# Diamonds in Klein geometry

Rafael Mancini Santos 111Currently at Divisão de Astrofísica, Instituto
Nacional de Pesquisas Espaciais - Inpe, São Paulo, Brazil
<EMAIL_ADDRESS>L. C. T. Brito<EMAIL_ADDRESS>Cleverson Filgueiras
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de
Lavras, Campus Universitário, Lavras,3037, Minas Gerais, Brasil

###### Abstract

Recently, there has been a suggestion that the Unruh effect might manifest in
metamaterials at accessible Unruh temperatures. In certain instances, the
class of metamaterials that could be instrumental for this observation
exhibits a Klein signature instead of a Minkowski one. Consequently,
confirming this effect in those materials necessitates a more meticulous
analysis. In this paper, we employ the path integral formulation of Quantum
Field Theory to investigate the analogue of the Unruh effect in Kleinian
geometry. We perform calculations for a scalar theory, provided we restrict
the action to a convenient subspace of the Kleinian spacetime. As a result, we
determine the diamond temperature for a static observer with a finite
lifetime. The outcome suggests that metamaterials could serve as a potential
system for observing diamond regions.

##  I Introduction

The exchange of ideas between Quantum Field Theory (QFT) and condensed matter
physics has been very fruitful over the years. The application of the concept
of spontaneously broken symmetry in particle physics Nambu:1960xd  and
superconductivity Anderson:1963pc  in the early 1960s represents the first
remarkable example. A more recent case is the discovery that two-dimensional
graphene is described at low energy by the massless Dirac equation graphene .
Besides these examples, there exists a wide range of ideas connecting particle
physics and condensed matter Wilczek:2016jqq . In this paper, we will
investigate analogous phenomena in QFT and condensed matter physics motivated
by a recent proposal that suggests the Unruh effect might occur in certain
classes of metamaterials Smolyaninov:2020noj .

The Unruh effect is a prediction of QFT Unruh:1976db , and therefore it can be
interpreted as a natural consequence of Quantum Mechanics and Special
Relativity. It states that uniformly accelerating observers experience the
Minkowski vacuum as a thermal bath with temperature $T=\hbar a/2\pi c\kappa$,
where $a$ is the observer’s acceleration, $\hbar$ is the reduced Planck
constant, $c$ is the speed of light, and $\kappa$ is the Boltzmann’s constant.
A detailed review on the fundamentals and applications of the Unruh effect can
be found in reference Crispino:2007eb .

Direct experimental verification of the Unruh effect is a difficult challenge,
as an acceleration of order $10^{20}\,\text{m/s}^{2}$ is required to achieve a
temperature of the order of $1{\text{K}}$. Nevertheless, there are good
reasons to believe that it is possible to detect the Unruh effect using both
quantum Martin-Martinez:2010gnz ; martin2013berry  and classical
PhysRevLett.118.161102  systems, with experiments that may be accessible to
the current technology. Also, the reported observation of thermalization at
the Unruh temperature caused by radiation emitted from accelerated positron
might be the first direct measure of the Unruh temperature in a high-energy
system PhysRevD.104.025015 .

Recently it was suggested that the Unruh effect might occur in hyperbolic
metamaterial waveguides Smolyaninov:2020noj . These materials have a high
degree of anisotropy, and photons propagating through them might achieve high
accelerations, which could be sufficient to put the Unruh temperature at
values accessible to experiments. However, it is possible to show that
hyperbolic metamaterials may behave effectively as a space with a Kleinian
signature as light propagates through it smolyaninov2010metric . In this case,
a careful analysis of the existence of the Unruh effect is necessary.

Investigation concerning the Kleinian signature goes back to alty1994kleinian
. Physical effects in optics and cosmology in the context of analog systems
were discussed in fumeron2015optics ; figueiredo2016modeling . It was only in
reference PhysRevD.103.044023  that the Kleinian geometry was investigated in
more detail. In particular, the authors show that acceleration restricts the
time coordinate of a classical particle to take values within an interval
determined by the magnitude of acceleration. As we will see, this restriction
is crucial for obtaining the analogous Unruh temperature in Klein geometry.
Besides, only phenomena in relativistic quantum mechanics were considered in
PhysRevD.103.044023 .

A traditional way to deduce the Unruh effect for free scalar fields in
Minkowski space is by using the canonical formalism of QFT ( see
Crispino:2007eb  for a pedagogical and wald1  for a more rigorous
presentation). In particular, Minkowski space is globally hyperbolic and
static (see wald2  for details about classical solutions of the Klein-Gordon
equation in hyperbolic spacetimes). These characteristics justify the
canonical formulation of QFT for free scalar fields in Minkowski spacetime. As
a consequence, it is possible to expand the quantum field in terms of creation
and annihilation operators and construct the Hilbert space as a Fock space
with a well-defined vacuum state that preserves translational symmetry. The
relevant point for our discussion is that Klein’s geometry, in which we are
interested, is ultrahyperbolic instead of hyperbolic Tegmark:1997jg  .

In this paper, we utilize the method of path integrals of QFT to investigate
and provide justification for the occurrence of an effect analogous to the
Unruh effect in an effective geometry with a Kleinian signature. In
particular, the method circumvents the mathematical challenge of constructing
a Hilbert space for QFT in ultrahyperbolic geometries. We calculate the
counterpart to the Unruh temperature in this scenario, the diamond
temperature, and discuss why metamaterials are a potential candidate for
providing insights into the presence of diamond regions Martinetti:2002sz ;
Chakraborty:2022qdr . Given the recent observation of photon acceleration in
tapered optical waveguides ge2021observation , we may have the possibility of
testing the occurrence of the Unruh or analogous effects in a setup similar to
the one outlined in reference Smolyaninov:2020noj .

The paper is organized as follows. In section II, we review the deduction of
the Unruh effect by using the path integral approach of QFT
unruh1984acceleration  and show how to obtain the diamond temperature with
this approach in the Minkowski space. In section III, we apply path integrals
to a geometry with a Kleinian signature to obtain the diamond temperature. In
section IV we discuss the connection with metamaterials. Finally, in section
V, we summarize our results and make some final remarks.

