# Degenerate operators on the half-line

G. Metafune L. Negro C. Spina Dipartimento di Matematica e Fisica “Ennio De
Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy. -mail:
<EMAIL_ADDRESS>di Matematica e Fisica “Ennio De
Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy. email:
<EMAIL_ADDRESS>di Matematica e Fisica“Ennio De Giorgi”,
Università del Salento, C.P.193, 73100, Lecce, Italy. e-mail:
<EMAIL_ADDRESS>
###### Abstract

We study elliptic and parabolic problems governed by the singular elliptic
operators

$y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)-V(y),\qquad\alpha\in\mathbb{R}$

in $\mathbb{R}_{+}$, where $V$ is a potential having non-negative real part.

Mathematics subject classification (2020): 35K67, 35B45, 47D07, 35J70, 35J75.

Keywords: degenerate elliptic operators, boundary degeneracy, vector-valued
harmonic analysis, maximal regularity.

##  1 Introduction

In this paper we study solvability and regularity of elliptic and parabolic
problems associated to the degenerate operators

$L=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)-V\quad{\rm and}\quad
D_{t}-L$

in the half-line $\mathbb{R}_{+}$.

Here $c,\alpha$ are real numbers and $V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)$ is a potential having
non-negative real part. The operator $B=D_{yy}+\frac{c}{y}D_{y}$ is a Bessel
operator and satisfies the scaling property

$I_{s}^{-1}BI_{s}=s^{2-{\alpha}}B,\quad I_{s}u(y)=u(sy).$

We study $L$ in the weighted spaces
$L^{p}_{m}:=L^{p}\left(\mathbb{R}^{+},y^{m}dy\right)$, $m\in\mathbb{R}$, and
we characterize all $m$ such that $L$ generates a $C_{0}$-semigroup. When
$V\geq 0$ we also prove that the generated semigroup is analytic and we show
that it has maximal regularity, which means that both $D_{t}v$ and $Lv$ have
the same regularity as $(D_{t}-L)v$. In the case $V(y)=y^{\alpha}$ we finally
characterize the domain of $L$.

We observe that the results already available for $B$, see [11, Section 3] and
also [10, 9, 8, 14, 7] for the $N$-d version of $B$, imply the corresponding
ones for $y^{\alpha}B$ in $L^{p}_{m}$ by a change of variables, as described
in Section 3. The change of variables varies the underlying measure and
explains why need the full scale of $L^{p}_{m}$ spaces.

More effort is needed to add the potential term. We consider first $B-V$ in
$L^{2}(\mathbb{R}_{+};y^{c}dy)$. We use form methods to construct an analytic
semigroup, we prove kernel bounds for complex times via Davies-Gaffney
estimates and provide a core. Then, with the methods of Section 3, we deduce
similar results for $y^{\alpha}B-V$ in $L^{2}(\mathbb{R}_{+};y^{c-\alpha}dy)$.
Next we prove that the semigroup can be extended to $L^{p}_{m}$ under sharp
conditions on $p$ and $m$. Finally we prove that for every $\epsilon>0$ the
family of operators

$\displaystyle\left\\{e^{z(y^{\alpha}B-V)}:\;z\in\Sigma_{\frac{\pi}{2}-\epsilon},\;0\leq
V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)\right\\}$

is $\mathcal{R}$-bounded in $L^{p}_{m}$, which implies the maximal regularity
of the semigroup when $V\geq 0$.

As a motivation for our investigation, we point out that, in the special case
$V(y)=y^{\alpha}$, all the results above play a crucial role in [13] in the
investigation of the degenerate operators

$\displaystyle\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right).$

Let us suppose, for simplicity, $b=0$, $\alpha_{1}=\alpha_{2}:=\alpha$.
Assuming that $y^{\alpha}(\Delta_{x}u+B_{y}u)=f$ and taking the Fourier
transform ${\cal F}u$ or $\hat{u}$ with respect to $x$ we obtain
$y^{\alpha}|\xi|^{2}\hat{u}(\xi,y)=-y^{\alpha}|\xi|^{2}(y^{\alpha}|\xi|^{2}-y^{\alpha}B_{y})^{-1}\hat{f}(\xi,y)$.
Therefore

$\displaystyle y^{\alpha}\Delta_{x}\mathcal{L}^{-1}$ $\displaystyle={\cal
F}^{-1}\left(y^{\alpha}|\xi|^{2}(y^{\alpha}|\xi|^{2}-y^{\alpha}B_{y})^{-1})\right){\cal
F}$

and the boundedness of $y^{\alpha}\Delta_{x}\mathcal{L}^{-1}$ is equivalent to
that of the multiplier

$\displaystyle\xi\in\mathbb{R}^{N}\to
y^{\alpha}|\xi|^{2}(y^{\alpha}|\xi|^{2}-y^{\alpha}By)^{-1}.$

For this reason we prove in Section 8 that certain multiplier associated to
$y^{\alpha}B-V$ satisfy a vector valued Mikhlin theorem. These results rely on
square function estimates which we deduce from kernel bounds and the following
equality, which allows to treat $\lambda$ or $|\xi|^{2}$ as spectral
parameters simultaneously

$\displaystyle\left(\lambda-y^{\alpha}B+|\xi|^{2}y^{\alpha}\right)^{-1}=\left(|\xi|^{2}-B+\frac{\lambda}{y^{\alpha}}\right)^{-1}\frac{1}{y^{\alpha}}.$

We restrict ourselves to $\alpha<2$ and consider $y^{\alpha}B$ with Neumann
boundary condition at $0$, namely $\lim_{y\to 0}y^{c}D_{y}u(y)=0$. This is
equivalent to require $y^{\alpha-1}D_{y}u\in L^{p}_{m}$, see [12, Proposition
5.11]. The restriction $\alpha<2$ is not really essential since one can deduce
from it the case $\alpha>2$, which requires a boundary condition at $\infty$,
using the change of variables described in Section 3.

Besides this, our strategy can be easily adapted to different boundary
conditions and to more general operators
$y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right)-V$. We do this
(in much more generality) in [13, Sections 7,8].

The paper is organized as follows. In Section 2 we briefly recall the harmonic
analysis background needed in the paper, as square function estimates,
$\mathcal{R}$-boundedness and a vector valued multiplier theorem.

In Section 3, we exploit an elementary change of variables, in a functional
analytic setting, to reduce our operators to the simpler case where
$\alpha=0$.

Section 4 is devoted to the study of the Bessel operator $y^{\alpha}B$. In
Sections 5, 6 and 7 we perturb the Bessel operator by adding the potential $V$
and we prove real and complex kernel estimates, generation results and maximal
regularity for $y^{\alpha}B-V$. Finally in Section 8 we treat the case
$V(y)=y^{\alpha}$ and characterize the domain of $y^{\alpha}B-y^{\alpha}$.

Notation. For $m\in\mathbb{R}$ we consider the measure $y^{m}dy$ in
$\mathbb{R}_{+}$ and we write $L^{p}_{m}$ for $L^{p}(\mathbb{R}_{+},y^{m}dy)$.
Similarly $W^{k,p}_{m}=\\{u\in L^{p}_{m}:\partial^{\alpha}u\in
L^{p}_{m}\quad|\alpha|\leq k\\}$. When we write $V\in
L^{q}_{loc}(\mathbb{R}^{+},y^{m}\,dy)$, we mean that $V\in
L^{q}([0,b],y^{m}\,dy)$ for every $b<\infty$.

We use
$\mathbb{C}^{+}=\\{\lambda\in\mathbb{C}:\textrm{\emph{Re}\,}\lambda>0\\}$ and,
for $|\theta|\leq\pi$, we denote by $\Sigma_{\theta}$ the open sector
$\\{\lambda\in\mathbb{C}:\lambda\neq 0,\ |Arg(\lambda)|<\theta\\}$.

##  2 Harmonic analysis and maximal regularity

The study of maximal regularity of parabolic problems of the form
$u_{t}=Au+f,u(0)=0$, where $A$ is the generator of an analytic semigroup on a
Banach space $X$, consists in proving estimates like

$\|u_{t}\|_{p}+\|Au\|_{p}\leq\|f\|_{p}$

where the $L^{p}$ norm is that of $L^{p}([0,T[;X)$. This can be interpreted as
closedness of $D_{t}-A$ on the intersection of the respective domains or,
equivalently, boundedness of the operator $A(D_{t}-A)^{-1}$ in
$L^{p}([0,T[;X)$.

Nowadays this strategy is well established and relies on Mikhlin vector-valued
multiplier theorems. Let us state the relevant definitions and main results we
need, referring the reader to [5], [16] or [6].

Let ${\cal S}$ be a subset of $B(X)$, the space of all bounded linear
operators on a Banach space $X$. ${\cal S}$ is $\mathcal{R}$-bounded if there
is a constant $C$ such that

$\|\sum_{i}\varepsilon_{i}S_{i}x_{i}\|_{L^{p}(\Omega;X)}\leq
C\|\sum_{i}\varepsilon_{i}x_{i}\|_{L^{p}(\Omega;X)}$

for every finite sum as above, where $(x_{i})\subset X,(S_{i})\subset{\cal S}$
and $\varepsilon_{i}:\Omega\to\\{-1,1\\}$ are independent and symmetric random
variables on a probability space $\Omega$. The smallest constant $C$ for which
the above definition holds is the $\mathcal{R}$-bound of $\mathcal{S}$,
denoted by $\mathcal{R}(\mathcal{S})$. It is well-known that this definition
does not depend on $1\leq p<\infty$ (however, the constant
$\mathcal{R}(\mathcal{S})$ does) and that $\mathcal{R}$-boundedness is
equivalent to boundedness when $X$ is an Hilbert space. When $X$ is an $L^{p}$
space (with respect to any $\sigma$-finite measure), testing
$\mathcal{R}$-boundedness is equivalent to proving square functions estimates,
see [6, Remark 2.9 ].

###### Proposition 2.1

Let ${\cal S}\subset B(L^{p}(\Sigma))$, $1<p<\infty$. Then ${\cal S}$ is
$\mathcal{R}$-bounded if and only if there is a constant $C>0$ such that for
every finite family $(f_{i})\in L^{p}(\Sigma),(S_{i})\in{\cal S}$

$\left\|\left(\sum_{i}|S_{i}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}.$

The best constant $C$ for which the above square functions estimates hold
satisfies $\kappa^{-1}C\leq\mathcal{R}(\mathcal{S})\leq\kappa C$ for a
suitable $\kappa>0$ (depending only on $p$). The proposition above
$\mathcal{R}$-boundedness follows from domination.

###### Corollary 2.2

Let ${\cal S},{\cal T}\subset B(L^{p}(\Sigma))$, $1<p<\infty$ and assume that
$\cal T$ is $\mathcal{R}$ bounded and that for every $S\in\cal S$ there exists
$T\in\cal T$ such that $|Sf|\leq|Tf|$ pointwise, for every $f\in
L^{p}(\Sigma)$. Then ${\cal S}$ is $\mathcal{R}$-bounded.

Let $(A,D(A))$ be a sectorial operator in a Banach space $X$; this means that
$\rho(-A)\supset\Sigma_{\pi-\phi}$ for some $\phi<\pi$ and that
$\lambda(\lambda+A)^{-1}$ is bounded in $\Sigma_{\pi-\phi}$. The infimum of
all such $\phi$ is called the spectral angle of $A$ and denoted by $\phi_{A}$.
Note that $-A$ generates an analytic semigroup if and only if
$\phi_{A}<\pi/2$. The definition of $\mathcal{R}$-sectorial operator is
similar, substituting boundedness of $\lambda(\lambda+A)^{-1}$ with
$\mathcal{R}$-boundedness in $\Sigma_{\pi-\phi}$. As above one denotes by
$\phi^{R}_{A}$ the infimum of all $\phi$ for which this happens; since
$\mathcal{R}$-boundedness implies boundedness, we have
$\phi_{A}\leq\phi^{R}_{A}$.

The $\mathcal{R}$-boundedness of the resolvent characterizes the regularity of
the associated inhomogeneous parabolic problem, as we explain now.

