begin{matrix}
    \mathcal{H}^{0,\gamma}_p({\mathbb D},E)\\ \oplus\\
    \mathcal{B}^{-\frac{1}{2},\gamma-\frac{1}{2}}_{pp}({\mathbb B},F)
   \end{matrix}
\end{equation}
is bijective for sufficiently large $|\eta|$, and the inverse coincides with the
parametrix. Also it follows that $B(\eta)\in{C}^{-\mu,0}({\mathbb D},\Sigma;\gamma;E)$ in
the sense of Section \ref{sectionh4.4.3}. As we shall show below even
$B(\eta)\in C^{-\mu,0}_{\rm hol}({\mathbb D},\Sigma;\gamma;E)$.
Next, we note that invertibility of \eqref{h6.A} is equivalent to the invertibility of
\begin{equation}\label{h6.B}
 \eta^\mu-A:\mathcal{H}^{\mu,\gamma+\mu}_p({\mathbb D},E)_T\longrightarrow
 \mathcal{H}^{0,\gamma}_p({\mathbb D},E);
\end{equation}
the inverse of \eqref{h6.B} is just $B(\eta)$. By Theorem \ref{h4.2.4} we can conclude
that the invertibility of \eqref{h6.B} then also is true for arbitrary $1<p<\infty$,
and the inverse again coincides with $B(\eta)$.

Now let $B(\eta)$ be as in \eqref{h4.D}. Decomposing $\widetilde{h}$ as in \eqref{h4.D.5} and
\eqref{h4.E} yields local symbols
$\widetilde{a}\in S^{-\mu}_{{\rm cl}}(\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}\times\Sigma)$.
By parametrix construction, the leading term is given by inversion of the
parameter-dependent principal symbol, i.e.\ $\widetilde a=\widetilde a_0+\widetilde a_1$ with
$\widetilde{a}_1(t,x,\tau,\xi,\eta)\in
S^{-\mu-1}_{\rm cl}(\overline{{\mathbb R}}_+\times{\mathbb R}^n_x\times{\mathbb R}^{1+n}_{(\tau,\xi)}\times\Sigma)$ and
 $$\widetilde{a}_0(t,x,\tau,\xi,\eta)=\chi(|(\tau,\xi,\eta)|)
   (\eta^{\mu}-\widetilde\sigma_\psi^\mu(A)(t,x,\tau,\xi))^{-1},$$
where $\chi$ is a 0-excision function. By ellipticity assumption (E1),
$(\eta^{\mu}-\widetilde\sigma_\psi^\mu(A)(t,x,\tau,\xi))^{-1}$ is defined for
$0\not=(\tau,\xi,\eta)\in{\mathbb R}^{1+n}\times\Sigma$. By homogeneity, it is clear that
 $$\text{\rm spec}(\widetilde\sigma_\psi^\mu(A)(t,x,\tau,\xi))\subset \{\lambda\in{\mathbb C}\mbox{\boldmath$\;|\;$\unboldmath}
   \mbox{$\frac{1}{c_0}$}|(\tau,\xi)|^\mu<|\lambda|<c_0|(\tau,\xi)|^\mu\}$$
for a suitable constant $c_0>1$. Thus
$(\eta^{\mu}-\widetilde\sigma_\psi^\mu(A)(t,x,\tau,\xi))^{-1}$ is defined on
 $$\Omega:=\{(\tau,\xi,\eta)\mbox{\boldmath$\;|\;$\unboldmath} (\tau,\xi,\eta)\in({\mathbb R}^{1+n}\times\Sigma)\setminus\{0\}
   \text{ or }|\eta|\ge c_0|\tau,\xi|\}.$$
As $(\eta^{\mu}-\widetilde\sigma_\psi^\mu(A)(t,x,\tau,\xi))^{-1}$ is positively homogeneous
of degree $-\mu$ in $(\eta,\tau,\xi)\in\Omega$, and $\Omega\cap\partial U_1(0)$ is
compact ($U_1(0)$ denoting the unit ball), the estimates \eqref{h4.H} with $\nu=-\mu$
follow.
\end{proof}