Through the rest of the paper we will use units in which $c=\hbar=\kappa=1$.

##  II Path integrals, Unruh effect and the diamond temperature

We begin our discussion with the generating functional for the Green function
in the path integral formulation of QFT:

$Z[J]=\rho\int D\phi e^{i\left(S+\int d^{4}xJ\phi\right)},$ (1)

where the normalization constant $\rho$ is chosen such that $Z[0]=1$. We are
interested in the theory of a scalar field $\phi$ whose dynamic is governed by
the classical action

$S=\int_{t_{1}}^{t_{2}}dt\int_{V}d^{3}x\mathcal{L},$ (2)

with the Lagrangian density

$\mathcal{L}=-\frac{1}{2}\left(g_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi+m^{2}\phi^{2}\right).$
(3)

Here, $m^{2}$ is a parameter with the dimension of mass squared, and
$g_{\mu\nu}$ is the metric for a flat geometry. In the present section, we are
interested in the Unruh effect in the Minkowski space. So we consider
$g_{\mu\nu}=(-1,1,1,1)$, leading to the Lagrangian density

$\mathcal{L}=\frac{1}{2}\left(\frac{\partial\phi}{\partial
t}\right)^{2}-\left(\nabla\phi\right)^{2}-m^{2}\phi^{2}.$ (4)

Also, we need the classical Lagrangian density as defined by a non-inertial
observer with constant acceleration $a$. It is obtained by the Rindler
transformation of coordinates

$x=r\cosh\left(a\eta\right)\,\,\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,\,\,t=r\sinh\left(a\eta\right),$ (5)

where $0<r<\infty$ and $-\infty<\eta<+\infty$ are the coordinates
correspondent to the geometry of the right wedge of the Rindler space:

$d^{2}s=-a^{2}r^{2}d\eta^{2}+dr^{2}+dy^{2}+dz^{2}.$ (6)

Thus, an observer with acceleration $a$ and fixed spacial coordinates $r,y$,
and $z$ measures the proper time $d\eta$ at $r=\frac{1}{a}$. Using the
transformation (5) in the Lagrangian density (3), we obtain the Lagrangian
density for such an observer:

$\mathcal{L}^{R}=\frac{1}{2}\,a\,r\,\left[\frac{1}{\left(ar\right)^{2}}\left(\frac{\partial\phi}{\partial\eta}\right)^{2}-\left(\frac{\partial\phi}{\partial
r}\right)^{2}-\left(\nabla_{\perp}\phi\right)^{2}-m^{2}\phi^{2}\right].$ (7)

Here, we have $\nabla_{\perp}=(\frac{\partial}{\partial
y},\frac{\partial}{\partial z})$.

Now, we turn our attention to quantum theory. We will consider the QFT in
terms of an imaginary time. By taking $t_{1}=-i\tau$ and $t_{2}=i\tau$ and
letting $\tau\rightarrow\infty$, the generating functional (1) corresponds to
the standard QFT at zero temperature. On the other hand, to introduce the
equilibrium temperature $T$ into the QFT defined by (1), we must consider the
time coordinate $t$ as an imaginary number and confine it to the interval

$t_{2}-t_{1}=i\beta,$ (8)

with $\beta=\frac{1}{T}$.

It is a standard procedure performing a change of coordinates
$t\rightarrow-i\tau$ in (1) such that the generating functional in the QFT at
finite temperature $T$ can be written as

$Z[J]=\rho\int D\phi e^{-\int_{-\beta/2}^{\beta/2}d\tau\int
d^{3}x\left(\mathcal{L}_{\tau}+J\phi\right)}.$ (9)

Here, we have

$\mathcal{L}_{\tau}=-\mathcal{L}(t=-i\tau),$ (10)

which means that on the right-hand side of this expression we need to replace
$t\rightarrow-i\tau$ in the Lagrangian density (3). Convergence of the
functional generator (9) is assured in Minkowski space when utilizing the
Euclidean action. For geometries where convergence does not occur for all
field configurations, like the Klein geometry we will consider in the next
section, the sum in the path integrals must take only the configurations for
which the functional generator is convergent.

Let us now apply the prescription outlined above to review the calculation of
the Unruh temperature in the Minkowski space unruh1984acceleration . Utilizing
the prescription as indicated in (10), we can derive the Lagrangian densities
for the inertial and non-inertial observers from (4) and (7), which are given
respectively by

$\mathcal{L}_{\tau}=\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial\tau}\right)^{2}+\left(\nabla\phi\right)^{2}+m^{2}\phi^{2}\right].$
(11)

and

$\mathcal{L}_{\tau}^{R}=\frac{1}{2}\left[\frac{1}{\left(ar\right)^{2}}\left(\frac{\partial\phi}{\partial\tau}\right)^{2}+\left(\frac{\partial\phi}{\partial
r}\right)^{2}+\left(\nabla_{\perp}\phi\right)^{2}+m^{2}\phi^{2}\right].$ (12)

The QFT in Rindler spacetime at temperature $T$ is defined by the generating
functional (9) with the action

$S_{\tau}^{R}=\frac{1}{2}\int_{0}^{\infty}dr\int_{-\beta/2}^{+\beta/2}d\tau\int_{x,y}\,a\,r\,\left[\frac{1}{\left(ar\right)^{2}}\left(\frac{\partial\phi}{\partial\tau}\right)^{2}+\left(\frac{\partial\phi}{\partial
r}\right)^{2}+\left(\nabla_{\perp}\phi\right)^{2}+m^{2}\phi^{2}\right],$ (13)

where the Lagrangian (12) has been employed. Here, we have introduced the
symbol $\int_{y,z}$ to denote integrals that do not change by the
transformation (5). Changing variables in (13) as

$x=r\cos(a\tau)\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,t=r\sin(a\tau),$ (14)

with $-\beta/2<\tau<+\beta/2$, we find that if we take

$\beta=\frac{2\pi}{a},$ (15)

we will have an integral in polar coordinates over the entire plane.