An analytic semigroup $(e^{-tA})_{t\geq 0}$ on a Banach space $X$ with
generator $-A$ has maximal regularity of type $L^{q}$ ($1<q<\infty$) if for
each $f\in L^{q}([0,T];X)$ the function $t\mapsto
u(t)=\int_{0}^{t}e^{-(t-s)A})f(s)\,ds$ belongs to $W^{1,q}([0,T];X)\cap
L^{q}([0,T];D(B))$. This means that the mild solution of the evolution
equation

$u^{\prime}(t)+Au(t)=f(t),\quad t>0,\qquad u(0)=0,$

is in fact a strong solution and has the best regularity one can expect. It is
known that this property does not depend on $1<q<\infty$ and $T>0$. A
characterization of maximal regularity is available in UMD Banach spaces,
through the $\mathcal{R}$-boundedness of the resolvent in a suitable sector
$\omega+\Sigma_{\phi}$, with $\omega\in\mathbb{R}$ and $\phi>\pi/2$ or,
equivalently, of the scaled semigroup $e^{-(A+\omega^{\prime})t}$ in a sector
around the positive axis. In the case of $L^{p}$ spaces it can be restated in
the following form, see [6, Theorem 1.11]

###### Theorem 2.3

Let $(e^{-tA})_{t\geq 0}$ be a bounded analytic semigroup in $L^{p}(\Sigma)$,
$1<p<\infty$, with generator $-A$. Then $T(\cdot)$ has maximal regularity of
type $L^{q}$ if and only if the set
$\\{\lambda(\lambda+A)^{-1},\lambda\in\Sigma_{\pi/2+\phi}\\}$ is
$\mathcal{R}$\- bounded for some $\phi>0$. In an equivalent way, if and only
if there are constants $0<\phi<\pi/2$, $C>0$ such that for every finite
sequence $(\lambda_{i})\subset\Sigma_{\pi/2+\phi}$, $(f_{i})\subset L^{p}$

$\left\|\left(\sum_{i}|\lambda_{i}(\lambda_{i}+A)^{-1}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}$

or, equivalently, there are constants $0<\phi^{\prime}<\pi/2$, $C^{\prime}>0$
such that for every finite sequence $(z_{i})\subset\Sigma_{\phi^{\prime}}$,
$(f_{i})\subset L^{p}$

$\left\|\left(\sum_{i}|e^{-z_{i}A}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C^{\prime}\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}.$

Finally we state a version of the operator-valued Mikhlin multiplier theorem
in the N-dimensional case, see [5, Theorem 3.25] or [6, Theorem 4.6].

###### Theorem 2.4

Let $1<p<\infty$, $M\in C^{N}(\mathbb{R}^{N}\setminus\\{0\\};B(L^{p}(\Sigma))$
be such that the set

$\left\\{|\xi|^{|\alpha|}D^{\alpha}_{\xi}M(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\alpha|\leq N\right\\}$

is $\mathcal{R}$-bounded. Then the operator $T_{M}={\cal F}^{-1}M{\cal F}$ is
bounded in $L^{p}(\mathbb{R}^{N},L^{p}(\Sigma))$, where $\cal{F}$ denotes the
Fourier transform.

We end this section with the following lemma on radially symmetric
multipliers.

###### Lemma 2.5

Let $1<p<\infty$, $m\in C^{N}(\mathbb{R}_{+};B(L^{p}(\Sigma))$ be such that
the set

$\left\\{s^{k}m^{(k)}(s):\ s\in\mathbb{R}_{+},\ k\leq N\right\\}$

is $\mathcal{R}$-bounded. For $a\in\mathbb{R}$ let
$M(\xi)=m\left(|\xi|^{a}\right)$. Then $M\in
C^{N}(\mathbb{R}^{N}\setminus\\{0\\};B(L^{p}(\Sigma))$ and

$\left\\{|\xi|^{|\alpha|}D^{\alpha}_{\xi}M(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\alpha|\leq N\right\\}$

is $\mathcal{R}$-bounded and

$\mathcal{R}\left\\{|\xi|^{|\alpha|}D^{\alpha}_{\xi}M(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\alpha|\leq N\right\\}\leq C(N)\mathcal{R}\left\\{s^{k}m^{(k)}(s):\
s\in\mathbb{R}_{+},\ k\leq N\right\\}.$

Proof. Let us observe preliminarily that for any multi-index $\alpha$ with
$0<|\alpha|\leq N$ one has

$\displaystyle D^{\alpha}_{\xi}M(\xi)$
$\displaystyle=\displaystyle\sum_{i=1}^{|\alpha|}h_{i,\alpha}(\xi)m^{(i)}\left(|\xi|^{a}\right)$
(1)

where $h_{i,\alpha}\in C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ are
homogeneous functions of degree $ia-|\alpha|$. Obviously $\eqref{eq mik comp}$
is valid for $|\alpha|=1$ since $\nabla M(\xi)=a\
m^{\prime}(|\xi|^{a})|\xi|^{a-2}\xi$ and follows by induction, since the
derivatives of $h_{i,\alpha}$ are homogeneous of degree $ia-|\alpha|-1$.

The proof of the lemma now follows by Corollary (2.2) since from (1) one has
for $f\in L^{p}(\Sigma)$

$\displaystyle|\xi|^{|\alpha|}|D^{\alpha}_{\xi}M(\xi)f|$
$\displaystyle\leq\displaystyle|\xi|^{|\alpha|}\sum_{i=1}^{|\alpha|}|h_{i,\alpha}(\xi)||m^{(i)}\left(|\xi|^{a}\right)f|\leq
C\displaystyle\sum_{i=1}^{|\alpha|}|\xi|^{ia}|m^{(i)}\left(|\xi|^{a}\right)f|.$

##  3 Degenerate operators and similarity transformations

We investigate when the operators

$B=D_{yy}+\frac{c}{y}D_{y},\qquad
y^{\alpha}B=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)$

can be transformed one into the other by means of change of variables. Here
$\alpha,c$ are unrestricted real coefficients.

For $\beta\in\mathbb{R}$, $\beta\neq-1$ let

$\displaystyle T_{\beta\,}u(y)$
$\displaystyle:=|\beta+1|^{\frac{1}{p}}u(y^{\beta+1}),\quad
y\in\mathbb{R}_{+}.$ (2)

Observe that

$T_{\beta\,}^{-1}=T_{-\frac{\beta}{\beta+1}\,}.$

###### Proposition 3.1

Let $1\leq p\leq\infty$, $k,\beta\in\mathbb{R}$, $\beta\neq-1$. The following
properties hold.

  * (i)

For every $m\in\mathbb{R}$, $T_{\beta\,}$ maps isometrically
$L^{p}_{\tilde{m}}$ onto $L^{p}_{m}$ where

$\tilde{m}=\frac{m-\beta}{\beta+1}.$

  * (ii)

For every $u\in W^{2,1}_{loc}\left(\mathbb{R}_{+}\right)$ one has

    * 1.

$y^{\alpha}T_{\beta\,}u=T_{\beta\,}(y^{\frac{\alpha}{\beta+1}}u)$, for any
$\alpha\in\mathbb{R}$;

    * 2.

$D_{y}T_{\beta\,}u=T_{\beta\,}\left((\beta+1)y^{\frac{\beta}{\beta+1}}D_{y}u\right)$,
$D_{yy}(T_{\beta\,}u)=T_{\beta\,}\Big{(}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)\beta
y^{\frac{\beta-1}{\beta+1}}D_{y}u\Big{)}$.

Proof. The proof of (i) follows after observing the Jacobian of $y\mapsto
y^{\beta+1}$ is $|1+\beta|y^{\beta}$. Then we compute

$\displaystyle D_{y}T_{\beta\,}u(y)=$
$\displaystyle|\beta+1|^{\frac{1}{p}}\left((\beta+1)y^{\beta}D_{y}u(y^{\beta+1})\right)=T_{\beta\,}\left((\beta+1)y^{\frac{\beta}{\beta+1}}D_{y}u\right)$

and similarly

$\displaystyle D_{yy}T_{\beta\,}u(y)=$ $\displaystyle
T_{\beta\,}\Big{(}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)\beta
y^{\frac{\beta-1}{\beta+1}}D_{y}u\Big{)}.$

###### Proposition 3.2

Let $T_{\beta\,}$ be the isometry above defined. The following properties
holds.

For every $u\in W^{2,1}_{loc}\left(\mathbb{R}_{+}\right)$ one has

$T_{\beta\,}^{-1}\Big{(}y^{\alpha}B\Big{)}T_{\beta\,}u=\Big{(}(\beta+1)^{2}y^{\frac{\alpha+2\beta}{\beta+1}}\tilde{B}\Big{)}u$

where $\tilde{B}$ is the operator defined as in (1) with parameter $c$
replaced, respectively, by

$\tilde{c}=\frac{c+\beta\left(c+1+\beta\right)}{(\beta+1)^{2}}.$

Proof. Using Proposition 3.1 we can compute

$\displaystyle{B}T_{\beta\,}u(y)$
$\displaystyle=T_{\beta\,}\Big{[}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)\beta
y^{\frac{\beta-1}{\beta+1}}D_{y}u+c(\beta+1)y^{\frac{\beta-1}{\beta+1}}D_{y}u-by^{-\frac{2}{\beta+1}}u\Big{]}$
$\displaystyle=T_{\beta\,}\Bigg{[}y^{\frac{2\beta}{\beta+1}}\Bigg{(}(\beta+1)^{2}D_{yy}u+\frac{(\beta+1)\left(\beta+c\right)}{y}D_{y}u-b\frac{u}{y^{2}}\Bigg{)}\Bigg{]}=T_{\beta\,}\left(y^{\frac{2\beta}{\beta+1}}\tilde{B}u\right)$

which implies

$\displaystyle T_{\beta\,}^{-1}\left(y^{\alpha}{B}\right)T_{\beta\,}u$
$\displaystyle=y^{\frac{\alpha+2\beta}{\beta+1}}\tilde{B}u.$

##  4 The Bessel operator $y^{\alpha}B^{n}$

In this section we consider for $\alpha<2$, $c\in\mathbb{R}$ the operator

$y^{\alpha}B=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)$

in the space $L^{p}_{m}$ under Neumann boundary conditions.

According to Proposition 3.2, we use the isometry

$T_{-\frac{\alpha}{2}}:L^{p}_{\tilde{m}}\to L^{p}_{m}\,\quad
T_{-\frac{\alpha}{2}}u(y)=\left|1-\frac{\alpha}{2}\right|^{\frac{1}{p}}u(y^{1-\frac{\alpha}{2}}),$

$\tilde{m}=\frac{m+\frac{\alpha}{2}}{1-\frac{\alpha}{2}}$, under which
$y^{\alpha}B$ becomes isometrically equivalent to
$T_{-\frac{\alpha}{2}}^{-1}\Big{(}y^{\alpha}B\Big{)}T_{-\frac{\alpha}{2}}=\left(1-\frac{\alpha}{2}\right)^{2}\tilde{B}$
where $\tilde{B}=D_{yy}+\frac{\tilde{c}}{y}D_{y}$ and
$\tilde{c}=\frac{c-\frac{\alpha}{2}}{1-\frac{\alpha}{2}}$.

All the results for $y^{\alpha}B$ in $L^{p}_{m}$ are then immediate
consequence of those of $\tilde{B}$ in $L^{p}_{\tilde{m}}$ already proved in
[11, Section 3] (see also [10, 9, 8, 14] for some analogous results for the
$N$-d version of $\tilde{B}$).

If $1<p<\infty$, we define

$W^{2,p}_{\mathcal{N}}(\alpha,m)=\left\\{u\in W^{2,p}_{loc}(\mathbb{R}_{+}):\
u,\ y^{\alpha}D_{yy}u,\ y^{\frac{\alpha}{2}}D_{y}u,\ y^{\alpha-1}D_{y}u\in
L^{p}_{m}\right\\}$

and refer to [12] where these spaces are studied in detail in
$\mathbb{R}^{N+1}_{+}$. The Neumann boundary condition is enclosed in the
requirement $y^{\alpha-1}D_{y}u\in L^{p}_{m}$. This last is redundant when
$(m+1)/p>1-\alpha$ and equivalent to $D_{y}u(y)\to 0$ as $y\to 0$, when
$(m+1)/p<1-\alpha$, see [12, Proposition 4.3]. Consequently we write
$y^{\alpha}B^{n}$ or, more pedantically $y^{\alpha}B^{n}_{m,p}$ if necessary,
for its realization in $L^{p}_{m}$.

###### Remark 4.1

The restriction $\alpha<2$ is not really essential since one can deduce from
it the case $\alpha>2$, which requires boundary condition at $\infty$, using
the change of variables described in Section 3 or directly from the equality
$T_{-\frac{\alpha}{2}}^{-1}\Big{(}y^{\alpha}B\Big{)}T_{-\frac{\alpha}{2}}=\left(1-\frac{\alpha}{2}\right)^{2}\tilde{B}$
which is valid for any $\alpha\neq 2$. However, here and in what follows, we
keep to it in order to simplify the exposition.

###### Theorem 4.2

If $0<\frac{m+1}{p}<c+1-\alpha$, then $y^{\alpha}B^{n}$ endowed with domain
$W^{2,p}_{\mathcal{N}}(\alpha,m)$

generates a bounded positive analytic semigroup of angle $\pi/2$ on
$L^{p}\left(\mathbb{R}_{+},y^{m}dy\right)$.