\section{Example: The Dirichlet and Neumann problems for the Laplacian}\label{sectionh7}
We equip ${\mathbb D}$ with a straight conical metric, i.e. a metric that coincides
 with $dt^2+t^2g$  on  ${]0,1[}\times X$ for a fixed  metric $g$ on $X$.
The associated scalar Laplacian $-\Delta$ is a Fuchs-type operator.
Near $t=0$ it can be written in the form
\begin{equation}\label{eq:delta}
- \Delta=-t^{-2}\,\{(t\partial_t)^2+(n-1)t\partial_t+\Delta_X\},
    \qquad n=\text{\rm dim}\,X,
\end{equation}
where $\Delta_X$ denotes the Laplacian on $X$ with respect to $g$. We let
 $$
 \mathcal{A}_D := \begin{pmatrix}-\Delta \\ \gamma_0\end{pmatrix}
 \mbox{ and }
 \mathcal{A}_N := \begin{pmatrix}-\Delta \\ t^{-1} \gamma_1\end{pmatrix}
  $$
be the Dirichlet and Neumann boundary value problems for $-\Delta$, respectively.
We denote by $-\Delta_{D}$ and $-\Delta_N$ the unbounded operators in
$\mathcal{H}^{0,\gamma}_p({\mathbb D})$, acting as $-\Delta$ on the domains
$\mathcal{C}^{\infty,\infty}({\mathbb D})_{\gamma_0}$ and $\mathcal{C}^{\infty,\infty}({\mathbb D})_{\gamma_1}$, respectively.
In the sequel we will  write $-\Delta_{D/N}$ to address both operators.
It is easy to see that both satisfy the ellipticity conditions (E1) and (E2).

Given a function space $\mathcal{F}$ we will use the notation $\mathcal{F}_D$ and $\mathcal{F}_N$
in place of $\mathcal{F}_{\gamma_0}$ and $\mathcal{F}_{\gamma_1}$ to denote the closed subspace
of $\mathcal{F}$ where $\gamma_0$ and $\gamma_1$, respectively, vanish.

\forget{
It was shown in \cite{CSS2} that their adjoint problems are
$$
\mathcal{A}_D^* = \begin{pmatrix}-\Delta \\ -i\gamma_0\end{pmatrix}
\mbox{ and }
\mathcal{A}_N^* = \begin{pmatrix}-\Delta \\ -i t^{-1} \gamma_1\end{pmatrix}.
$$
We denote by $-\Delta_{D}$ and $-\Delta_N$ the unbounded operators in
$\mathcal{H}^{0,\gamma}_p({\mathbb D})$, acting as $-\Delta$ on the domains
$\mathcal{C}^{\infty,\infty}({\mathbb D})_{\gamma_0}$ and $\mathcal{C}^{\infty,\infty}({\mathbb D})_{\gamma_1}$, respectively.
In the sequel we will  write $-\Delta_{D/N}$ to address both operators.
Given a function space $\mathcal{F}$ we will also use the notation
$\mathcal{F}_D$ and $\mathcal{F}_N$ in place of $\mathcal{F}_{\gamma_0}$ and $\mathcal{F}_{\gamma_1}$
to denote the closed subspace of  $\mathcal{F}$ where $\gamma_0$
and $\gamma_1$, respectively, vanish.
}

\subsection{Closed extensions of the Dirichlet and Neumann Laplacians on ${\mathbb D}$}

According to the definition in Section \ref{sectionh6}, the
principal conormal symbols of $\mathcal{A}_D$ and $\mathcal{A}_N$ are
  $$\sigma^2_M(\mathcal{A}_D)(z)=
  \begin{pmatrix}-z^2+(n-1)z-\Delta_X
  \\
  \gamma_0
  \end{pmatrix}
  \mbox{ and }
 \sigma^2_M(\mathcal{A}_N)(z)=
  \begin{pmatrix}-z^2+(n-1)z-\Delta_X
  \\
  \gamma_1
  \end{pmatrix}.
  $$
They are invertible, unless $-z^2 + (n-1)z =\lambda_j$ for one of the eigenvalues
$\lambda_0>\lambda_1>\ldots$ of the boundary problems $\Delta_{X, \gamma_0}$ and
$\Delta_{X, \gamma_1}$, respectively (recall that $\lambda_0 < 0$ for the  Dirichlet
problem, while $\lambda_0=0$ for the Neumann problem). This is the case for
 $z=q_j^+$ or $z=q_j^-$ with
  \begin{equation*
   q_j^\pm=\mbox{$\frac{n-1}{2}\pm
   \sqrt{\big(\frac{n-1}{2}\big)^2-\lambda_j}$},
   \qquad j\in{\mathbb N}_0.
  \end{equation*}
We shall now study the minimal extension of $\Delta_{D/N}$ in
$\mathcal{H}^{0,\gamma}_p({\mathbb D})$. We shall require that
\begin{equation}\label{c1}
1-\sqrt{\left(\mbox{$\frac{n-1}{2}$}\right)^2-\lambda_0} < \gamma <
-1+\sqrt{\left(\mbox{$\frac{n-1}{2}$}\right)^2-\lambda_0}.
\end{equation}
Of course, this only makes sense, if
\begin{equation}\label{c2}
\left(\mbox{$\frac{n-1}{2}$}\right)^2-\lambda_0>1.
\end{equation}