Actually, using (14) and (15) in (13), we obtain

$S_{\tau}^{R}=\frac{1}{2}\int_{-\infty}^{+\infty}dt\int_{-\infty}^{+\infty}dx\int_{y,z}\left[\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}+\left(\nabla_{\perp}\phi\right)^{2}+m^{2}\phi^{2}\right].$ (16)

Remarkably, this is the action $S_{\tau}=\int d^{4}x\mathcal{L}_{\tau}$ that
defines the generating function for a QFT at zero temperature with the
Lagrangian density (11), for an inertial observer. Hence, the QFT at zero
temperature for an inertial observer is equivalent to the QFT at temperature
$T=\frac{a}{2\pi}$ for a Rindler observer.

Interestingly, the method can be used to calculate the temperature when the
QFT at zero temperature is restricted to a finite region of spacetime. In
particular, if we restrict the spacetime in (2) to be the right wedge of the
light cone, the angle $a\tau$ in (14) must take values in the interval
$-\frac{\pi}{4a}<\tau<\frac{\pi}{4a}$. So, instead of (15), the choice of
$\beta$ in this case is given by

$\beta_{D}=\frac{\pi}{2a},$ (17)

which is equivalent to the temperature, $T_{D}=\frac{2a}{\pi}$. As we will
discuss in section 4, this is the temperature experienced by a static observer
with a finite lifetime $\mathcal{T}=\frac{1}{a}$ in a diamond region
Martinetti:2002sz .

##  III The diamond temperature in Klein geometry

In the previous section, we outlined a general methodology to obtain the Unruh
temperature for a Rindler observer and for a observer with a finite lifetime
in Minkowski spacetime. Now, we will use this methodology to focus on our
primary objective of determining the analogous result for Klein Geometry.

The Klein geometry corresponds to the metric $g_{\mu\nu}=(-1,-1,1,1)$, with
the line element

$d^{2}s=-dt^{2}-dx^{2}+dy^{2}+dz^{2}.$ (18)

Using this metric in the Lagrangian density (3), we have the theory of a
scalar field in the Klein space:

$\mathcal{L}^{K}=\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}-\left(\nabla_{\perp}\phi\right)^{2}-m^{2}\phi^{2}\right].$ (19)

The coordinate transformation to a system with constant acceleration $a$ in
Klein space can be described by

$x=r\cos a\eta\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,t=r\sin a\eta,$ (20)

where $0<r<\infty$ and $-\frac{\pi}{4}<a\eta<+\frac{\pi}{4}$. Notably, the
first relevant point for the discussion of the Unruh effect in the case of
Klein space is that the “time” coordinate $\eta$ of the accelerated observer
is initially confined. Furthermore, the coordinates do not cover the full
plane but only the right wedge of spacetime. As we will see, the coordinate
$\eta$ is defined in the interval $(-\frac{\pi}{4a},\frac{\pi}{4a})$ to ensure
equivalence between the QFT at zero temperature and the QFT at the finite
temperature. The restriction of time in the motion of an accelerated particle
was previously obtained in a classical context in PhysRevD.103.044023 . In the
context of the equivalence between zero and finite-temperature QFTs we
consider here, it is crucial to impose a restricted interval of time. The
necessity to take this restriction as coinciding with the right wedge of
spacetime will become clear soon.

The Lagrangian density for the accelerated observer in Klein space is given
by:

$\mathcal{L}^{(a)}=\frac{1}{2}\,a\,r\,\Bigg{[}\frac{1}{(ar)^{2}}\left(\frac{\partial\phi}{\partial\eta}\right)^{2}+\left(\frac{\partial\phi}{\partial
r}\right)^{2}-(\nabla_{\perp}\phi)^{2}\Bigg{]},$ (21)

which is obtained by using the coordinates transformation (20) in (19).

In the QFT defined by the generating functional (9), we require the Lagrangian
densities (19) and (21) in the imaginary time. These Lagrangian densities are
given respectively by

$\mathcal{L}^{(K)}_{\tau}=\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial\tau}\right)^{2}-\left(\frac{\partial\phi}{\partial
x}\right)^{2}+(\nabla_{\perp}\phi)^{2}\right]$ (22)

and

$\mathcal{L}^{(a)}_{\tau}=\frac{1}{2}\,a\,r\,\Bigg{[}\frac{1}{(ar)^{2}}\left(\frac{\partial\phi}{\partial\tau}\right)^{2}-\left(\frac{\partial\phi}{\partial
r}\right)^{2}+(\nabla_{\perp}\phi)^{2}\Bigg{]}.$ (23)

By using (8) and the action for the accelerated frame,

$S^{a}_{\tau}=\int_{-\frac{\pi}{4a}}^{+\frac{\pi}{4a}}d\tau\int_{0}^{\infty}dr\int_{y,z}\mathcal{L}^{(a)}_{\tau},$
(24)

in the generating function (9) we obtain a QFT at the temperature
$T_{D}=\frac{2a}{\pi}$, which is the diamond temperature obtained from (17).
To demonstrate that this action is equivalent to the action defined by the
Lagrangian density (22) of the ”inertial” frame in Klein space, we need to
make the following change of coordinates:

$x=r\cosh(a\tau)\,\,\,\,\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,\,\,\,\,t=r\coth\left(\frac{\pi}{4}\right)\sinh(a\tau).$
(25)

The factor $\coth\left(\frac{\pi}{4}\right)$ in the above expression for the
$t$ coordinate enforce $x=\pm t$ at the limit
$\tau\rightarrow\pm\frac{\pi}{4a}$. Using the coordinate transformation (25)
in the action (24), we obtain

$S^{(K)}_{\tau}=\int_{\mathcal{M}}d^{4}x\mathcal{L}^{(K)}_{\tau},$ (26)

where

$\mathcal{M}:=\\{(t,x,y,z)\in\mathbb{R}^{2,2}\,;\,x>|t|\\}$ (27)

is the Klein space for the inertial theory, at zero temperature. The symbol
$\mathbb{R}^{2,2}$ is the space defined from the Klenian metric (18). As we
said before, $\mathcal{M}$ covers only the right wedge of the spacetime.

In conclusion, the equivalence between the zero-temperature QFT and the QFT at
temperature $T_{D}=\frac{2a}{\pi}$ holds only when we restrict the space to
the subspace defined in (27). This restriction is necessary to establish an
analogous of the Unruh temperature, the diamond temperature $T_{D}$, when the
space has a Klein signature.