Proof. We use the identity
$T_{-\frac{\alpha}{2}}^{-1}\Big{(}y^{\alpha}B^{n}\Big{)}T_{-\frac{\alpha}{2}}=\left(1-\frac{\alpha}{2}\right)^{2}\tilde{B^{n}}$
and apply [11, Proposition 3.3] in $L^{p}_{\tilde{m}}$. Note that
$D(y^{\alpha}B_{m,p}^{n})=T_{-\frac{\alpha}{2}}D(\tilde{B}_{\tilde{m},p}^{n})$
which means

$\displaystyle u\in D(y^{\alpha}B_{m,p}^{n})\qquad\Longleftrightarrow\qquad
v(y):=u(y^{\frac{2}{2-\alpha}})\in D(\tilde{B}_{\tilde{m},p}^{n}).$

Under the hypothesis of Theorem 4.2, the domain of $y^{\alpha}B^{n}$ consists
of all functions in the maximal domain satisfying a Neumann condition at $0$,
see [12, Proposition 4.6, 4.7], that is

$D(y^{\alpha}B_{m,p}^{n})=\left\\{u\in W^{2,p}_{loc}(\mathbb{R}_{+}):u,\
y^{\alpha}Bu\in L^{p}_{m}\text{\;\;and\;\;}\lim_{y\to
0}y^{c}D_{y}u=0\right\\}$

(the condition $\lim_{y\to 0}y^{c}D_{y}u=0$ can be deleted in the range
$0<\frac{m+1}{p}\leq c-1$). When $c\geq 1$ the domain can also be described
involving a Dirichlet, rather than Neumann, boundary condition

$\displaystyle D(y^{\alpha}B_{m,p}^{n})=$ $\displaystyle\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}_{+}):u,\ y^{\alpha}Bu\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}y^{c-1}u=0\right\\},\qquad\text{\hskip
4.30554ptif $c>1$};$ $\displaystyle D(y^{\alpha}B_{m,p}^{n})=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}_{+}):u,\ y^{\alpha}Bu\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}u\in\mathbb{C}\right\\},\qquad\text{\hskip
23.68048ptif $c=1$}.$

We close this section by describing a core which does not depend on
$\alpha,m,p$ and on the coefficients of the operator.

###### Proposition 4.3

If $0<\frac{m+1}{p}<c+1-\alpha$, then a core for $y^{\alpha}B^{n}$ is

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)):u\ {\rm constant\ in\ a\
neighborhood\ of}\ 0\right\\}.$

Proof. The proof immediately follows by observing that, by [11, Proposition
5.4], $\mathcal{D}$ is a core when $\alpha=0$, that is for
$\tilde{B}^{n}_{\tilde{m},p}$, and the isometry $T_{-\frac{\alpha}{2}}$ leaves
invariant $\mathcal{D}$ since $\alpha<2$.

###### Remark 4.4

We point out that, by the proof of [11, Proposition 5.4] or by [12, Remark
4.14], it follows that if $u\in D(y^{\alpha}B^{n}_{m,p})$ has support in
$[0,b]$, then there exists a sequence
$\left(u_{n}\right)_{n\in\mathbb{N}}\in\mathcal{D}$ such that $\mbox{supp
}u_{n}\subseteq[0,b]$ and $u_{n}\to u$ in $D(y^{\alpha}B^{n}_{m,p})$.

##  5 The operator $B^{n}-V$

We start our investigation by adding a potential $0\leq V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c}\,dy\right)$ to $B^{n}$. Here we prove
kernel bounds and construct a core.

###  5.1 Kernel bounds

For $c+1>0$ and $0\leq V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c}\,dy\right)$
we prove upper bounds for the heat kernel of $B^{n}-V$, following the method
used in [3, Sections 3,4].

Setting $H^{1}_{c}=\\{u\in L^{2}_{c},u^{\prime}\in L^{2}_{c}\\}$, we recall
that from [11, Section 2] the operator $B^{n}_{c,2}$ is associated to the non-
negative, symmetric and closed form in $L^{2}_{c}$

$\displaystyle\mathfrak{a}(u,v)$
$\displaystyle:=\int_{0}^{\infty}D_{y}uD_{y}\overline{v}\,y^{c}dy,\qquad
D(\mathfrak{a})=H^{1}_{c}.$

We consider the perturbed form $\mathfrak{a}_{V}$ in $L^{2}_{c}$ defined by

$\displaystyle\mathfrak{a}_{V}(u,v)=\mathfrak{a}(u,v)+\left\langle
Vu,v\right\rangle_{L^{2}_{c}}=\int_{\mathbb{R}_{+}}\left(D_{y}uD_{y}\overline{v}+Vu\overline{v}\right)y^{c}\,dy$
$\displaystyle D(\mathfrak{a}_{V})=D(\mathfrak{a})\cap
L^{2}\left(\mathbb{R}^{+},Vy^{c}\,dy\right)$ (3)

and define $B^{n}-V$ in $L^{2}_{c}$ as the operator associated to the form
$\mathfrak{a}_{V}$

$\displaystyle D(B^{n}-V)$ $\displaystyle=\\{u\in D(\mathfrak{a}_{V}):\exists
f\in L^{2}_{c}\ {\rm such\ that}\
\mathfrak{a}_{V}(u,v)=\int_{0}^{\infty}f\overline{v}y^{c}\,dy\ {\rm for\
every}\ v\in D(\mathfrak{a}_{V})\\},$ $\displaystyle B^{n}u-Vu$
$\displaystyle=-f.$

The positivity of $V$ implies that the norm induced by the form
$\mathfrak{a}_{V}$ is stronger then the one induced by $\mathfrak{a}$: as an
immediate consequence one deduces that $\mathfrak{a}_{V}$ is closed. By
standard theory on sesquilinear forms we have the following result.

###### Proposition 5.1

If $c+1>0$, $0\leq V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c}\,dy\right)$,
then $\mathfrak{a}_{V}$ is a non-negative, symmetric and closed form in
$L^{2}_{c}$. Its associated operator $-B^{n}+V$ is non-negative and self-
adjoint and $B^{n}-V$ generates a contractive analytic semigroup
$\left\\{e^{z(B^{n}-V)}:\ z\in\mathbb{C}_{+}\right\\}$ in $L^{2}_{c}$.
Moreover:

  * (i)

The semigroup $\left(e^{t(B^{n}-V)}\right)_{t\geq 0}$ is sub-markovian (i.e.
it is positive and $L^{\infty}$-contractive) and it is dominated by
$e^{tB^{n}}$, that is

$\displaystyle|e^{t(B^{n}-V)}f|\leq e^{tB^{n}}|f|,\qquad t>0,\quad f\in
L^{2}_{c}.$

  * (ii)

$\left(e^{t(B^{n}-V)}\right)_{t\geq 0}$ is a semigroup of integral operators
and its heat kernel $p_{V}$, taken with respect the measure $\rho^{c}d\rho$,
satisfies

$\displaystyle 0\leq p_{V}(t,y,\rho)\leq
Ct^{-\frac{1}{2}}\rho^{-c}\left(\frac{\rho}{t^{\frac{1}{2}}}\wedge
1\right)^{c}\exp\left(-\frac{|y-\rho|^{2}}{\kappa t}\right).$

Proof. The first claim follows from the property of $\mathfrak{a}_{V}$.
$e^{t(B^{n}-V)}$ is sub-markovian from [15, Corollary 2.17]. The domination
property follows from [15, Corollary 2.21]. (ii) is a consequence of [2,
Proposition 1.9] since $e^{t(B^{n}-V)}$ is dominated by the positive integral
operator $e^{tB^{n}}$ whose kernel satisfies the stated estimate, see [11,
Proposition 2.8] where, however, the kernel is written with respect to the
Lebesgue measure.

To extend the above heat kernel estimates to the half plane $\mathbb{C}_{+}$
we need the following lemma.

###### Lemma 5.2

Let $c+1>0$ and for $y_{0},r>0$

$\displaystyle Q_{c}(y_{0},r):=\int_{[y_{0},y_{0}+r]}y^{c}dy.$

Then one has

$\displaystyle Q_{c}(y_{0},r)\simeq
r^{c+1}\left(\frac{y_{0}}{r}\right)^{c}\left(\frac{y_{0}}{r}\wedge
1\right)^{-c},\qquad r,y_{0}>0.$

In particular the function $Q_{c}$ satisfies, for some constants $C\geq 1$,
the doubling condition

$\displaystyle\frac{Q_{c}(y_{0},s)}{Q_{c}(y_{0},r)}\leq
C\left(\frac{s}{r}\right)^{1\vee(c+1)},\qquad\forall y_{0}>0,\quad 0<r<s.$

Proof. A scaling argument immediately yields
$Q_{c}(y_{0},r)=r^{c+1}Q_{c}\left(\frac{y_{0}}{r},1\right)$ and we may
therefore assume $r=1$. The local integrability of $y^{c}$ implies that
$Q_{c}(y_{0},1)$ is continuous as a function of $y_{0}$ and moreover
$Q_{c}(y_{0},1)\to\int_{(0,1)}y^{c}dy>0$ as $y_{0}\to 0$. Therefore if
$y_{0}\leq 1$ then

$\displaystyle Q_{c}(y_{0},1)\simeq 1.$

On the other hand if $y_{0}>1$ then $y\simeq y_{0}$ for any
$y\in(y_{0},y_{0}+1)$ which implies

$\displaystyle Q_{c}(y_{0},1)=\int_{(y_{0},y_{0}+1)}y^{c}dy\simeq y_{0}^{c}.$

The last two inequalities yields
$Q_{c}(y_{0},1)\simeq\left(y_{0}\right)^{c}\left(y_{0}\wedge 1\right)^{-c}$.
The doubling condition follows from the previous estimates and the fact that
for $0<r<s$ one has

$\displaystyle\frac{Q_{c}(y_{0},s)}{Q_{c}(y_{0},r)}\leq
C\begin{cases}\left(\frac{s}{r}\right)^{c+1},\qquad&\text{if}\quad\frac{y_{0}}{s}\leq\frac{y_{0}}{r}\leq
1;\\\\[4.30554pt]
\frac{s}{r}(\frac{s}{y_{0}})^{c},\qquad&\text{if}\quad\frac{y_{0}}{s}\leq
1<\frac{y_{0}}{r};\\\\[4.30554pt] \frac{s}{r},\qquad&\text{if}\quad
1\leq\frac{y_{0}}{s}\leq\frac{y_{0}}{r}\end{cases}$

(note that in the range $\frac{y_{0}}{s}\leq 1<\frac{y_{0}}{r}$ one has
$(\frac{s}{y_{0}})^{c}\leq 1$ if $c<0$ and
$(\frac{s}{y_{0}})^{c}\leq\left(\frac{s}{r}\right)^{c}$ if $c\geq 0$).

###### Proposition 5.3

Let $c+1>0$, $0\leq V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c}\,dy\right)$.
The semigroup $\left\\{e^{z(B^{n}-V)}:\ z\in\mathbb{C}_{+}\right\\}$ consists
of integral operators

$e^{z(B^{n}-V)}f(y)=\int_{0}^{\infty}p_{V}(z,y,\rho)f(\rho)\,\rho^{c}d\rho,\quad
f\in L^{2}_{c},\quad y>0.$

Furthermore for every $\epsilon>0$ there exist $k_{\epsilon},C_{\epsilon}>0$
such that, for every $z\in\Sigma_{\frac{\pi}{2}-\epsilon}$ and $y,\rho>0$,

$\displaystyle|p_{V}(z,x,y)|\leq
C_{\epsilon}|z|^{-\frac{1}{2}}\rho^{-c}\left(\frac{\rho}{|z|^{\frac{1}{2}}}\wedge
1\right)^{c}\exp\left(-\frac{|y-\rho|^{2}}{\kappa_{\epsilon}|z|}\right).$

Proof. Using the previous lemma we rewrite Proposition 5.1 (ii) as

$\displaystyle 0\leq p_{V}(t,y,\rho)\leq
C\frac{1}{Q_{c}(\rho,\sqrt{t})}\exp\left(-\frac{|y-\rho|^{2}}{\kappa
t}\right).$

Furthermore by [4, Theorem 3.3], $e^{t(B^{n}-V)}$ satisfies the Davies-Gaffney
estimates

$\displaystyle|\langle
e^{t(B^{n}-V)}f_{1},f_{2}\rangle|\leq\exp\left(-\frac{r^{2}}{4t}\right)\|f_{1}\|_{L^{2}_{c}}\|f_{2}\|_{L^{2}_{c}}$

for all $t>0$, $U_{1}$, $U_{2}$ open subsets of $(0,+\infty)$,
$r:=d(U_{1},U_{2})=\min\\{|x-y|:x\in U_{1},y\in U_{2}\\}$ and $f_{i}$ in
$L^{2}(U_{i},y^{c}dy)$. By [4, Corollary 4.4] and Lemma 5.2 we then obtain for
$z\in\Sigma_{\frac{\pi}{2}-\epsilon}$ and $y,\rho>0$

$\displaystyle|p_{V}(z,y,\rho)|$ $\displaystyle\leq
C_{\epsilon}\frac{1}{\left(Q_{c}(y,\sqrt{|z|}\right)^{\frac{1}{2}}\left(Q_{c}(\rho,\sqrt{|z|}\right)^{\frac{1}{2}}}\exp\left(-\frac{|y-\rho|^{2}}{\kappa_{\epsilon}|z|}\right)$
$\displaystyle\leq
C_{\epsilon}^{\prime}|z|^{\frac{c+1}{2}}\left(\frac{y}{\sqrt{|z|}}\right)^{-\frac{c}{2}}\left(1\wedge\frac{y}{\sqrt{|z|}}\right)^{\frac{c}{2}}\left(\frac{\rho}{\sqrt{|z|}}\right)^{-\frac{c}{2}}\left(1\wedge\frac{\rho}{\sqrt{|z|}}\right)^{\frac{c}{2}}\exp\left(-\frac{|y-\rho|^{2}}{\kappa_{\epsilon}|z|}\right).$

This is an equivalent form (after modifying the constant in the exponential)
of the estimate in the statement, by [11, Lemma 10.2] with
$\gamma_{1}=\gamma_{2}=-\frac{c}{2}$.