\begin{theorem}\label{thm:dom}
Assume \eqref{c1} and \eqref{c2}.
For $1<p<\infty$, the minimal and maximal extensions coincide both
for the Dirichlet and the Neumann Laplacians
on $\mathcal{H}^{0,\gamma}_p({\mathbb D})$, and their domain is $\mathcal{H}^{2,\gamma+2}_p({\mathbb D})_{D/N}$.
In case $\gamma=0$ and $p=2$, the $($minimal$)$ extension is self-adjoint.
\end{theorem}
\begin{proof}
This is an immediate consequence of Proposition 6.1 of \cite{CSS2}, since condition
\eqref{c1} implies that the conormal symbols of both $\mathcal{A}_{D}$ and $\mathcal{A}_{N}$ are
invertible for all $z$ with $\frac{n+1}{2}-\gamma-2 \le \Re z \le \frac{n+1}{2}-\gamma$.
The self-adjointness is a consequence of Theorem 4.5 in \cite{CSS2}, since the adjoint
problems of $\mathcal{A}_{D}$ and $\mathcal{A}_{N}$ are $\spalte{-\Delta}{-i\gamma_0}$ and
$\spalte{-\Delta}{-i t^{-1}\gamma_1}$, respectively.
\end{proof}

\begin{remark}\label{rem:E4}
\begin{itemize}
 \item[(a)] Assumption \eqref{c1} implies that
  both $-\Delta_D$ and $-\Delta_N$, considered as unbounded operators on
  $\mathcal{H}^{0,\gamma}_p({\mathbb D})$, satisfy the ellipticity condition {\rm(E3)}.
 \item[(b)] In the Dirichlet case,  condition \eqref{c2}
  is always true for $n\ge 3$; depending on $\lambda_0$ $($i.e.\ on $X$ and $g)$,
  it might also hold for $n=1$ or $n=2$.
  In the Neumann case, condition \eqref{c2} only holds for   $n > 3$.
\end{itemize}
\end{remark}

\subsection{Domains of the model cone operator}
Recall that $\widehat{\Delta}_{D/N,\min}$ denotes the closure in
$\mathcal{K}^{0,\gamma}_p(X^\wedge)$ of the model cone operator $\widehat{\Delta}_{D/N}$
considered with domain $\mathcal{S}^\infty(X^\wedge)_{D/N}$, while
 $$\mathcal{D}(-\widehat{\Delta}_{D/N,\max})=\{u\in\mathcal{K}^{2,\gamma}_p(X^\wedge)_{D/N} \mbox{\boldmath$\;|\;$\unboldmath}
   -\widehat{\Delta}u \in \mathcal{K}^{0,\gamma}_p(X^\wedge) \}$$
defines the closed operator $\widehat{\Delta}_{D/N,\min}$. Note that always
$\mathcal{K}^{2,\gamma+2}_p(X^\wedge)_{D/N}\subset\mathcal{D}(-\widehat{\Delta}_{D/N,\min})$.
\forget{
In fact, $\mathcal{K}^{2,\gamma+2}_p(X^\wedge)_{D/N}$ is invariant under
multiplication by cut-off functions, since the boundary condition does
not contain $t$-derivatives. Hence, the statement follows from
\cite{CSS2}, Proposition 4.2, for functions supported close to the singularity.
For functions supported away from $\{t=0\}$ we recall that
$\mathcal{K}^{2,\gamma+2}_p(X^\wedge)$ coincides with the cone Sobolev
space $H^2_{p,\textrm{cone}}(X^\wedge)$,
and we deduce the result as in the standard case, cf.\ Section 1.6 in \cite{Grub}.
}