##  IV Diamond regions and metamaterials

Diamond is the name of a region encompassing the intersection of the lightcone
of the future of the initial point and the lightcone of the past of the final
point in a finite spacetime trajectory. In Minkowski spacetime, one can
interpret the diamond as the region causally connected to a static observer
with a finite lifetime $\mathcal{T}$ Martinetti:2002sz ; Chakraborty:2022qdr .
As shown in the seminal paper Martinetti:2002sz , an observer in this region
sees a thermalization process. Remarkably, this thermalization does not vanish
even for non-accelerating observers. In this case, the temperature is given by
$T_{D}=\frac{2}{\pi\mathcal{T}}$, which is called the diamond temperature.

There is a general procedure mapping the diamond region in the Rindler wedge
Chakraborty:2022qdr , which is exactly the right wedge of the spacetime that
we used before to establish the diamond temperature using the path integral
approach of QFT. The interesting point we have shown is that a diamond
temperature exists even when we consider a Klenian geometry instead of a
Minkowski one. In particular, constraints in Klenian geometry must be
necessarily imposed if we want to obtain an analogous for the Unruh
temperature in this case.

The physical interpretation of the diamond temperature in Klein space follows
the same interpretation in Minkowski spacetime. So, it is associated with some
“observer” with a finite lifetime. The natural restrictions occurring in a
geometry with Klein signature seem as if this geometry were the natural one to
observe diamond regions. Actually, the map between the right wedge and the
diamond region exists since the conformal symmetry can also be defined for the
flat Klein space Atanasov:2021oyu .

Interestingly, the fact that spacetimes with two-time coordinates have
inherent instabilities has been discussed previously Tegmark:1997jg . The
instabilities may be related, for example, to decay channels of some
elementary particles that do not exist in a space with just one-time
coordinate. Although there is no evidence about more than one time coordinate
in spacetime, geometries like the Klein provides an interesting background to
study processes where some kind of “decay” can be induced by geometry.
Actually, as we can see from the metric signature introduced in section 3,
Klein space can be interpreted in the realm of physics with two-time
coordinates. So, it is interesting to ask if some physical system could mimic
the Klein geometry, at least as an effective geometry. Anisotropic
metamaterials seem like the more prominent physical system for this purpose.

Metamaterials are an exciting subject of research cui ; Smolyaninov2017 .
Interest in these materials is increasing due to the advent of technologies
that make it possible to artificially engineer the structure of materials to
obtain unusual properties, such as negative values of electric permittivity,
which is described as tensor. This way, we find interesting physical
possibilities. Specifically, anisotropic metamaterials can be engineered to
possess a negative permittivity component, utilizing specific fabrication
techniquesPrintMeta ; metaTheory ; expe . In this case, the permittivity
tensor for such metamaterials can be expressed as:

$\varepsilon_{ij}=\begin{pmatrix}\varepsilon_{1}&0&0\\\ 0&\varepsilon_{1}&0\\\
0&0&\varepsilon_{2}\ \end{pmatrix}\;.$ (28)

Assuming $\epsilon_{1}>0$ and $\epsilon_{2}<0$, this tensor serves as an
effective metric for electromagnetic waves. Consequently, an incident
electromagnetic wave manifests an ’extraordinary’ component that propagates
akin to a real scalar field over a Kleinian spacetime. The dispersion relation
in this context is given by:

$\frac{\omega^{2}}{c^{2}}=\frac{k_{x}^{2}}{\epsilon_{1}}+\frac{k_{y}^{2}}{\epsilon_{1}}-\frac{k_{z}^{2}}{|\epsilon_{2}|}\;,$
(29)

where ${\bf k}=(k_{x},k_{y},k_{z})$ represents the wave vector, $\omega$
denotes the angular frequency, and $c$ is the speed of light.

In addition to optic metamaterials, other instances of anisotropic materials
are explored. In a study focused on nematic liquid crystals fumeron2015optics
, the behavior of light rays is investigated within a distinct scenario. The
effective metric in cylindrical coordinates is expressed as:

$ds_{\text{eff}}^{2}=-c^{2}dt^{2}+\varepsilon_{0}dr^{2}+\varepsilon_{e}r^{2}d\phi^{2}+\varepsilon_{0}dz^{2}\;.$
(30)

This configuration assumes permittivities $\epsilon_{0}>0$ and
$\epsilon_{e}<0$, aligning with a Klein-like spacetime signature. Furthermore,
it is noteworthy that Kleinian spacetime can also be simulated in the realm of
electronic metamaterialsfigueiredo2016modeling .

Besides, metamaterials have been used to obtain analogous effects arising from
General Relativity Leonhardt:2006ai , quantum gravity Smolyaninov:2022lio ,
cosmology Smolyaninov:2011ah , and to investigate modifications in Planck’s
law for thermal radiation BiehsPRL2015 . They also serve as a possible
background that could enhance the Unruh effect Smolyaninov:2020noj . This last
proposal was that motivated the present investigation.

Actually, it has been suggested that Kleinian geometry can be established in
certain types of metamaterials smolyaninov2010metric . So, it is interesting
to search for physical processes in these metamaterials that resemble some
kind of ”decay,” which could be interpreted as a manifestation of a
thermalization process occurring in a diamond region. Of course, it is not our
intention to discuss this matter here. The point is that since metamaterials
can mimic Klein geometry, and there is an analogous effect to the Unruh
temperature in these geometries, the diamond temperature, these materials
naturally can be useful as a system for investigating diamond regions.

##  V Concluding Remarks

In conclusion, this paper has explored the analogue of the Unruh effect in a
Kleinian geometry, which is the ”diamond temperature” associated with a finite
spacetime trajectory. The path integral formulation of Quantum Field Theory
(QFT) was employed to establish this temperature, which was found to be
$T_{D}=\frac{2a}{\pi}$ for a static observer with a finite lifetime
$\mathcal{T}=\frac{1}{a}$ in the diamond region. The study also highlighted
the importance of restricting the spacetime geometry to a specific subspace
for the existence of the diamond temperature.