###### Remark 5.4

We remark that in [4], the authors work in an abstract metric measure space
$(M,d,\mu)$ and assume that the heat kernel $p$ associated with a semigroup
$e^{-zL}$, where $L$ is a non-negative self-adjoint operator on
$L^{2}(M,d\mu)$, is continuous with respect to the space variables. In such a
case, in fact,

$\displaystyle\underset{x\in U_{1},y\in
U_{2}}{sup}|p(z,x,y)|=\mbox{sup}\\{\int_{M}e^{-zL}f_{1}\overline{f_{2}}\,d\mu,\quad\|f_{1}\|_{L^{1}(U_{1},d\mu)}=\|f_{2}\|_{L^{1}(U_{2},d\mu)}=1\\}.$

In our setting the continuity assumption on $p$ can be avoided since the
proofs of [4, Theorem 4.1, Corollary 4.4] hold only assuming that for a.e.
$x,y\in M$

$\displaystyle p(z,x,y)=\lim_{s\to
0}\int_{M}e^{-zL}f_{1}\overline{f_{2}}\,d\mu=\lim_{s\to
0}\frac{1}{\mu(B(x,s)\mu(B(y,s)}\int_{B(x,s)\times
B(y,s)}p(z,\bar{x},\bar{y}),d\mu(\bar{x})d\mu(\bar{y}),$

where $f_{1}=\frac{\chi_{B(x,s)}}{\mu(B(x,s))}$,
$f_{2}=\frac{\chi_{B(y,s)}}{\mu(B(y,s))}$. This holds, outside a set of zero
measure, when the measure $\mu$ is doubling, by the Lebesgue differentiation
theorem.

###  5.2 A core for $B^{n}-V$

We prove that under mild hypotheses the set

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)):u\ {\rm constant\ in\ a\
neighborhood\ of}\ 0\right\\}$

is a core for $B^{n}-V$ in $L^{2}_{c}$. Note that this is true when $V=0$, by
Proposition 4.3. We need some elementary lemmas. Unless explicitly stated, we
only assume that $0\leq V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c}\,dy\right)$.

###### Lemma 5.5

Assume that $0\leq V\in L^{2}_{loc}(\mathbb{R}_{+},y^{c}\,dy).$ Then
$D(\mathfrak{a}_{V})=H^{1}_{c}\cap
L^{2}\left(\mathbb{R}^{+},Vy^{c}\,dy\right)$ is dense in $H^{1}_{c}$.

Proof. By Proposition 4.3, $\mathcal{D}$ is dense in $D(B^{n})$ with respect
to the graph norm. Moreover, since $V\in L^{2}_{c}$ locally,
$\mathcal{D}\subset D(\mathfrak{a}_{V})$. The claim follows from the density
of $D(B^{n})$ in $H^{1}_{c}$.

###### Lemma 5.6

Let $u\in H^{1}_{c}$ such that $Vu\in L^{2}_{c}$. Then $u\in D(B^{n})$ if and
only if $u\in D(B^{n}-V)$. Moreover

$(B^{n}-V)u=Bu-Vu.$

Proof. Let $u\in D(B^{n})$. Then $u\in D(\mathfrak{a})$ and there exists $f\in
L^{2}_{c}$ such that

$\mathfrak{a}(u,v)=\int_{0}^{\infty}D_{y}uD_{y}\overline{v}y^{c}\,dy=\int_{0}^{\infty}f\overline{v}y^{c}\,dy$

for every $v\in H^{1}_{c}$. Setting $g=f+Vu\in L^{2}_{c}$ we have

$\mathfrak{a}_{V}(u,v)=\int_{0}^{\infty}(D_{y}uD_{y}\overline{v}+Vu\overline{v})y^{c}\,dy=\int_{0}^{\infty}(f+Vu)\overline{v}y^{c}\,dy$

for every $v\in H^{1}_{c}$ and, in particular, for every $v\in
D(\mathfrak{a}_{V})\subseteq H^{1}_{c}$. Therefore $u\in D(B^{n}-V)$.
Conversely, if $u\in D(B^{n}-V)$, then $u\in D(\mathfrak{a}_{V})$ and there
exists $g\in L^{2}_{c}$ such that

$\mathfrak{a}_{V}(u,v)=\int_{0}^{\infty}(D_{y}uD_{y}\overline{v}+Vu\overline{v})y^{c}\,dy=\int_{0}^{\infty}g\overline{v}y^{c}\,dy$

for every $v\in D(\mathfrak{a}_{V}).$ Setting $f=g-Vu\in L^{2}_{c}$ we have
that

$\mathfrak{a}(u,v)=\int_{0}^{\infty}f\overline{v}y^{c}\,dy$

for every $v\in D(\mathfrak{a}_{V})$, hence for every $v\in H^{1}_{c}$, by
Lemma 5.5.

###### Lemma 5.7

Let $u\in D(B^{n}-V)$ and $\eta$ be a smooth function such that $\eta=1$ for
$0\leq y\leq 1$ and $\eta=0$ for $y\geq 2$. Then $\eta u\in D(B^{n}-V)$ and

$(B^{n}-V)(\eta u)=\eta(B^{n}-V)u+2D_{y}\eta
D_{y}u+uD_{yy}\eta+cu\frac{D_{y}\eta}{y}.$

Proof. Let $u\in D(B^{n}-V)$, then $\eta u\in D(\mathfrak{a}_{V})$ and,
setting $f=(B^{n}-V)u$,

$\displaystyle\mathfrak{a}_{V}(\eta u,v)$
$\displaystyle=\int_{0}^{\infty}(D_{y}(\eta u)D_{y}\overline{v}+V\eta
u\overline{v})y^{c}\,dy=$
$\displaystyle\int_{0}^{\infty}(D_{y}uD_{y}(\eta\overline{v})+Vu\eta\overline{v}+uD_{y}\eta
D_{y}\overline{v}-D_{y}uD_{y}\eta\overline{v})y^{c}\,dy=$
$\displaystyle-\int_{0}^{\infty}\eta
f\overline{v}y^{c}\,dy-\int_{0}^{\infty}D_{y}uD_{y}\eta\overline{v}y^{c}\,dy+\int_{0}^{\infty}uD_{y}\eta
D_{y}\overline{v}y^{c}\,dy$ $\displaystyle=-\int_{0}^{\infty}\eta
f\overline{v}y^{c}\,dy-\int_{0}^{\infty}D_{y}uD_{y}\eta\overline{v}y^{c}\,dy-\int_{0}^{\infty}\overline{v}D_{y}(uD_{y}\eta
y^{c})\,dy$ $\displaystyle=-\int_{0}^{\infty}\eta
f\overline{v}y^{c}\,dy-2\int_{0}^{\infty}D_{y}uD_{y}\eta\overline{v}y^{c}\,dy-\int_{0}^{\infty}\overline{v}uD_{yy}\eta
y^{c}\,dy-\int_{0}^{\infty}\frac{cu}{y}\overline{v}D_{y}\eta y^{c}\,dy$

for every $v\in D(\mathfrak{a}_{V})$.

###### Lemma 5.8

Let $u\in D(B^{n}-V)$. Then there exists $(u_{k})\subseteq D(B^{n}-V)$ with
compact support such that $(u_{k})\to u$ in $D(B^{n}-V)$.

Proof. Let $\eta$ be a smooth function such that $\eta=1$ for $0\leq y\leq 1$
and $\eta=0$ for $y\geq 2$. Setting
$\eta_{k}(y)=\eta\left(\frac{y}{k}\right)$, by Lemma 5.7, $u_{k}=\eta_{k}u\in
D(B^{n}-V)$ and

$(B^{n}-V)(\eta_{k}u)=\eta_{k}(B^{n}-V)u+2D_{y}\eta_{k}D_{y}u+uD_{yy}\eta_{k}+\frac{cu}{y}D_{y}\eta_{k}.$

Then $u_{k}\to u$, $\eta_{k}(B^{n}-V)u\to(B^{n}-V)u$ in $L^{2}_{c}$ by
dominated convergence and, since $D_{y}\eta_{k}=0$ in $[0,1]$,

$\left|D_{y}\eta_{k}D_{y}u+uD_{yy}\eta_{k}+\frac{cu}{y}D_{y}\eta_{k}\right|\leq
C\left(\frac{|u|}{k}+\frac{|u|}{k^{2}}+\frac{|D_{y}u|}{k}\right)\chi_{[k,\infty[}\to
0.$

Lemma 5.7 shows that functions with compact support are a core for $B^{n}-V$.
To show that $\mathcal{D}$ is a core we need more information on the behavior
near $y=0$ of functions in the domain of $B^{n}-V$.

We start by recalling some well-known facts about the modified Bessel
functions $I_{\nu}$ and $K_{\nu}$ which constitute a basis of solutions of the
modified Bessel equation

$z^{2}\frac{d^{2}v}{dz^{2}}+z\frac{dv}{dz}-(z^{2}+\nu^{2})v=0,\quad\textrm{\emph{Re}\,}z>0.$

We recall that for $\textrm{\emph{Re}\,}z>0$ one has

$I_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu}\sum_{m=0}^{\infty}\frac{1}{m!\,\Gamma(\nu+1+m)}\left(\frac{z}{2}\right)^{2m},\quad
K_{\nu}(z)=\frac{\pi}{2}\frac{I_{-\nu}(z)-I_{\nu}(z)}{\sin\pi\nu},$

where limiting values are taken for the definition of $K_{\nu}$ when $\nu$ is
an integer. The basic properties of these functions we need are collected in
the following lemma, see e.g., [1, Sections 9.6 and 9.7].

###### Lemma 5.9

For $\nu>-1$, $I_{\nu}$ is increasing and $K_{\nu}$ is decreasing (when
restricted to the positive real half line). Moreover they satisfy the
following properties if $z\in\Sigma_{\pi/2-\varepsilon}$.

  * (i)

$I_{\nu}(z)\neq 0$ for every $\textrm{\emph{Re}\,}z>0$.

  * (ii)

$I_{\nu}(z)\approx\frac{1}{\Gamma(\nu+1)}\left(\frac{z}{2}\right)^{\nu},\quad\text{as
}|z|\to 0,\qquad I_{\nu}(z)\approx\frac{e^{z}}{\sqrt{2\pi
z}}(1+O(|z|^{-1}),\quad\text{as }|z|\to\infty$.

  * (iii)

If $\nu\neq 0$,
$K_{\nu}(z)\approx\frac{\nu}{|\nu|}\frac{1}{2}\Gamma(|\nu|)\left(\frac{z}{2}\right)^{-|\nu|},\qquad
K_{0}(z)\approx-\log z,\qquad\text{as }|z|\to 0$
$K_{\nu}(z)\approx\sqrt{\frac{\pi}{2z}}e^{-z},\quad\text{as }|z|\to\infty$.

  * (iv)

$I_{\nu}^{\prime}(z)=I_{\nu+1}(z)+\frac{\nu}{z}I_{\nu}(z)$,
$K_{\nu}^{\prime}(z)=-K_{\nu+1}(z)+\frac{\nu}{z}K_{\nu}(z)$, for every
$\textrm{\emph{Re}\,}z>0$.