\begin{lemma}\label{lemma:model}
Under the assumptions of Theorem {\rm \ref{thm:dom}}, the minimal and
maximal extensions coincide both for the model Dirichlet and Neumann
Laplacians on $\mathcal{K}^{0,\gamma}_p(X^\wedge)$ and
$$
\mathcal{D}(-\widehat{\Delta}_{D/N,\min})
=\mathcal{D}(-\widehat{\Delta}_{D/N,\max})=
\mathcal{K}^{2,\gamma+2}_p(X^\wedge)_{D/N}.
$$
\end{lemma}

\begin{proof}
For $u\in \mathcal{D}(-\widehat{\Delta}_{D,\max})$ we have
$u\in\mathcal{K}^{2,\gamma}_p(X^\wedge)_D$ and
$-\widehat{\Delta}u\in\mathcal{K}^{0,\gamma}_p(X^\wedge)$. Then
$(1-\omega)u \in \mathcal{K}^{2,\infty}_p(X^\wedge)_D$, and
$-\widehat{\Delta} (1-\omega)u \in \mathcal{K}^{0,\infty}_p(X^\wedge)$.
Hence, $(1-\omega)u \in \mathcal{D}(-\widehat{\Delta}_{D,\max})$, thus
$\omega u \in \mathcal{D}(-\widehat{\Delta}_{D,\max})$. As we may consider
$\omega u$ as an element of $\mathcal{H}^{2,\gamma}_p({\mathbb D})$, and since $\Delta$ and
$\widehat{\Delta}$ have the same form close to $t=0$, we also have
$\omega u\in\mathcal{D}(-\Delta_{D,\max})$. By Theorem \ref{thm:dom},
$\omega u\in \mathcal{D}(-\Delta_{D,\min})=\mathcal{H}^{2,\gamma+2}_p({\mathbb D})_D$. Therefore,
$u=\omega u + (1-\omega)u \in \mathcal{K}^{2,\gamma+2}_p(X^\wedge)_D\subset
\mathcal{D}(-\widehat{\Delta}_{D,\min})$.
We conclude that $\mathcal{D}(-\widehat{\Delta}_{D,\min})=
\mathcal{D}(-\widehat{\Delta}_{D,\max})=\mathcal{K}^{2,\gamma+2}_p(X^\wedge)_D$, and
the proof of the statement for $-\widehat{\Delta}_D$ is complete. The argument for
$-\widehat{\Delta}_N$ is the same.
\end{proof}

\begin{corollary}\label{selfadjoint}
For $\gamma=0$ and $p=2$ the closure $\Delta_{D/N,\min}$ is self-adjoint.
\end{corollary}
\begin{proof}
Let $\widehat{\Delta}_{F}$ denote the Friedrichs extension of $-\widehat{\Delta}_{D/N}$.
By construction, one has
 $$\mathcal{D}(\widehat{\Delta}_{D/N,\min})\subset
   \mathcal{D}(\widehat{\Delta}_{F})\subset\mathcal{D}((\widehat{\Delta}_{D/N,\min})^*).$$
Let $u\in\mathcal{D}((\widehat{\Delta}_{D/N,\min})^*)$ be given.
Using the special parametrix $\widehat{R}$ from ii) in Section \ref{sectionh6}
(for $\widehat{A}=-\widehat{\Delta}$, $T=D/N$, and $\lambda_0=-1$), and the fact that
$R^*(\widehat{\Delta}_{D/N,\min})^*\subset(\widehat{\Delta}_{D/N}R)^*$,
we deduce that $u\in\mathcal{K}^{2,0}_2(X^\wedge)$. Then, arguing as in the proof of
Theorem 4.5 in \cite{CSS2} to verify there the identity (4.5), we conclude that
$u\in\mathcal{D}(\widehat{\Delta}_{D/N,\max})$. It follows from Lemma \ref{lemma:model} that
$\widehat{\Delta}_{F}=\widehat{\Delta}_{D/N,\min}$.
\end{proof}

\begin{theorem}\label{thm:hinfty}
Assume \eqref{c1} and \eqref{c2}.
Then, for $1<p<\infty$, both the Dirichlet and Neumann Laplacian fulfill the
ellipticity conditions {\rm(E1)} to {\rm(E4)} of Section {\rm\ref{sectionh6}}.
\end{theorem}
\begin{proof}
It was noted above that (E1), (E2), and (E3) hold for
$-\Delta_{D/N}$. It remains to check (E4).
By spectral invariance, cf.\ Proposition \ref{h6.3},  we
may assume $p=2$.