Moreover, we suggest that hyperbolic metamaterials may serve as a suitable
system to observe diamond regions, given their potential to mimic Kleinian
geometry. This proposition opens up the possibility of experimental
investigations into the concept of diamond temperatures in the context of
metamaterials. As is known, metamaterials are artificial materials designed to
possess properties not found in natural materials. Some types of metamaterials
include negative-index metamaterials, plasmonic metamaterials, acoustic
metamaterials, magnetic metamaterials, elastic metamaterials, terahertz
metamaterials, and quantum metamaterials 3dmeta . They offer a significant
platform for exploring analogous phenomena in high-energy physics,
gravitation, astrophysics, and more. However, our work has demonstrated that
the mapping between these branches and the metamaterials must be approached
with caution, as this mapping must provide an adequate description of the
problem.

In summary, our research provides valuable insights into the interplay between
quantum field theory, spacetime geometry, and metamaterials, potentially
paving the way for exciting new experimental directions in the study of
thermal effects in various analogous physical systems.

## Declaration of Competing Interest

The authors declare no conflict of interest.

## Acknowledgement

This work was partially supported by the Brazilian agencies CNPq and FAPEMIG.
C. Filgueiras and L.C.T.Brito acknowledges FAPEMIG Grant No. APQ 02226/22. C.
Filgueiras acknowledges CNPq Grant No. 310723/2021-3. R. M. Santos
acknowledges FAPEMIG Grant No. 13681/2021-3.

## Data Availability Statement

No Data associated in the manuscript

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# Anisotropic Sobolev spaces with weights

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 Sobolev spaces with weights in the half-space
$\mathbb{R}^{N+1}_{+}=\\{(x,y):x\in\mathbb{R}^{N},y>0\\}$, adapted to the
singular elliptic operators

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

Mathematics subject classification (2020): 46E35, 35J70, 35J75.

Keywords: weighted Sobolev spaces, degenerate elliptic operators.

##  1 Introduction

Elliptic and parabolic problems associated to the degenerate operators

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

in the half-space $\mathbb{R}^{N+1}_{+}=\\{(x,y):x\in\mathbb{R}^{N},y>0\\}$ or
in $(0,\infty)\times\mathbb{R}^{N+1}_{+}$ lead quite naturally to the
introduction of weighted Sobolev spaces which are anisotropic if
$\alpha_{1}\neq\alpha_{2}$. The aim of this paper is to provide the functional
analytic properties of these Sobolev spaces needed in [11] and in [10] in the
$1$-d case, where we prove existence, uniqueness and regularity of elliptic
and parabolic problems governed by the operators above. We also refer to [1,
5, 6, 7, 8, 15] for the analogous results concerning the $N$-d version of
$D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}$.

For $m\in\mathbb{R}$ we consider the measure $y^{m}dxdy$ in
$\mathbb{R}^{N+1}_{+}$ and write $L^{p}_{m}$ for
$L^{p}(\mathbb{R}_{+}^{N+1};y^{m}dxdy)$. Given $p>1$,
$\alpha_{1}\in\mathbb{R}$, $\alpha_{2}<2$, we define the Sobolev space

$\displaystyle W^{2,p}(\alpha_{1},\alpha_{2},m)$ $\displaystyle=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):\ u,\ y^{\alpha_{1}}D_{x_{i}x_{j}}u,\
y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u,y^{\alpha_{2}}D_{yy}u,\
y^{\frac{\alpha_{2}}{2}}D_{y}u\in L^{p}_{m}\right\\}$

which is a Banach space equipped with the norm

$\displaystyle\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}=$
$\displaystyle\|u\|_{L^{p}_{m}}+\sum_{i,j=1}^{n}\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\sum_{i=1}^{n}\|y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u\|_{L^{p}_{m}}$
$\displaystyle+\|y^{\alpha_{2}}D_{yy}u\|_{L^{p}_{m}}+\|y^{\frac{\alpha_{2}}{2}}D_{y}u\|_{L^{p}_{m}}.$

Next we add a Neumann boundary condition for $y=0$ in the form
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$ and set

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):\ y^{\alpha_{2}-1}D_{y}u\ \in L^{p}_{m}\\}$

with the norm

$\|u\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}.$

We consider also an integral version of the Dirichlet boundary condition,
namely a weighted summability requirement for $y^{-2}u$ and introduce

$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):y^{\alpha_{2}-2}u\in L^{p}_{m}\\}$

with the norm

$\|u\|_{W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-2}u\|_{L^{p}_{m}}.$

Note that $\alpha_{1},\alpha_{2}$ are not assumed to be positive. The
restriction $\alpha_{2}<2$ is not really essential since one can deduce from
it the case $\alpha_{2}>2$, using the change of variables described in the
next section. However, we keep it both to simplify the exposition and because
$\mathcal{L}$ is mainly considered for $\alpha_{2}<2$.

No requirement is made for the mixed derivatives $D_{x_{i}y}u$ to simplify
some arguments. However, the weighted integrability of the mixed derivatives
is automatic under the condition of Proposition 4.1.

Sobolev spaces with weights are well-known in the literature, see e.g. [3],
[14, Chapter 6], [2] and [13] for the non-anisotropic case. Variants of
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$, usually defined as the
closure of compactly supported functions in
$W^{2,p}(\alpha_{1},\alpha_{2},m)$, can be found in the above papers .
However, we have not been able to find anything about
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

Let us briefly describe the content of the paper. In Section 2 we show that,
by a change of variables, the spaces
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ and
$W^{2,p}_{\mathcal{N}}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})$,
$\tilde{\alpha}_{1}=\frac{\alpha_{1}}{\beta+1},\quad\tilde{\alpha}_{2}=\frac{\alpha_{2}+2\beta}{\beta+1},\quad\tilde{m}=\frac{m-\beta}{\beta+1}$
are isomorphic. This observation simplifies many proofs but requires the full
scale of $L^{p}_{m}$ spaces, according to the general strategy of to study the
operator $\mathcal{L}$. Hardy inequalities and traces for $y=0$ are studied in
Section 3. The main properties of the spaces
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ are proved in Section 4
together with a density result for smooth functions having zero $y$-derivative
in a strip around $y=0$, which is crucial in the study of the operator
$\mathcal{L}$. The space $W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ is
studied in Section 5.