Note that

$|I_{\nu}(z)|\simeq
C_{\nu,\epsilon}(1\wedge|z|)^{\nu+\frac{1}{2}}\frac{e^{Rez}}{\sqrt{|z|}},\qquad
z\in\Sigma_{\frac{\pi}{2}-\epsilon}$ (4)

for suitable constants $C_{\nu,\epsilon}>0$ which may be different in lower an
in the upper estimate.

The following estimates of the resolvent operator of $B^{n}-V$ is a
consequence of the domination property stated in Proposition 5.1.

###### Proposition 5.10

Let $c+1>0$ and $\lambda>0$. Then, for every $f\in L^{2}_{c}$,

$(\lambda-B^{n}+V)^{-1}f=\int_{0}^{\infty}G(\lambda,y,\rho)f(\rho)\rho^{c}d\rho$

with

$0\leq G(\lambda,y,\rho)\leq G^{n}(\lambda,y,\rho)$

where

$G^{n}(\lambda,y,\rho):=\begin{cases}y^{\frac{1-c}{2}}\rho^{\frac{1-c}{2}}\,I_{\frac{c-1}{2}}(\sqrt{\lambda}\,y)K_{{\frac{|1-c|}{2}}}(\sqrt{\lambda}\,\rho)\quad
y\leq\rho\\\\[6.45831pt]
y^{\frac{1-c}{2}}\rho^{\frac{1-c}{2}}\,I_{\frac{c-1}{2}}(\sqrt{\lambda}\,\rho)K_{\frac{|1-c|}{2}}(\sqrt{\lambda}\,y)\quad
y\geq\rho,\end{cases}$ (5)

is the integral kernel (taken with respect to the measure $\rho^{c}d\rho$) of
the operator $(\lambda-B^{n})^{-1}$.

Proof. Writing $(\lambda-B^{n}+V)^{-1}=\int_{0}^{\infty}e^{-\lambda
t}e^{t(B^{n}-V)}dt$ and using property (i) of Proposition 5.1 we get that

$\displaystyle|(\lambda-B^{n}+V)^{-1}f|\leq(\lambda-B^{n})^{-1}|f|,\qquad\lambda>0,\quad
f\in L^{2}_{c}.$

This yields the domination $G(\lambda,y,\rho)\leq G^{n}(\lambda,y,\rho)$ (the
existence of the kernel follows by [2, Proposition 1.9] as in Proposition
5.1). Formula (5) is proved in [11, Proposition 2.4].

We now prove local pointwise estimates for functions in the domain of
$B^{n}-V$.

###### Proposition 5.11

Let $c+1>0$. Then there exists $C>0$, independent of $V$, such that for every
$u\in D(B^{n}-V)$ and $0<y<1$

  * (i)

if $-1<c<3$

$\displaystyle|u(y)|\leq
C\left(\|u\|_{L^{2}_{c}}+\|(B-V)u\|_{L^{2}_{c}}\right),$

  * (ii)

if $c=3$

$\displaystyle|u(y)|\leq
C\left(\|u\|_{L^{2}_{c}}+\|(B-V)u\|_{L^{2}_{c}}\right)|\log y|^{\frac{1}{2}},$

  * (iii)

if $c>3$

$\displaystyle|u(y)|\leq
C\left(\|u\|_{L^{2}_{c}}+\|(B-V)u\|_{L^{2}_{c}}\right)y^{\frac{3-c}{2}}.$

Proof. Let $u\in D(B^{n}-V)$ and $f=u-(B^{n}-V)u\in L^{2}_{c}$ so that
$u=(I-B^{n}+V)^{-1}f$. Let us distinguish between the following cases and
always take $0<y<1$.

  * (i)

If $-1<c<1$, Lemma 5.9 implies that for $y\leq 1$

$\displaystyle G(1,y,\rho)\simeq\begin{cases}1,&\quad\rho<1,\\\\[4.30554pt]
\rho^{-\frac{c}{2}}e^{-\rho},&\quad 1<\rho.\end{cases}$

Then one has

$\displaystyle|u(y)|$
$\displaystyle\leq\int_{0}^{\infty}G(1,y,\rho)|f(\rho)|\rho^{c}d\rho\leq
C\left(\int_{0}^{1}|f(\rho)|\rho^{c}d\rho+\int_{1}^{\infty}\rho^{-\frac{c}{2}}e^{-\rho}|f(\rho)|\,\rho^{c}d\rho\right)$
$\displaystyle\leq
C\left(\|f\|_{L^{2}_{c}((0,1))}+\|\rho^{-\frac{c}{2}}e^{-\rho}\|_{L^{2}_{c}((1,\infty))}\|f\|_{L^{2}_{c}((1,\infty))}\right)\leq
C\|f\|_{L^{2}_{c}}.$

  * (ii)

If $c=1$, Lemma 5.9 gives for $y\leq 1$

$\displaystyle G(1,y,\rho)\simeq\begin{cases}|\log
y|\leq|\log\rho|,&\quad\rho<y<1,\\\\[4.30554pt] |\log\rho|,&\quad
y<\rho<1,\\\\[4.30554pt] \rho^{-\frac{1}{2}}e^{-\rho},&\quad
1<\rho.\end{cases}$

Then analogously

$\displaystyle|u(y)|$ $\displaystyle\leq
C\left(\int_{0}^{1}|\log\rho||f(\rho)|\rho
d\rho+\int_{1}^{\infty}\rho^{\frac{1}{2}}e^{-\rho}|f(\rho)|d\rho\right)$
$\displaystyle\leq
C\left(\|\log\rho\|_{L^{2}_{c}((0,1))}\|f\|_{L^{2}_{c}((0,1))}+\|\rho^{-\frac{1}{2}}e^{-\rho}\|_{L^{2}_{c}((1,\infty))}\|f\|_{L^{2}_{c}((1,\infty))}\right)\leq
C\|f\|_{L^{2}_{c}}.$

  * (iii)

Let now $1<c$. Then Lemma 5.9 implies that for $y\leq 1$

$\displaystyle
G(1,y,\rho)\simeq\begin{cases}y^{1-c}\leq\rho^{1-c},&\quad\rho<y<1,\\\\[4.30554pt]
\rho^{1-c},&\quad y<\rho<1,\\\\[4.30554pt] \rho^{-\frac{c}{2}}e^{-\rho},&\quad
1<\rho.\end{cases}$

If $c<3$ one has

$\displaystyle|u(y)|$ $\displaystyle\leq
C\left(\int_{0}^{1}\rho^{1-c}|f(\rho)|\rho^{c}d\rho+\int_{1}^{\infty}\rho^{-\frac{c}{2}}e^{-\rho}|f(\rho)|\rho^{c}d\rho\right)$
$\displaystyle\leq
C\left(\|\rho^{1-c}\|_{L^{2}_{c}((0,1))}\|f\|_{L^{2}_{c}((0,1))}+\|\rho^{-\frac{c}{2}}e^{-\rho}\|_{L^{2}_{c}((1,\infty))}\|f\|_{L^{2}_{c}((1,\infty))}\right)\leq
C\|f\|_{L^{2}_{c}}.$

If $c=3$ then we get

$\displaystyle|u(y)|$ $\displaystyle\leq
C\left(y^{-2}\int_{0}^{y}|f(\rho)|\rho^{3}d\rho+\int_{y}^{1}\rho^{-2}|f(\rho)|\rho^{3}d\rho+\int_{1}^{\infty}\rho^{-\frac{3}{2}}e^{-\rho}|f(\rho)|\rho^{3}d\rho\right)$
$\displaystyle\leq
C\|f\|_{L^{2}_{c}}\left(y^{-2}\left(\int_{0}^{y}\rho^{3}d\rho\right)^{\frac{1}{2}}+\left(\int_{y}^{1}\rho^{-4}\rho^{3}d\rho\right)^{\frac{1}{2}}+\|\rho^{-\frac{3}{2}}e^{-\rho}\|_{L^{2}_{c}((1,\infty))}\right)$
$\displaystyle\leq C\|f\|_{L^{2}_{c}}\left(1+|\log y|^{\frac{1}{2}}\right)$

and finally if $c>3$

$\displaystyle|u(y)|$ $\displaystyle\leq
C\left(y^{1-c}\int_{0}^{y}|f(\rho)|\rho^{c}d\rho+\int_{y}^{1}\rho^{1-c}|f(\rho)|\rho^{c}d\rho+\int_{1}^{\infty}\rho^{-\frac{c}{2}}e^{-\rho}|f(\rho)|\rho^{c}d\rho\right)$
$\displaystyle\leq
C\|f\|_{L^{2}_{c}}\left(y^{1-c}\left(\int_{0}^{y}\rho^{c}d\rho\right)^{\frac{1}{2}}+\left(\int_{y}^{1}\rho^{2-2c}\rho^{c}d\rho\right)^{\frac{1}{2}}+\|\rho^{-\frac{c}{2}}e^{-\rho}\|_{L^{2}_{c}((1,\infty))}\right)$
$\displaystyle\leq C\|f\|_{L^{2}_{c}}y^{\frac{3-c}{2}}.$

We can now show that, under stronger assumptions, the potential term $V$ can
be seen as a perturbation of $B^{n}$ near 0, that is $Vu\in L^{2}_{c}$ for
every $u\in D(B_{n})$ having compact support. In particular we prove that
$\mathcal{D}$ is a core for $B^{n}-V$.

###### Proposition 5.12

Let $c+1>0$ and assume that

  * (i)

$c<3$ and $V\in L^{2}_{loc}\left(\mathbb{R}^{+},y^{c}dy\right)$ or

  * (ii)

$c=3$ and $V|\log y|^{\frac{1}{2}}\in
L^{2}_{loc}\left(\mathbb{R}^{+},y^{c}dy\right)$ or

  * (iii)

$c>3$ and $Vy^{\frac{3-c}{2}}\in
L^{2}_{loc}\left(\mathbb{R}^{+},y^{c}dy\right)$.

If $\mathcal{C}_{r}:=\\{u\in L^{2}_{c}:\mbox{supp }u\subseteq[0,r]\\}$ then
$D(B^{n}-V)\cap\mathcal{C}_{r}=D(B^{n})\cap\mathcal{C}_{r}$ with equivalence
of norms

$\displaystyle\|u\|_{D(B^{n}-V)}$
$\displaystyle\simeq\|u\|_{D(B^{n})},\quad\forall u\in
D(B^{n})\cap\mathcal{C}_{r}.$

Finally,

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)):u\ {\rm constant\ in\ a\
neighborhood\ of}\ 0\right\\}$

is a core for $B^{n}-V$.

Proof. Let $u\in\mathcal{C}_{r}$. Then the hypotheses on $V$ and Proposition
5.11 imply that $Vu\in L^{2}_{c}$ and $\|Vu\|_{L^{2}_{c}}\leq
C\|u-(B-V)u\|_{L^{2}_{c}}$. Then by Lemma 5.6 $u\in D(B^{n}-V)$ if and only if
$u\in D(B^{n})$. This show the equality
$D(B^{n}-V)\cap\mathcal{C}_{r}=D(B^{n})\cap\mathcal{C}_{r}$. Using Proposition
5.11 again we also have $\|Vu\|_{L^{2}_{c}}\leq C_{1}\|u-Bu\|_{L^{2}_{c}}$ for
any $u\in D(B^{n})\cap\mathcal{C}_{r}$, which proves the equivalence of the
graph norms. Finally, let $u\in D(B^{n}-V)$. We have to prove that $u$ can be
approximated in the graph norm with functions belonging to $\mathcal{D}$.
Using Lemma 5.8 we may suppose, without any loss of generality, that
$\mbox{supp\,}u\subseteq(0,r)$. Then, by Proposition 4.3, there exist
$(u_{n})\subset\mathcal{D}$ such that $u_{n}\to u$ in the graph norm
$\|\cdot\|_{D(B^{n})}$. We may also assume, after multiplying by a suitable
cut-off function, that $\mbox{supp\,}u_{n}\subseteq(0,2r)$ for every $n$. Then
the previous point implies that $Vu_{n}\to Vu$ in $L^{2}_{c}$, too.