For $\gamma=0$, the Laplacians $-\widehat{\Delta}_{D/N,\min}$ are self-adjoint
with domain $\mathcal{K}^{2,2}_2(X^\wedge)_{D/N}$.
\forget{
and positive. Hence, the associated Friedrichs extensions are
selfadjoint. As the Friedrichs extension is between the minimal and
the maximal, and these coincide by Lemma \ref{lemma:model}, its
domain is $\mathcal{K}^{2,2}_2(X^\wedge)_{D/N}$.
}
For $\lambda\notin\overline{{\mathbb R}}_+$, we therefore conclude that
\begin{equation}\label{eq:selfadj}
-\widehat{\Delta}_{D/N}-\lambda \;\colon\;
\mathcal{K}^{2,2}_2(X^\wedge)_{D/N} \longrightarrow
\mathcal{K}^{0,0}_2(X^\wedge)
\end{equation}
is invertible. Clearly, \eqref{eq:selfadj} shows the injectivity of
$-\widehat{\Delta}_{D/N}-\lambda$ on
$\mathcal{K}^{2,\gamma+2}_2(X^\wedge)_{D/N}\subset\mathcal{K}^{2,2}_2(X^\wedge)_{D/N}$
for $\gamma>0$.

Next suppose $\gamma<0$, $u\in\mathcal{K}^{2,\gamma+2}_2(X^\wedge)_{D/N}$ and
$u\in\ker \{ -{\widehat\Delta}_{D/N} -\lambda \colon
\mathcal{K}^{2,\gamma+2}_2(X^\wedge)_{D/N}\rightarrow \mathcal{K}^{0,\gamma}_2(X^\wedge)\}$.
Then $u$ is in the maximal domain of $-{\widehat\Delta}_{D/N} -\lambda$, considered
as an unbounded operator on $\mathcal{K}^{0,\gamma+2}_2(X^\wedge)$.
Since condition (E4) holds, $u$ also is in the associated minimal domain, which is
$\mathcal{K}^{2,4+\gamma}_2(X^\wedge)_{D/N}$. We can iterate this argument $j$ times,
until $\gamma+2j > 0$. Then, we conclude that $u=0$, by the previous step.
Thus $-{\widehat\Delta}_{D/N} -\lambda$ is injective on
$\mathcal{K}^{2,\gamma+2}_2(X^\wedge)_{D/N}$ for all $\gamma$ satisfying the hypotheses
and $\lambda\notin\overline{{\mathbb R}}_+$. Finally, we note that the adjoint of
$$
-{\widehat\Delta}_{D/N} -\lambda \; \colon\;
\mathcal{K}^{2,\gamma+2}_2(X^\wedge)_{D/N} \longrightarrow
\mathcal{K}^{0,\gamma}_2(X^\wedge)
$$
with respect to the scalar product of $\mathcal{K}^{0,0}_2(X^\wedge)$ is
$$
-{\widehat\Delta}_{D/N} -\overline{\lambda} \; \colon\;
\mathcal{K}^{2,-\gamma+2}_2(X^\wedge)_{D/N} \longrightarrow
\mathcal{K}^{0,-\gamma}_2(X^\wedge).
$$
Hence, also the adjoint is injective for $\lambda\notin\overline{{\mathbb R}}_+$, so
$-{\widehat\Delta}_{D/N} -\lambda$
is bijective, as claimed.
\end{proof}

\subsection{Maximal $L_p$ regularity of the Cauchy problem for
Dirichlet and Neumann Laplacians}

As a consequence of Theorem \ref{thm:hinfty} we get the following
result on the
solvability of the Cauchy Problem for the Dirichlet and Neumann
Laplacians.