##  2 A useful change of variables

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

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

Observe that

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

###### Lemma 2.1

The following properties hold for $1\leq p\leq\infty$.

  * (i)

$T_{k,\beta\,}$ maps isometrically $L^{p}_{\tilde{m}}$ onto $L^{p}_{m}$ where

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

  * (ii)

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

    * 1.

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

    * 2.

$D_{x_{i}x_{j}}(T_{k,\beta\,}u)=T_{k,\beta}\left(D_{x_{i}x_{j}}u\right)$,
$D_{x_{i}}(T_{k,\beta\,}u)=T_{k,\beta}\left(D_{x_{i}}u\right)$;

    * 3.

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

    * 4.

$D_{xy}T_{k,\beta\,}u=T_{k,\beta\,}\left(ky^{-\frac{1}{\beta+1}}D_{x}u+(\beta+1)y^{\frac{\beta}{\beta+1}}D_{xy}u\right)$

Proof. The proof of (i) follows after observing the Jacobian of
$(x,y)\mapsto(x,y^{\beta+1})$ is $|1+\beta|y^{\beta}$. To prove (ii) one first
observes that any $x$-derivatives commutes with $T_{k,\beta}$. Then we compute

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

and similarly

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

Let us specialize the above lemma to

$\displaystyle T_{0,\beta}$ $\displaystyle:L^{p}_{\tilde{m}}\to
L^{p}_{m},\qquad\tilde{m}=\frac{m-\beta}{\beta+1}$

to transform Sobolev spaces with different exponents.

###### Proposition 2.2

Let $p>1$, $m,\alpha_{1},\alpha_{2}\in\mathbb{R}$ with $\alpha_{2}<2$. Then
one has

$\displaystyle
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=T_{0,\beta}\left(W^{2,p}_{\mathcal{N}}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})\right),\qquad\tilde{\alpha}_{1}=\frac{\alpha_{1}}{\beta+1},\quad\tilde{\alpha}_{2}=\frac{\alpha_{2}+2\beta}{\beta+1}.$

In particular, by choosing $\beta=-\frac{\alpha_{2}}{2}$ one has

$\displaystyle
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=T_{0,-\frac{\alpha_{2}}{2}}\left(W^{2,p}_{\mathcal{N}}(\tilde{\alpha},0,\tilde{m})\right),\qquad\tilde{\alpha}=\frac{2\alpha_{1}}{2-\alpha_{2}},\quad\tilde{m}=\frac{m+\frac{\alpha_{2}}{2}}{1-\frac{\alpha_{2}}{2}}.$

Proof. Given $\tilde{u}\in
W^{2,p}_{\mathcal{N}}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})$ let us
set
$u(x,y)=(T_{0,\beta}\tilde{u})(x,y)=|\beta+1|^{1/p}\tilde{u}(x,y^{\beta+1})$.
Everything follows from the equalities of Lemma 2.1,

  * (i)

$y^{\alpha_{1}}D_{x_{i}x_{j}}u=T_{0,\beta}\left(y^{\tilde{\alpha}_{1}}D_{x_{i}x_{j}}\tilde{u}\right)$,
$y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u=T_{0,\beta}\left(y^{\frac{\tilde{\alpha}_{1}}{2}}D_{x_{i}}\tilde{u}\right)$;

  * (ii)

$y^{\frac{\alpha_{2}}{2}}D_{y}u=(1+\beta)T_{0,\beta}\left(y^{\tilde{\alpha}_{2}}D_{y}\tilde{u}\right)$,
$y^{\alpha_{2}-1}D_{y}u=(1+\beta)T_{0,\beta}\left(y^{\tilde{\alpha}_{2}-1}D_{y}\tilde{u}\right)$;

  * (iii)

$y^{\alpha_{2}}D_{yy}u=(1+\beta)T_{0,\beta}\left[(1+\beta)y^{\tilde{\alpha}_{2}}D_{yy}\tilde{u}+\beta
y^{\tilde{\alpha}_{2}-1}D_{y}\tilde{u}\right]$.

###### Remark 2.3

Note that in the above proposition is essential to deal with
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$. Indeed in general the
isometry $T_{0,\beta}$ does not transform
$W^{2,p}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})$ into
$W^{2,p}(\alpha_{1},\alpha_{2},m)$, because of identity (iii) above.

##  3 Hardy inequalities and traces

In this section we prove some weighted Hardy inequalities and investigate
trace properties of function $u$ such that $y^{\beta}D_{y}u\in L^{p}_{m}$.

The following result is standard but we give a proof to settle ”almost
everywhere” issues.

###### Lemma 3.1

Let $u\in L^{1}_{loc}(\mathbb{R}^{N+1}_{+})$ be such that $D_{y}u\in
L^{1}(\mathbb{R}^{N+1}_{+})$. Then there exists $v$ such that $v=u$ almost
everywhere and $v(\cdot,y)\in L^{1}_{loc}(\mathbb{R}^{N})$ for every $y\geq 0$
and

$v(x,y_{2})-v(x,y_{1})=\int_{y_{1}}^{y_{2}}D_{y}u(x,s)\ ds$

for every $0\leq y_{1}<y_{2}\leq\infty$ and almost every $x\in Q$.

Proof. For a.e. $x\in\mathbb{R}^{N}$ the function $u(x,\cdot)$ is absolutely
continuous and then

$u(x,y_{2})-u(x,y_{1})=\int_{y_{1}}^{y_{2}}D_{y}u(x,s)\ ds$

for a.e. $y_{1},y_{2}$. It is therefore sufficient to define
$v(x,y)=\int_{c}^{y}D_{y}u(x,s)\,ds+u(x,c)$, if $c$ is chosen in such a way
that $u(\cdot,c)\in L^{1}_{loc}(\mathbb{R}^{N})$.

Properties of functions $u\in L^{p}_{m}$ such that $D_{y}u\in L^{p}_{m}$ have
been proved in [9, Appendix B]. Here we exploit the more general property
$y^{\beta}D_{y}u\in L^{p}_{m}$.