##  6 The operator $y^{\alpha}B^{n}-V$ in $L^{2}_{c-\alpha}$

We consider now for $c\in\mathbb{R}$, $\alpha<2$, and $0\leq V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\,dy\right)$ the operator

$y^{\alpha}B^{n}-V=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)-V$

in the space $L^{2}_{c-\alpha}$. As in Section 4 we use the isometry
$T_{-\frac{\alpha}{2}}u(y)=\left|1-\frac{\alpha}{2}\right|^{\frac{1}{p}}u(y^{1-\frac{\alpha}{2}})$,

$T_{-\frac{\alpha}{2}}:L^{2}_{\tilde{c}}\to
L^{2}_{c-\alpha},\qquad\tilde{c}=\frac{c-\frac{\alpha}{2}}{1-\frac{\alpha}{2}},$

under which $y^{\alpha}B-V$ becomes similar to

$\displaystyle
T_{-\frac{\alpha}{2}}^{-1}\Big{(}y^{\alpha}B-V\Big{)}T_{-\frac{\alpha}{2}}=\left(1-\frac{\alpha}{2}\right)^{2}\left(\tilde{B}-\tilde{V}\right)$

where $\tilde{B}=D_{yy}+\frac{\tilde{c}}{y}D_{y}$ and
$\tilde{V}(y)=(1-\frac{\alpha}{2})^{-2}V\left(y^{\frac{2}{2-\alpha}}\right)\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{\tilde{c}}\,dy\right)$.

Defining

$D(y^{\alpha}B^{n}-V):=T_{-\frac{\alpha}{2}}\left(D(\tilde{B^{n}}-\tilde{V})\right)$

one obtains that, when $c>-1+\alpha$, $y^{\alpha}B^{n}-V$ generates a
contractive analytic semigroup $\left\\{e^{z(y^{\alpha}B^{n}-V)}:\
z\in\mathbb{C}_{+}\right\\}$ in $L^{2}_{c-\alpha}$ which satisfies

$\displaystyle
e^{z\left(y^{\alpha}B-V\right)}=T_{-\frac{\alpha}{2}}\left(e^{z\left(1-\frac{\alpha}{2}\right)^{2}\left(\tilde{B}-\tilde{V}\right)}\right)T_{-\frac{\alpha}{2}}^{-1}.$
(6)

We state the properties obtained so far, together with a density result which
is a restating of Proposition 5.12 under the isometry $T_{-\frac{\alpha}{2}}$.

###### Proposition 6.1

Let $c+1-\alpha>0$ and $0\leq V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)$. Then the operator
$y^{\alpha}B^{n}-V$ generates a contractive analytic semigroup in
$L^{2}_{c-\alpha}$. If, in addition,

  * (i)

$c<3-\alpha$ and $V\in L^{2}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)$ or

  * (ii)

$c=3-\alpha$ and $V|\log y|^{\frac{1}{2}}\in
L^{2}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)$ or

  * (iii)

$c>3-\alpha$ and $Vy^{\frac{3-c-\alpha}{2}}\in
L^{2}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)$

then

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)):u\ {\rm constant\ in\ a\
neighborhood\ of}\ 0\right\\}$

is a core for $y^{\alpha}B^{n}-V$ in $L^{2}_{c-\alpha}$.

###### Remark 6.2

If $V(y)=y^{\alpha}$, then $V$ always satisfies (ii) and (iii) when $c\geq
3-\alpha$. Instead, if $c<3-\alpha$, we need $c+1-|\alpha|>0$.

Let $\mathfrak{a}_{\tilde{V}}$ be the form in $L^{2}_{\tilde{c}}$, defined in
(5.1), associated to $\tilde{B}^{n}-\tilde{V}$. In $L^{2}_{c-\alpha}$ we
introduce the form $\mathfrak{a}_{\alpha,V}$ which is the image of
$\mathfrak{a}_{\tilde{V}}$ under the isometry $T_{0,-\frac{\alpha}{2}}$, that
is

$\displaystyle\mathfrak{a}_{\alpha,V}(u,v):=\mathfrak{a}_{\tilde{V}}\left(T_{-\frac{\alpha}{2}}^{-1}u,T_{-\frac{\alpha}{2}}^{-1}v\right)=\int_{\mathbb{R}_{+}}\left(y^{\alpha}D_{y}uD_{y}\overline{v}+Vu\overline{v}\right)y^{c-\alpha}\,dy,$
$\displaystyle
D(\mathfrak{a}_{\alpha,V}):=T_{-\frac{\alpha}{2}}D(\mathfrak{a}_{\tilde{V}})=\left\\{u\in
L^{2}_{c-\alpha}:u^{\prime}\in L^{2}_{c}\right\\}\cap
L^{2}\left(\mathbb{R}^{+},Vy^{c-\alpha}\,dy\right).$ (7)

To keep consistency of notation we often write
$\mathfrak{a}_{0,V}=\mathfrak{a}_{V}$. By construction, $y^{\alpha}B^{n}-V$ is
the operator associated to the form $\mathfrak{a}_{\alpha,V}$ in
$L^{2}_{c-\alpha}$

$\displaystyle D(y^{\alpha}B^{n}-V)$ $\displaystyle=\\{u\in
D(\mathfrak{a}_{\alpha,V}):\exists f\in L^{2}_{c-\alpha}\quad{\rm such\ that}\
$ $\displaystyle\hskip
107.63855pt\mathfrak{a}_{\alpha,V}(u,v)=\int_{0}^{\infty}f\overline{v}y^{c-\alpha}\,dy\
{\rm for\ every}\ v\in D(\mathfrak{a}_{\alpha,V})\\},$ $\displaystyle
y^{\alpha}B^{n}u-Vu$ $\displaystyle=-f.$

The next lemma, which follows from the considerations above, will be used
later to relate the resolvents of $y^{\alpha}B^{n}-y^{\alpha}$ and
$B^{n}-y^{-\alpha}$.

###### Lemma 6.3

Let $\mathfrak{a}_{\alpha,y^{\alpha}}$ and $\mathfrak{a}_{y^{-\alpha}}$ be the
sesquilinear forms associated respectively to the operator
$y^{\alpha}B^{n}-y^{\alpha}$ in $L^{2}_{c-\alpha}$ and $B^{n}-y^{-\alpha}$ in
$L^{2}_{c}$. Then

$\displaystyle\mathfrak{a}_{\alpha,y^{\alpha}}(u,v)=\int_{\mathbb{R}_{+}}\left(D_{y}uD_{y}\overline{v}+u\overline{v}\right)y^{c}\,dy,\qquad\mathfrak{a}_{y^{-\alpha}}(u,v)=\int_{\mathbb{R}_{+}}\left(D_{y}uD_{y}\overline{v}+y^{-\alpha}u\overline{v}\right)y^{c}\,dy.$

on the common form domain

$\displaystyle
D(\mathfrak{a}_{\alpha,y^{\alpha}})=D(\mathfrak{a}_{y^{-\alpha}})=\left\\{u\in
L^{2}_{c-\alpha}\cap L^{2}_{c}:u^{\prime}\in L^{2}_{c}\right\\}$

Note that the above operators act in different Hilbert spaces; in particular
their domains are different. However, the form domains coincide.

##  7 The operator $y^{\alpha}B^{n}-V$ in $L^{p}_{m}$

Here we investigate properties of $y^{\alpha}B-V$, $\alpha<2$, in $L^{p}_{m}$
when $0<\frac{m+1}{p}<c+1-\alpha$.

We introduce the family of integral operators
$\left(S_{\alpha}^{\beta}(t)\right)_{t>0}$ on $L^{p}_{m}$

$S_{\alpha}^{\beta}(t)f(y):=t^{-\frac{1}{2}}\,\int_{\mathbb{R}_{+}}\left(\frac{\rho}{t^{\frac{1}{2-\alpha}}}\wedge
1\right)^{-\beta+\frac{\alpha}{2}}\exp\left(-\frac{|y^{1-\frac{\alpha}{2}}-\rho^{1-\frac{\alpha}{2}}|^{2}}{\kappa
t}\right)f(\rho)\rho^{-\frac{\alpha}{2}}\,d\rho$

and note that

$S_{\alpha}^{\beta}(t)=T_{-\frac{\alpha}{2}}\circ
S_{0}^{\tilde{\beta}}(t)\circ
T_{-\frac{\alpha}{2}}^{-1},\qquad\tilde{\beta}=\frac{\beta-\frac{\alpha}{2}}{1-\frac{\alpha}{2}}.$

As usual
$T_{-\frac{\alpha}{2}}u(y)=\left|1-\frac{\alpha}{2}\right|^{\frac{1}{p}}u(y^{1-\frac{\alpha}{2}})$
is an isometry from $L^{p}_{\tilde{m}}$ onto $L^{p}_{m}$,
$\tilde{m}=\frac{m+\frac{\alpha}{2}}{1-\frac{\alpha}{2}}$. Here $\kappa$ is a
positive constant but we omit the dependence on it. The following result has
been proved for $\alpha=0$ in [11].

###### Lemma 7.1

Let $m\in\mathbb{R}$ and let $p\in(1,\infty)$ such that
$0<\frac{m+1}{p}<1-\alpha-\beta$. The families
$\left(S_{\alpha}^{\beta}(t)\right)_{t\geq 0}$ and
$\\{\Gamma(\lambda)=\int_{0}^{\infty}\lambda e^{-\lambda
t}S_{\alpha}^{\beta}(t)\,dt,\quad\lambda>0\\}$ are $\mathcal{R}$-bounded in
$L^{p}_{m}$.

Proof. Since the $\mathcal{R}$-boundedness is preserved under isometries, from
$S_{\alpha}^{\beta}(t)=T_{-\frac{\alpha}{2}}\circ
S_{0}^{\tilde{\beta}}(t)\circ T_{-\frac{\alpha}{2}}^{-1}$ we may assume that
$\alpha=0$ (note that $0<\frac{m+1}{p}<-\beta+1-\alpha$ is equivalent to
$0<\frac{\tilde{m}+1}{p}<-\tilde{\beta}+1$). The first result is then a
consequence of [11, Theorem 7.7]. The family

$\Gamma(\lambda)=\int_{0}^{\infty}\lambda e^{-\lambda
t}S_{\alpha}^{\beta}(t)\,dt,\quad\lambda>0$

is $\mathcal{R}$-bounded by [6, Corollary 2.14].

We can now prove our main results for the operator $y^{\alpha}B-V$.

###### Theorem 7.2

Let $0\leq V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\,dy\right)$. For
any $p\in(1,\infty)$ such that $0<\frac{m+1}{p}<c+1-\alpha$, the semigroup
$e^{z(y^{\alpha}B^{n}-V)}$ initially defined on $L^{2}_{c-\alpha}$ extends to
a bounded analytic semigroup on $L^{p}_{m}$ of angle $\pi/2$ which consists of
integral operators. Moreover the generated semigroup has maximal regularity
and the following properties hold.

  * (i)

For every $\epsilon>0$ there exist $C=C(\epsilon,\alpha)>0$ (independent of
$V$) such that

$\displaystyle\left|e^{z(y^{\alpha}B^{n}-V)}f\right|\leq
CS_{\alpha}^{-c}(|z|)|f|,\quad f\in L^{p}_{m},\quad|\arg
z|<\frac{\pi}{2}-\epsilon.$

  * (ii)

For every $\epsilon>0$ the families of operators

$\displaystyle\left\\{e^{z(y^{\alpha}B^{n}-V)}:\;z\in\Sigma_{\frac{\pi}{2}-\epsilon},\;0\leq
V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)\right\\},$
$\displaystyle\left\\{\lambda\left(\lambda-y^{\alpha}B^{n}+V\right)^{-1}:\;\lambda\in\Sigma_{\pi-\epsilon}:\;0\leq
V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right)\right\\}$

are $\mathcal{R}$-bounded in $L^{p}_{m}$.

Proof.  By Proposition 5.3 and (6), (i) holds for any $f\in L^{2}_{c-\alpha}$.
The boundedness of $e^{z(y^{\alpha}B^{n}-V)}$ in $L^{p}_{m}$ follows from the
previous lemma and (i) extends to $L^{p}_{m}$. The semigroup law is inherited
from $L^{2}_{c-\alpha}$ via a density argument and we have only to prove the
strong continuity at $0$. Using the isometry $T_{-\frac{\alpha}{2}}$, we may
suppose that $\alpha=0$. Let $f,g\in C_{c}^{\infty}(0,\infty)$. Then as $z\to
0$, $z\in\Sigma_{\frac{\pi}{2}-\epsilon}$,

$\int_{0}^{\infty}(e^{z(B^{n}-V)}f)\,g\,y^{m}dy=\int_{0}^{\infty}(e^{z(B^{n}-V)}f)\,g\,y^{m-c}y^{c}dy\to\int_{0}^{\infty}fgy^{m-c}y^{c}dy=\int_{0}^{\infty}fgy^{m}dy,$

by the strong continuity of $e^{z(B^{n}-V)}$ in $L^{2}_{c}$. By density and
uniform boundedness of the family
$(e^{z(B^{n}-V)})_{z\in\Sigma_{\frac{\pi}{2}-\epsilon}}$ this holds for every
$f\in L^{p}_{m}$, $g\in L^{p^{\prime}}_{m}$. The semigroup is then weakly
continuous, hence strongly continuous.