\begin{theorem}\label{cauchy}
Let $\Delta$ be the Laplacian as described above, $1<p<\infty$, and assume \eqref{c1}
and \eqref{c2}. Then the initial boundary value problems
\begin{equation}\label{cauchy1}
   u'(\tau)-\Delta u(\tau)=f(\tau),\quad 0\le \tau\le T;\qquad u(0)=0,\quad
   \gamma_j u = 0,
\end{equation}
$j=0,1$, have  a unique solution
  $$u\in W^1_r\left([0,T],\mathcal{H}^{0,\gamma}_p({\mathbb D})\right)\,\cap\,
    L_r\left([0,T],\mathcal{H}^{2,\gamma+2}_q({\mathbb D})_{D/N}\right)$$
for each
  $$f\in L_r\left([0,T],\mathcal{H}^{0,\gamma}_p({\mathbb D})_{D/N}\right),\qquad
    1<r<\infty.$$
Furthermore, $u$, $u'$, and $\Delta u$ depend continuously on $f$.
\end{theorem}

\begin{proof}
Solving \eqref{cauchy1} is equivalent to solving
$v'(\tau)-(\Delta_{D/N}-c)v(\tau)=e^{c\tau}f(\tau)$, $v(0)=0$, $\gamma_j v=0$,
for some $c>0$.

The operators $-\Delta_{D/N}$ are closed with minimal (and maximal) domain equal to
$\mathcal{H}^{2,\gamma+2}_p({\mathbb D})_{D/N}$. According to Theorem \ref{thm:hinfty}, they
satisfy conditions (E1)-(E4) for each sector $\Lambda$ not containing ${\mathbb R}_+$.
Applying Theorem \ref{thm:himin}, we deduce that $-\Delta_{D/N} + c$ has a
bounded $H_\infty$-calculus for sufficiently large $c$, and  Theorem \ref{cauchy}
immediately follows from Dore and Venni's theorem, cf.\ Theorem 3.2 in \cite{DoVe}.
\end{proof}

Nazarov \cite{Naza} has studied the Dirichlet and the Neumann problem for the Laplacian
on infinite cones and wedges in Euclidean space. He shows results on maximal regularity
in weighted Sobolev spaces using the explicit Green's function. He obtains restrictions
on the weight which are similar to \eqref{c1}. They do not coincide, however, since he
works on a different scale of spaces.

\begin{small}
\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


The property committee gives oversight to the ongoing maintenance, both short term and long term, of the church property. Their task is to develop plans and make recommendations to the church council on property issues and help leadership in raising and assessing the potential funding for property and facility projects. This committee works very closely with the 20/20 Club (a fund-raising arm for the congregation) concerning larger projects that arise and the funding of those projects.
I want to thank all of the members of the property committee for their service and for their commitment to keeping this wonderful church home in the best condition possible. We also need to thank the FLC Foundation and the 20/20 Club for their assistance in funding a number of these projects. We truly appreciate their partnership and commitment.
The mission of the property committee is to continually look towards the future needs of the congregation and to keep the facility in the best possible condition so as to preserve it for many generations to come. If you are interested in joining, please contact Terry Yoney at 701-235-7389 or tyoney@flcfargo.org.