###### Proposition 3.2

Let $C:=\left|\frac{m+1}{p}-(1-\beta)\right|^{-1}$. The following properties
hold for $u\in L^{1}_{loc}(\mathbb{R}^{N+1}_{+})$ such that
$y^{\beta}D_{y}u\in L^{p}_{m}$.

  * (i)

If $\frac{m+1}{p}<1-\beta$ then $D_{y}u\in L^{1}\left(Q\times[0,1]\right)$ for
any cube $Q$ of $\mathbb{R}^{N}$; in particular $u$ has a trace $u(\cdot,y)\in
L^{1}_{loc}(\mathbb{R}^{N})$ for every $0\leq y\leq 1$. Moreover setting
$u_{0}(x)=\lim_{y\to 0}u(x,y)$ one has

$\displaystyle\|y^{\beta-1}(u-u_{0})\|_{L^{p}_{m}}\leq
C\|y^{\beta}D_{y}u\|_{L^{p}_{m}}.$

If moreover $u\in L^{p}_{m}$ then $u(\cdot,y)\in L^{p}(\mathbb{R}^{N})$ for
every $0\leq y\leq 1$.

  * (ii)

If $\frac{m+1}{p}>1-\beta$ then $D_{y}u\in
L^{1}\left(Q\times[1,\infty[\right)$ for any cube $Q$ of $\mathbb{R}^{N}$; in
particular $u$ has a finite trace $u_{\infty}(x)=\lim_{y\to\infty}u(x,y)\in
L^{1}_{loc}\left(\mathbb{R}^{N}\right)$ and

$\displaystyle\|y^{\beta-1}(u-u_{\infty)}\|_{L^{p}_{m}}\leq
C\|y^{\beta}D_{y}u\|_{L^{p}_{m}}.$ (2)

If moreover $u\in L^{p}_{m}$ then $u_{\infty}\in L^{p}(\mathbb{R}^{N})$ and
$u_{\infty}=0$ if $m\geq-1$.

Proof. To prove (i) let $f(x,y):=y^{\beta}D_{y}u(x,y)$. If $Q$ is a cube of
$\mathbb{R}^{N}$ then since $\frac{m+1}{p}>1-\beta$ one has

$\displaystyle\int_{Q\times[0,1]}|D_{y}u|dxdy$
$\displaystyle=\int_{Q\times[0,1]}|D_{y}u|y^{\beta}y^{-\beta-m}y^{m}dxdy$
$\displaystyle\leq\|y^{\beta}D_{y}u\|_{L^{p}_{m}}\left(\int_{0}^{1}y^{-(\beta+m)p^{\prime}+m}\right)^{\frac{1}{p^{\prime}}}|Q|^{\frac{1}{p^{\prime}}}=C(Q,b,p)\|y^{\beta}D_{y}u\|_{L^{p}_{m}}.$

In particular by Lemma 3.1, $u$ has a finite trace $u(\cdot,y)\in
L^{1}_{loc}\left(\mathbb{R}^{N}\right)$ for every $0\leq y\leq 1$. Setting
$u_{0}(x)=u(x,0)=\lim_{y\to 0}u(x,y)$ we can write

$\displaystyle
y^{\beta-1}\left(u(x,y)-u_{0}(x)\right)=y^{\beta-1}\int_{0}^{y}f(x,s)s^{-\beta}\,ds:=(H_{1}f)(y).$

By [9, Lemma 10.3, (i)], the operator $H_{1}$ is bounded on
$L^{p}_{m}(\mathbb{R}_{+})$ when $\frac{m+1}{p}<1-\beta$, hence

$\displaystyle\|y^{\beta-1}\left(u(x,\cdot)-u_{0}(x)\right)\|_{L^{p}_{m}(\mathbb{R}_{+})}\leq
C\|y^{\beta}D_{y}u(x,\cdot)\|_{L^{p}_{m}(\mathbb{R}_{+})}.$

Claim (i) then follows by raising to the power $p$ and integrating with
respect to $x$. To prove that $u(\cdot,y)\in L^{p}(\mathbb{R}^{N})$ we proceed
analogously: since $u\in L^{p}_{m}$ then $u(\cdot,y)\in L^{p}(\mathbb{R}^{N})$
for a.e. $y\in[0,1]$. Without any loss of generality we suppose $u(\cdot,1)\in
L^{p}(\mathbb{R}^{N})$ and we write for any $y_{0}\in[0,1]$

$u(x,y_{0})=u(x,1)-\int_{y_{0}}^{1}D_{y}u(x,s)\
ds=u(x,1)-\int_{y_{0}}^{1}s^{\beta}D_{y}u(x,s)s^{\frac{m}{p}}s^{-\beta-\frac{m}{p}}\
ds.$

Then using Hôlder inequality

$\displaystyle|u(x,y_{0})|$
$\displaystyle\leq|u(x,1)|+\left(\int_{y_{0}}^{1}\left|s^{\beta}D_{y}u(x,s)\right|^{p}s^{m}\
ds\right)^{\frac{1}{p}}\left(\int_{y_{0}}^{1}s^{(-\beta-\frac{m}{p})p^{\prime}}\
ds\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq|u(x,1)|+C\left(\int_{0}^{1}\left|s^{\beta}D_{y}u(x,s)\right|^{p}s^{m}\
ds\right)^{\frac{1}{p}}.$

Raising to the power $p$ and integrating with respect to $x$ we obtain

$\displaystyle\|u(\cdot,y_{0})\|_{L^{p}(\mathbb{R}^{N})}\leq
C\left(\|u(\cdot,1)\|_{L^{p}(\mathbb{R}^{N})}+\left\|y^{\beta}D_{y}u\right\|_{L^{p}_{m}}\right).$

The proof of (ii) is similar writing

$\displaystyle
y^{\beta-1}\left(u(x,y)-u_{\infty}(x)\right)=-y^{\beta-1}\int_{y}^{\infty}f(x,s)s^{-\beta}\,ds:=-(H_{2}f)(y)$

and applying [9, Lemma 10.3, (ii)]. If $u\in L^{p}_{m}$ and $m\geq-1$, then
$|u(x,\cdot)|^{p}$ is not summable with respect to $y^{m}\,dy$ for every $x$
where $u_{\infty}(x)\neq 0$, hence $u_{\infty}=0$ a.e.