The $\mathcal{R}$-boundedness of e${}^{z(y^{\alpha}B^{n}-V)}$ follows then by
domination from Lemma 7.1, see Corollary 2.2. To prove the
$\mathcal{R}$-boundedness of the resolvent family, for
$\lambda\in\Sigma_{\pi-\epsilon}\setminus\\{0\\}$ let
$\theta=\frac{|\mbox{arg}\lambda|}{\mbox{arg}\lambda}\left(\frac{\pi}{2}-\frac{\epsilon}{2}\right)$
so that
$\mu:=e^{-i\theta}\lambda\in\Sigma_{\frac{\pi}{2}-\frac{\epsilon}{2}}$. Then

$\displaystyle\left|\lambda\left(\lambda-y^{\alpha}B^{n}+V\right)^{-1}f\right|$
$\displaystyle=\left|\mu\left(\mu-e^{-i\theta}(y^{\alpha}B^{n}-V)\right)^{-1}f\right|=\left|\int_{0}^{\infty}\mu
e^{-\mu t}e^{-i\theta t(y^{\alpha}B^{n}-V)}f\,dt\right|$ $\displaystyle\leq
C\int_{0}^{\infty}|\mu|e^{-\textrm{\emph{Re}\,}\mu
t}S^{-c}_{\alpha}(t)|f|\,dt\leq
C\int_{0}^{\infty}|\lambda|e^{-|\lambda|\sin\frac{\epsilon}{2}t}S^{-c}_{\alpha}(t)|f|\,dt.$

The $\mathcal{R}$-boundedness of the second family in (ii) now follows from
[6, Corollary 2.14] and the maximal regularity of the semigroup from Theorem
2.3.

In our investigation of degenerate Nd problems, see [13], we need also a
weaker version of the result above for potentials having non-negative real
part. We formulate it in the next proposition.

###### Proposition 7.3

Let $V\in L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\,dy\right)$ be a
potential having non-negative real part. Then for any $1<p<\infty$ such that
$0<\frac{m+1}{p}<c+1-\alpha$, $y^{\alpha}B^{n}-V$ generates a
$C_{0}$-semigroup on $L^{p}_{m}$. The generated semigroup consists of integral
operators and the following estimates hold

$\displaystyle\left|e^{t(y^{\alpha}B^{n}-V)}f\right|$ $\displaystyle\leq
e^{ty^{\alpha}B^{n}}|f|,\hskip 55.97205ptf\in L^{p}_{m},\quad t\geq 0$

In particular the families of operators

$\displaystyle\left\\{e^{t(y^{\alpha}B^{n}-V)}:\;t\geq 0,\;V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right),\
\textrm{\emph{Re}\,}V\geq 0\right\\},$
$\displaystyle\left\\{\lambda\left(\lambda-y^{\alpha}B^{n}+V\right)^{-1}:\;\lambda>0,\;V\in
L^{1}_{loc}\left(\mathbb{R}^{+},y^{c-\alpha}\right),\
\textrm{\emph{Re}\,}V\geq 0\right\\}$

are $\mathcal{R}$-bounded in $L^{p}_{m}$.

Proof. Using the isometry $T_{0,-\frac{\alpha}{2}}$ we may assume that
$\alpha=0$. Let us treat first the symmetric case in $L^{2}_{c}$. The
generation results can be proved as in Proposition 5.1 (where we assumed
$V\geq 0$). If $\mathfrak{a}$ is the form associated with $B^{n}$, then
$B^{n}-V$ is associated to ${\mathfrak{a}}_{V}:=\mathfrak{a}(u,v)+\langle
Vu,v\rangle_{L^{2}_{c}}$ and, by the standard theory on sesquilinear forms,
$B^{n}-V$ generates a $C_{0}$-semigroup on $L^{2}_{c}$.

The domination properties follow from [15, Theorem 2.21]. Let $u,v\in
D\left({\mathfrak{a}}_{V}\right)=D(\mathfrak{a})\cap
L^{2}\left(\mathbb{R}^{+},|V|y^{c}\,dy\right)$ such that $u\bar{v}\geq 0$.
Since $e^{tB^{n}}$ is positive one has
$\textrm{\emph{Re}\,}\mathfrak{a}(u,v)\geq\mathfrak{a}(|u|,|v|)$. Moreover

$\displaystyle\textrm{\emph{Re}\,}{\mathfrak{a}}_{V}(u,v)=\textrm{\emph{Re}\,}\mathfrak{a}(u,v)+\int_{0}^{\infty}\textrm{\emph{Re}\,}V\,u\bar{v}\,y^{c}dy\geq\textrm{\emph{Re}\,}\mathfrak{a}(|u|,|v|)$

which by [15, Theorem 2.21] again implies the stated domination of the
generated semigroups (one easily verifies that
$D\left({\mathfrak{a}}_{V}\right)$ is an ideal of $D(\mathfrak{a})$ since this
last is an ideal in itself, by the positivity of $e^{tB^{n}}$, see [15,
Proposition 2.20]). The extrapolation on $L^{p}_{m}$ follows as in Theorem
7.2. The domination of the resolvent is a straightforward consequence of that
of the semigroup. The $\mathcal{R}$-boundedness of the semigroup follows by
domination from the $\mathcal{R}$-boundedness of $(e^{tB^{n}})_{t\geq 0}$
proved in Theorem 7.2. The $\mathcal{R}$-boundedness of the resolvent follows
as in Theorem 7.2.

##  8 The operator $y^{\alpha}B^{n}-y^{\alpha}$

We end the paper by thoroughly investigating the special case
$V(y)=y^{\alpha}$, keeping $\alpha<2$. We prove, in particular, that the
domain of $y^{\alpha}B-V$ is $D(y^{\alpha}B)\cap D(V)$, under slightly more
restrictive hypotheses than those of Theorem 7.2.

As explained in the Introduction this case plays a crucial role in [13] in the
investigation of the degenerate operators

$\displaystyle\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right),\qquad\alpha_{1},\alpha_{2}\in\mathbb{R}$

in the spaces $L^{p}\left(\mathbb{R}^{N+1}_{+},y^{m}dxdy\right)$. In
particular we prove in Propositions 8.3 and 8.4 that the multipliers

$\displaystyle\xi\in\mathbb{R}^{N}\to N_{\lambda}(\xi)$
$\displaystyle=\lambda(\lambda-y^{\alpha}By+y^{\alpha}|\xi|^{2})^{-1},$
$\displaystyle\xi\in\mathbb{R}^{N}\to M_{\lambda}(\xi)$
$\displaystyle=y^{\alpha}|\xi|^{2}(\lambda-y^{\alpha}By+y^{\alpha}|\xi|^{2})^{-1}$

satisfy the hypothesis of Theorem 2.4.

We start with the following Lemma.

###### Lemma 8.1

Assume that $c+1>0$ and $c+1-\alpha>0$, that is $B^{n}$ generates a
$C_{0}$-semigroup in $L^{2}_{c}$ and $y^{\alpha}B^{n}$ generates a
$C_{0}$-semigroup in $L^{2}_{c-\alpha}$. If $\lambda\in\mathbb{C}^{+}$ and
$\mu>0$, then

$\displaystyle\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}f=\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}\left(\frac{f}{y^{\alpha}}\right),\qquad\forall
f\in C^{\infty}_{c}((0,\infty)).$

Proof. Under the assumptions $y^{\alpha}B^{n}-\mu y^{\alpha}$ and
$B^{n}-\lambda y^{-\alpha}$ generate a semigroup on $L^{2}_{c-\alpha}$ and
$L^{2}_{c}$, respectively, see Theorem 7.2. Since
$\textrm{\emph{Re}\,}\lambda>0$, $\mu>0$ both resolvents are well defined but
map to different spaces.

Let $\mathfrak{a}_{\alpha,\mu y^{\alpha}}$ , $\mathfrak{a}_{\lambda
y^{-\alpha}}$ be the forms associated to $y^{\alpha}B^{n}-\mu y^{\alpha}$ in
$L^{2}_{c-\alpha}$ and $B^{n}-\lambda y^{-\alpha}$ in $L^{2}_{c}$

$\displaystyle\mathfrak{a}_{\alpha,\mu
y^{\alpha}}(u,v)=\int_{\mathbb{R}_{+}}\left(D_{y}uD_{y}\overline{v}+\mu
u\overline{v}\right)y^{c}\,dy,\qquad\mathfrak{a}_{\lambda
y^{-\alpha}}(u,v)=\int_{\mathbb{R}_{+}}\left(D_{y}uD_{y}\overline{v}+\lambda
y^{-\alpha}u\overline{v}\right)y^{c}\,dy.$

By Lemma 6.3 they are defined on the common domain

$\displaystyle\mathcal{F}:=\left\\{u\in L^{2}_{c-\alpha}\cap
L^{2}_{c}:u^{\prime}\in L^{2}_{c}\right\\}$

Given $f\in C^{\infty}_{c}((0,\infty))$ let
$u:=\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}\left(\frac{f}{y^{\alpha}}\right)$.
In order to prove that the equality $u=\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}f$ holds, we have to show that $u\in\mathcal{F}$ and
that for every $v\in\mathcal{F}$, $u$ satisfies the weak equality

$\displaystyle\int_{0}^{\infty}f\overline{v}y^{c-\alpha}\,dy$
$\displaystyle=\int_{0}^{\infty}\lambda
u\overline{v}y^{c-\alpha}\,dy+\mathfrak{a}_{\alpha,\mu
y^{\alpha}}(u,v)=\int_{0}^{\infty}(\lambda
y^{-\alpha}u\overline{v}+D_{y}uD_{y}\overline{v}+\mu u\overline{v})y^{c}\,dy.$
(8)

By construction $u$ is in the domain of $B^{n}-\lambda y^{-\alpha}$ which is
contained in $\mathcal{F}$ and satisfies

$\displaystyle\int_{0}^{\infty}\frac{f}{y^{\alpha}}\overline{v}y^{c}\,dy$
$\displaystyle=\int_{0}^{\infty}\mu
u\overline{v}y^{c}\,dy+\mathfrak{a}_{\alpha,\lambda
y^{-\alpha}}(u,v)=\int_{0}^{\infty}(\mu
u\overline{v}+D_{y}uD_{y}\overline{v}+\lambda
y^{-\alpha}u\overline{v})y^{c}\,dy,$

which is the same as (8).

In the next results we relate the resolvent of $y^{\alpha}B^{n}-y^{\alpha}$
with that of $B^{n}-\frac{1}{y^{\alpha}}$. We shall assume both the conditions
$0<\frac{m+1}{p}<c+1-\alpha$ and $-\alpha<\frac{m+1}{p}<c+1-\alpha$ (that is
$\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$). The first guarantees that
$y^{\alpha}B^{n}$ is a generator in $L^{p}_{m}$ and the second that $B^{n}$ is
a generator in $L^{p}_{m+\alpha p}$.

###### Corollary 8.2

Assume that $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. If
$\lambda\in\mathbb{C}^{+}$ and $\mu>0$, then

  * (i)

for every $f\in L^{p}_{m}$

$\displaystyle\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}f=\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}\left(\frac{f}{y^{\alpha}}\right)\in
L^{p}_{m+\alpha p}\cap L^{p}_{m};$

  * (ii)

the operator $y^{\alpha}\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}$ is bounded in $L^{p}_{m}$;

  * (iii)

the operator
$\frac{1}{y^{\alpha}}\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}$
is bounded in $L^{p}_{m+\alpha p}$.

Proof. Equality (i) is proved in Lemma 8.1 for any $f\in
C^{\infty}_{c}((0,\infty))$. Since $\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}$ is bounded form $L^{p}_{m}$ into itself and
$\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}\left(\frac{\cdot}{y^{\alpha}}\right)$
is bounded from $L^{p}_{m}$ to $L^{p}_{m+\alpha p}$, by density, (i) holds for
every $f\in L^{p}_{m}$. Parts (ii), (iii) are consequence of (i).

In the next propositions we prove the boundedness of the multipliers
$N_{\lambda}$ and $M_{\lambda}$. We start with $M_{\lambda}$, used in [13] to
characterize the domain of $\mathcal{L}=y^{\alpha}(\Delta_{x}+B_{y})$.

###### Proposition 8.3

Assume that $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ and let for
$\lambda\in\mathbb{C}^{+}$, $\xi\neq 0$

$M_{\lambda}(\xi)=|\xi|^{2}y^{\alpha}\left(\lambda-y^{\alpha}B^{n}+|\xi|^{2}y^{\alpha}\right)^{-1}\in\mathcal{B}(L^{p}_{m}).$

Then the family
$\left\\{|\xi|^{|\beta|}D^{\beta}_{\xi}(M_{\lambda})(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\beta|\leq N,\lambda\in\mathbb{C}^{+}\right\\}$ is $\mathcal{R}$-bounded in
$L^{p}_{m}$.