text:
Photo and Video Chronology – Kīlauea – January 7, 2021
The eruption in Halema'uma'u at Kīlauea's summit on the Island of Hawai'i continues, with the west vent erupting lava into the lava lake. Gas emissions and seismic activity at the summit remain elevated. HVO field crews—equipped with specialized safety gear and PPE—monitor the current eruption from within the closed area of Hawai'i Volcanoes National Park with NPS permission.
HVO scientists collect detailed data to assess hazards and understand how the eruption is evolving at Kīlauea's summit, all of which are shared with the National Park Service and emergency managers. Access to this hazardous area is by permission from, and in coordination with, Hawai'i Volcanoes National Park.
The eruption from the west vent in Halema'uma'u crater continues at Kīlauea's summit. Spatter being ejected from the vent has built another spatter cone, while lava entering into the lava lake is still forming a dome fountain. The lava lake level continues to slowly rise, and as of January 6 at 7:35 p.m. HST the lake was measured at 194 m (636 ft) deep. HVO scientists continue to monitor the eruption within an area of Hawai'i Volcanoes National Park that remains closed to the public for safety reasons. View looking to the northeast. USGS photo taken by L. DeSmither.
Telephoto video of lava flow and fountaining in Halema'uma'u lava lake; spatter bursts from the active west fissure vent in Halema'uma'u.
Clip 1: The lava lake in Halema'uma'u Crater is being fed by two sources from the western fissure today, the tubed over lava stream entering the lava lake at the margin (initiated yesterday) and the small dome fountain (maximum height 5 meters or 16 ft).
Clip 2: The southwest vent of the western fissure is producing periodic spatter bursts that have built the cone up to nearly the height of the previously active cone to the right. In this video, spatter that has built up at the top of the cone collapses, widening the opening.
Hannah Dietterich, Alaska Volcano Observatory
The activity at the Halema'uma'u west vent area on January 5, 2021, was dynamic. In the foreground, a "dome fountain" upwells from beneath the lake surface, pulsing with a bubble-like geometry. Just behind it, an effusive lava flow emerges from the base of the vent structure. A mid-elevation vent is also active, emitting another effusive lava flow. Finally, the higher elevation vent(s) exhibit intermittent weak strombolian activity, where incandescent spatter is thrown out onto the vent structure. All four of these eruption sources were active in the ongoing Halema'uma'u eruption throughout the afternoon and evening hours.
Kendra Lynn, USGS Hawaiian Volcano Observatory
Aerial image of the active west vent area and the largest island in Halemaʻumaʻu's growing lava lake. During the overflight of Kīlauea's summit today (January 7, 2021) at approximately 10:30 a.m. HST, HVO geologists noted that the dome fountain, which had been persistent near the base of the west vent area, had subsided and was no longer present. USGS photo by K. Mulliken.
This photo, taken during an overflight of Kīlauea's summit this morning (January 7, 2021), shows the southeast margin of Halemaʻumaʻu's growing lava lake. The photo shows the boundary between the active portion of the lava lake (left) and the subtle levee around the lake perimeter that allows the lake to be slightly perched. The photo shows that a margin around the lake (right side of photo) is lower than the active portion. USGS photo by K. Mulliken.
During an overflight of Kīlauea summit today (January 7, 2021) at approximately 10:30 a.m. HST, HVO geologists captured this image of the growing lava lake within Halemaʻumaʻu. Sunny weather allowed for clear views of Mauna Loa, to the west. USGS photo by M. Patrick.
This thermal image, looking northwest, shows the distribution of activity on the lava lake at Kīlauea's summit. The lake is slightly perched and surrounded along its perimeter by a lower ledge of cooler lava. The western fissure, in the wall of Halema'uma'u crater, is visible in the upper left portion of the image, just above the lake. Today, the eastern end of the lake (right side of image) was undergoing crustal foundering, in which crustal plates sink and fluid lava resurfaces the lake. USGS image by M. Patrick.
Hawaiian Volcano Observatory
Email: askHVO@usgs.gov

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to use Objective-C sources' ExplicitInit class? In the Objc source code, I found the following code. What is the meaning of this code and how to understand it?
objc/Project Headers/DenseMapExtras.h line:38
template <typename Type>
class ExplicitInit {
    alignas(Type) uint8_t _storage[sizeof(Type)];

public:
    template <typename... Ts>
    void init(Ts &&... Args) {
        new (_storage) Type(std::forward<Ts>(Args)...);
    }

    Type &get() {
        return *reinterpret_cast<Type *>(_storage);
    }
};

Below is my test code:
class MyC{
public:
    long l1;
    long l2;
    MyC(long _l1, long _l2){
        l1 = _l1;
        l2 = _l2;
    }
};

int main(){
    MyExplicitInit<MyC> e1 {};
    e1.init();
    return 0;
}

The compiler prompts the following error:

A:
In the Objc source code, I found the following code. What is the
meaning of this code and how to understand it?

To me it looks like a kind-of-factory which can be used as an alternative to the Construct On First Use Idiom. An instantiated class here represents a storage for an instance you can initialise and request when needed. As far as I understand it's not supposed to be used for local variables (it doesn't make much sense, despite being technically possible) and this is also suggested by the comments of code section with the said class template:

// We cannot use a C++ static initializer to initialize certain globals because
// libc calls us before our C++ initializers run. We also don't want a global
// pointer to some globals because of the extra indirection.
//
// ExplicitInit / LazyInit wrap doing it the hard way

For the error you are experiencing:

No matching operator new function for non-allocating placement new expression;
include <new>

Assuming that you just added that piece of code somewhere in your own sources, the problem here is that you didn't include the <new> header. As simple as that - the error just says that you need to add #include <new> since so-called placement new is not part of the "default" C++, it's an overloaded operator declared in this header.
Second, your init function expects arguments that matches one of the existing (non-aggregate) constructors of the given class, so you are expected to pass arguments which are either match the constructor parameters or can be implicitly converted to them: e1.init(1l, 2l)
A complete example looks something like this:
#include <_types/_uint8_t.h>
#include <new>

namespace objc {

    template <typename Type>
    class ExplicitInit {
        alignas(Type) uint8_t _storage[sizeof(Type)];

    public:
        template <typename... Ts>
        void init(Ts &&... Args) {
            new (_storage) Type(std::forward<Ts>(Args)...);
        }

        Type &get() {
            return *reinterpret_cast<Type *>(_storage);
        }
    };

};

struct sample_struct {
    long l1, l2;

    sample_struct(long _l1, long _l2): l1{_l1}, l2{_l2} {}
};

sample_struct& getInstance(bool should_init = false) {
    static objc::ExplicitInit<sample_struct> factory;
    if (should_init) {
        factory.init(1l, 2l);
    }
    return factory.get();
}

redpajama_set_name: RedPajamaStackExchange


Recently someone gave me a 10-year-old (or so) laptop that had been written off as dead. It was sold with Windows Vista, and it would no longer boot. I installed Ubuntu Linux on it and found that it works great. Bottom line: Free laptop.
I'm a Mac loyalist and a conscientious objector to anything from Microsoft (though I believe that Microsoft products have gotten much better now that they've lost their monopoly and competition has forced them to improve their software). I've been a Unix user since about 1985, and I first used Linux in the early 1990s. Linux has come a long way.
A laptop is not something that I particularly need. But, on those relatively rare occasions when I travel, a laptop is nice to have. Laptops of this vintage can be bought on eBay for as little as $40 if you catch a bargain. In choosing an older laptop to run Linux, you want one new enough to have a dual-core 64-bit processor and 4 GB of memory. An older laptop may be heavy, but they're cheap. Older batteries can be a problem, but the battery in my newly acquired laptop will run for about an hour. Most of the time, though, even when traveling, you can find a place to plug the laptop into the wall. You'll want a laptop with built-in WIFI.
Learning to use Linux may be a tad more difficult than learning to use a Mac or a Windows machine. But Linux has gotten much easier to use, with a pretty graphical interface. Probably the biggest challenge that most people would face in bringing up an older laptop on Linux is installing Linux. That's not something that I want to get into here in detail, because you'll find many tutorials if you Google for it. But the simplest route is to download a Linux installer on another computer and then copy the installer to a USB thumb drive that is configured to be bootable. You boot the laptop off the USB thumb drive and run the Linux installer. Once you've installed Linux, the sailing is much easier.
I am using Ubuntu Linux 18.04, which is the newest version of Ubuntu Linux at present. Ubuntu Linux comes with LibreOffice already installed. LibreOffice is an open-source suite of office software that is, as far as I know, pretty much 100 percent compatible with Microsoft Office. It's as easy to use as Microsoft Office. It will open all your existing Microsoft Office files. And if you use LibreOffice for word processing, you can send your files to users of Microsoft Office and they'll be able to open the files just fine.
Ubuntu Linux also comes with the FireFox web browser installed, and Thunderbird for email. If you need software that is not pre-installed, there is a long list of open-source applications that Ubuntu will download and install for you.
This was written by daltoni. Posted on Thursday, January 17, 2019, at 12:34 pm. Filed under Technology. Bookmark the permalink. Follow comments here with the RSS feed.	Trackbacks are closed, but you can post a comment.


I did it! I packed my first debian package and it's holding GSDesktop Helper for Linux. So for an easy installation, you can now try these commands.
And you can launch "helper" from menu (inside "Accessories") and have even more fun with it.