Proof. Let $m_{\lambda}(\mu)=\mu y^{\alpha}\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}$, $\mu>0$.

Using Lemma 2.5 it suffices to show that the
family$\left\\{\mu^{k}D^{k}_{\mu}(m_{\lambda})(\mu):\mu>0,\ k\leq
N,\lambda\in\mathbb{C}^{+}\right\\}$ is $\mathcal{R}$-bounded in $L^{p}_{m}$.

The map $Tf=f/y^{\alpha}$ is an isometry of $L^{p}_{m}$ onto $L^{p}_{m+\alpha
p}$ and by Corollary 8.2,

$m_{\lambda}(\mu)=T^{-1}\mu\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}T.$

The family

$\left\\{\mu^{k}D^{k}_{\mu}(\Gamma_{\lambda})(\mu):\mu>0,\ k\leq
N,\lambda\in\mathbb{C}^{+}\right\\},\qquad\Gamma_{\lambda}(\mu)=\mu\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}$

is $\mathcal{R}$-bounded in $L^{p}_{m+\alpha p}$. Indeed,

$\Gamma_{\lambda}(\mu)=\int_{0}^{\infty}\mu e^{-\mu
t}e^{t(B^{n}-\frac{\lambda}{y^{\alpha}})}\,dt$

and $\left\\{e^{t(B^{n}-\frac{\lambda}{y^{\alpha}})}:\,t\geq 0,\
\lambda\in\mathbb{C}^{+}\right\\}$ is $\mathcal{R}$-bounded in
$L^{p}_{m+\alpha p}$, by Theorem 7.3. The $\mathcal{R}$-boundedness of the
derivatives follows either by the resolvent equation or by differentiating the
last equation under the integral and using [6, Corollary 2.14]. In fact, if
$h(\mu,t)=\mu e^{-\mu t}$, then

$\mu^{k}\int_{0}^{\infty}|D^{k}_{\mu}h(\mu,t)|dt\leq C_{k},\quad\mu>0.$

Next we deal with $N_{\lambda}$ which is crucial in [13] for the proof that
$\mathcal{L}=y^{\alpha}(\Delta_{x}+B_{y})$ generates an analytic semigroup.

###### Proposition 8.4

Assume that $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ and let for
$\lambda\in\mathbb{C}^{+}$, $\xi\neq 0$

$N_{\lambda}(\xi)=(\lambda-y^{\alpha}B^{n}+|\xi|^{2}y^{\alpha})^{-1}\in\mathcal{B}\left(L^{p}_{m}\right).$

Then the family

$\left\\{|\xi|^{|\beta|}D^{\beta}_{\xi}(\lambda
N_{\lambda})(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\ |\beta|\leq
N,\lambda\in\mathbb{C}^{+}\right\\}$

is $\mathcal{R}$-bounded in $L^{p}_{m}$.

Proof. For $\mu>0$ let $n_{\lambda}(\mu)=\left(\lambda-y^{\alpha}B^{n}+\mu
y^{\alpha}\right)^{-1}$. Using Lemma 2.5 we have to show that the family

$\displaystyle\left\\{\mu^{k}D^{k}_{\mu}(n_{\lambda})(\mu):\mu>0,\ k\leq N,\
\lambda\in\mathbb{C}^{+}\right\\}$ (9)

is $\mathcal{R}$-bounded in $L^{p}_{m}$.

Theorem 7.2 with $V(y)=\mu y^{\alpha}$ and Proposition 8.3 imply that the
families

$\displaystyle\left\\{\lambda n_{\lambda}(\mu):\mu>0,\
\lambda\in\mathbb{C}^{+}\right\\},\qquad\left\\{\mu
y^{\alpha}n_{\lambda}(\mu):\mu>0,\ \lambda\in\mathbb{C}^{+}\right\\}$ (10)

are $\mathcal{R}$-bounded in $L^{p}_{m}$.

We have that $n_{\lambda}(\cdot)\in
C^{1}\left(\mathbb{R}_{+},\mathcal{B}\left(L^{p}_{m}\right)\right)$ and

$\displaystyle
D_{\mu}(n_{\lambda}(\mu))=-n_{\lambda}(\mu)y^{\alpha}n_{\lambda}(\mu).$ (11)

Indeed setting $A=\lambda-y^{\alpha}B^{n}_{y}$, $V=y^{\alpha}$ we have

$\displaystyle\frac{n_{\lambda}(\mu+h)-n_{\lambda}(\mu)}{h}$
$\displaystyle=\frac{(A+(\mu+h)V)^{-1}-(A+\mu V)^{-1}}{h}$
$\displaystyle=(A+\mu V)^{-1}\frac{(A+\mu V)(A+(\mu+h)V)^{-1}-I}{h}$
$\displaystyle=-(A+\mu V)^{-1}\,V\,(A+(\mu+h)V)^{-1}$

which tends to $-n_{\lambda}(\mu)\,y^{\alpha}\,n_{\lambda}(\mu)$ as $h\to 0$
in the norm of $\mathcal{B}\left(L^{p}_{m}\right)$ since, by Corollary 8.2,

$\mu\mapsto V(A+\mu)V)^{-1}=\mu
y^{\alpha}\left(\mu-B^{n}+\frac{\lambda}{y^{\alpha}}\right)^{-1}\frac{1}{y^{\alpha}}$

is continuous from $(0,\infty)$ to $\mathcal{B}\left(L^{p}_{m}\right)$. This
shows (11) and then $n_{\lambda}(\cdot)\in
C^{\infty}\left(\mathbb{R}_{+},\mathcal{B}\left(L^{p}_{m}\right)\right)$ and

$\displaystyle
D^{k}_{\mu}(n_{\lambda}(\mu))=a_{k}n_{\lambda}(\mu)\left(y^{\alpha}n_{\lambda}(\mu)\right)^{k},\qquad
a_{1}=-1,\quad a_{k+1}=-(k+1)a_{k}.$ (12)

Formula (12) follows by induction after observing that, since
$y^{\alpha}n_{\lambda}(\mu)$ and its derivative
$D_{\mu}\left(y^{\alpha}n_{\lambda}\right)=-\left(y^{\alpha}\,n_{\lambda}(\mu)\right)^{2}$
commute, then

$D_{\mu}\left(y^{\alpha}n_{\lambda}(\mu)\right)^{k}=kD_{\mu}\left(y^{\alpha}n_{\lambda}(\mu)\right)\left(y^{\alpha}n_{\lambda}(\mu)\right)^{k-1}=-k\left(y^{\alpha}n_{\lambda}(\mu)\right)^{k+1}.$

The $\mathcal{R}$-boundedness of the family (9) then follows from the
$\mathcal{R}$-boundedness of the families (10) since

$\displaystyle\mu^{k}D^{k}_{\mu}(\lambda n_{\lambda}(\mu))=a_{k}\lambda
n_{\lambda}(\mu)\left(\mu y^{\alpha}n_{\lambda}(\mu)\right)^{k+1}.$

In order to characterize the domain of $y^{\alpha}B^{n}-y^{\alpha}$, we denote
by

$D(y^{\alpha})=\left\\{u\in L^{p}_{m}:\ y^{\alpha}u\in L^{p}_{m}\right\\}$

the domain of the potential $V=y^{\alpha}$ in $L^{p}_{m}$. Recalling that
Theorem 4.2 assures that $D(y^{\alpha}B^{n})=W^{2,p}_{\mathcal{N}}(\alpha,m)$,
we consider, for $0<\frac{m+1}{p}<c+1-\alpha$, the Banach space

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha,m)\cap D(y^{\alpha})=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}_{+}):\ u,\ y^{\alpha}u,\ y^{\alpha}D_{yy}u,\
y^{\frac{\alpha}{2}}D_{y}u,\ y^{\alpha-1}D_{y}u\in L^{p}_{m}\right\\}$

endowed with norm
$\|y^{\alpha}Bu\|_{L^{p}_{m}}+\|y^{\alpha}u\|_{L^{p}_{m}}+\|u\|_{L^{p}_{m}}$.

###### Theorem 8.5

Let $\alpha<2$, $\mu>0$, $c\in\mathbb{R}$. Then for any $1<p<\infty$ such that
$\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ the operator $L=y^{\alpha}B^{n}-\mu
y^{\alpha}$ with domain $W^{2,p}_{\mathcal{N}}(\alpha,m)\cap D(y^{\alpha})$
generates a bounded analytic semigroup in $L^{p}_{m}$ which has maximal
regularity. Moreover,

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)):u\ {\rm constant\ in\ a\
neighborhood\ of}\ 0\right\\}$

is a core for $y^{\alpha}B^{n}-\mu y^{\alpha}$.

Proof. The generation properties as well as the maximal regularity follows
from Theorem 7.2. Without any loss of generality we may assume that $\mu=1$.
We prove preliminarily that $\mathcal{D}$ is dense in
$W^{2,p}_{\mathcal{N}}(\alpha,m)\cap D(y^{\alpha})=D(y^{\alpha}B^{n})\cap
D(y^{\alpha})$. Let $u\in W^{2,p}_{\mathcal{N}}(\alpha,m)\cap D(y^{\alpha})$;
up to using a standard cut-off argument we may suppose that $\mbox{supp
}u\subseteq[0,b]$ for some $b>0$. Using Remark 4.4, let
$(u_{n})_{n\in\mathbb{N}}\subseteq\mathcal{D}$ such that $\mbox{supp
}u_{n}\subseteq[0,b]$ and $u_{n}\to u$ in $W^{2,p}_{\mathcal{N}}(\alpha,m)$.
Then by [12, Proposition 3.2 (ii)]

$\displaystyle\|y^{\alpha}(u_{n}-u)\|_{L^{p}_{m}}\leq
C\|y^{\alpha+1}(D_{y}u_{n}-D_{y}u)\|_{L^{p}_{m}}\leq
Cb^{2}\|y^{\alpha-1}D_{y}(u_{n}-u)\|_{L^{p}_{m}}$

which tends to $0$ as $n\to\infty$. This proves the density of $\mathcal{D}$.

Let us now characterize the domain. By definition
$D(y^{\alpha}B^{n}-y^{\alpha})=(1-y^{\alpha}B^{n}+y^{\alpha})^{-1}\left(L^{p}_{m}\right)$.
Let $u=(1-y^{\alpha}B^{n}+y^{\alpha})^{-1}f$ with $f\in L^{p}_{m}$. Using
Corollary 8.2 (ii) we obtain

$\displaystyle\|y^{\alpha}u\|_{L^{p}_{m}}+\|y^{\alpha}Bu\|_{L^{p}_{m}}\leq
C\left(\|\mathcal{(}y^{\alpha}B-y^{\alpha})u\|_{L^{p}_{m}}+\|u\|_{L^{p}_{m}}\right)$
(13)

which proves the inclusion $D(y^{\alpha}B^{n}-y^{\alpha})\subseteq
D(y^{\alpha}B^{n})\cap D(y^{\alpha})$. To prove the reverse property we
observe that, since the graph norm of $y^{\alpha}B^{n}-y^{\alpha}$ is clearly
weaker than the norm of $D(y^{\alpha}B^{n})\cap D(y^{\alpha})$, inequality
(13) again shows that they are equivalent on $D(y^{\alpha}B^{n}-y^{\alpha})$,
in particular on $\mathcal{D}$ which is dense in $D(y^{\alpha}B^{n})\cap
D(y^{\alpha})$, by the previous step. Therefore
$D(y^{\alpha}B^{n}-y^{\alpha})=D(y^{\alpha}B^{n})\cap D(y^{\alpha})$ and in
particular $\mathcal{D}$ is a core.

We remark that Theorem 7.2 assures that $y^{\alpha}B^{n}-y^{\alpha}$ generates
a semigroup on $L^{p}_{m}$ under the milder assumption
$0<\frac{m+1}{p}<c+1-\alpha$ and $c+1>0$. However, the hypothesis
$(m+1)/p+\alpha>0$ must be added when $\alpha<0$ in order that
$\mathcal{D}\subset D(y^{\alpha})$.

The same method yields the domain of $B^{n}-\frac{1}{y^{\alpha}}$, using
Corollary 8.2 (iii) with $m$ replaced by $m-\alpha p$ .

###### Corollary 8.6

If $\alpha^{+}<\frac{m+1}{p}<c+1$, then the domain of
$B^{n}-\frac{1}{y^{\alpha}}$ is $W^{2,p}_{\mathcal{N}}(0,m)\cap
D(\frac{1}{y^{\alpha}})$.

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