In the next lemma we show that $u$ has has a logarithmic singularity for $y\to
0,\infty$, when $\frac{m+1}{p}=1-\beta$.

###### Lemma 3.3

If $\frac{m+1}{p}=1-\beta$ and $u,y^{\beta}D_{y}u\in L^{p}_{m}$, then

$\left(\int_{\mathbb{R}^{N}}|u(x,y)|^{p}\,dx\right)^{\frac{1}{p}}\leq\|u(\cdot,1)\|_{L^{p}(\mathbb{R}^{N})}+|\log
y|^{\frac{1}{p^{\prime}}}\|y^{\beta}D_{y}\|_{L^{p}_{m}}.$ (3)

Proof. Let $\frac{m+1}{p}=1-\beta$ and set $f=y^{\beta}D_{y}\in L^{p}_{m}$.
Then for $y\in(0,1)$ one has

$\displaystyle u(x,y)$ $\displaystyle=u(x,1)-\int_{y}^{1}D_{y}u(x,s)\
ds=u(x,1)-\int_{y}^{1}s^{-\beta}f(x,s)\ ds$
$\displaystyle=u(x,1)-\int_{y}^{1}s^{-\beta-m}f(x,s)s^{m}\ ds.$

Therefore, since $(-\beta-m)p^{\prime}+m=-1$, Hölder inequality yields

$\displaystyle|u(x,y)|$
$\displaystyle\leq|u(x,1)|+\left(\int_{y}^{1}s^{(-\beta-m)p^{\prime}}s^{m}\
ds\right)^{\frac{1}{p^{\prime}}}\left(\int_{y}^{1}|f(x,s)|^{p}s^{m}\
ds\right)^{\frac{1}{p}}$ $\displaystyle\leq|u(x,1)|+|\log
y|^{\frac{1}{p^{\prime}}}\|f(x,\cdot)\|_{L^{p}\left((0,1),y^{m}dy\right)}.$

The inequality for $y>1$ is similar.

Since $u\in L^{p}_{m}$ then, as in Proposition 3.2, we can suppose
$u(\cdot,1)\in L^{p}(\mathbb{R}^{N})$ and raising to the power $p$ and
integrating with respect to $x$ we conclude the proof.

We also need some elementary interpolative inequalities; the first generalizes
[12, Lemma 4.3] (see also [4]).

###### Lemma 3.4

For $m,\beta\in\mathbb{R}$, $1<p<\infty$ there exist $C>0,\varepsilon_{0}>0$
such that for every $u\in W^{2,p}_{loc}((0,\infty))$,
$0<\varepsilon<\varepsilon_{0}$,

$\|y^{\beta-1}u^{\prime}\|_{L^{p}_{m}(\mathbb{R}_{+})}\leq
C\left(\varepsilon\|y^{\beta}u^{\prime\prime}\|_{L^{p}_{m}(\mathbb{R}_{+})}+\frac{1}{\varepsilon}\|y^{\beta-2}u\|_{L^{p}_{m}(\mathbb{R}_{+})}\right).$

Proof.  Changing $\beta$ we may assume that $m=0$. We use the elementary
inequality

$\int_{a}^{b}|u^{\prime}(y)|^{p}\,dy\leq
C\left(\varepsilon^{p}(b-a)^{p}\int_{a}^{b}|u^{\prime\prime}(y)|^{p}\,dy+\frac{1}{\varepsilon^{p}(b-a)^{p}}\int_{a}^{b}|u(y)|^{p}\,dy\right)$
(4)

for $\varepsilon\leq\varepsilon_{0}$, where $\varepsilon_{0},C$ are the same
as for the unit interval (this follows by scaling). We apply this inequality
to each interval $I_{n}=[2^{n},2^{n+1}[$, $n\in\mathbb{Z}$ and multiply by
$2^{n(\beta-1)p}$ thus obtaining since $y\approx 2^{n}$ in $I_{n}$

$\int_{I_{n}}y^{(\beta-1)p}|u^{\prime}(y)|^{p}\,dy\leq\tilde{C}\left(\varepsilon^{p}\int_{I_{n}}y^{\beta
p}|u^{\prime\prime}(y)|^{p}\,dy+\frac{1}{\varepsilon^{p}}\int_{I_{n}}y^{(\beta-2)p}|u(y)|^{p}\,dy\right).$

The thesis follows summing over $n$.

###### Lemma 3.5

For $m,\beta<2$, $1<p<\infty$ there exist $C>0,\varepsilon_{0}>0$ such that
for every $u\in W^{2,p}_{loc}((1,\infty))$, $0<\varepsilon<\varepsilon_{0}$,

$\|y^{\frac{\beta}{2}}u^{\prime}\|_{L^{p}_{m}((1,\infty))}\leq
C\left(\varepsilon\|y^{\beta}u^{\prime\prime}\|_{L^{p}_{m}((1,\infty))}+\frac{1}{\varepsilon}\|u\|_{L^{p}_{m}((1,\infty))}\right).$

Proof. We use (4) in $(a_{n},a_{n+1})$ where $a_{n}=n^{1+\frac{\gamma}{2}}$,
so that $a_{n+1}-a_{n}\approx n^{\frac{\gamma}{2}}$. We multiply both sides by
$n^{(1+\frac{\gamma}{2})(m+\frac{\beta p}{2})}\approx y^{m+\frac{\beta p}{2}}$
in $(a_{n},a_{n+1})$ and sum over $n$. Choosing $\gamma\geq 0$ in such a way
that $\beta=\frac{2\gamma}{2+\gamma}$, the thesis follows.

##  4 The space $W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$

Let $p>1$, $m,\alpha_{1}\in\mathbb{R}$, $\alpha_{2}<2$. We recall that

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):\ y^{\alpha_{2}-1}D_{y}u\ \in L^{p}_{m}\\}$

with the norm

$\|u\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}.$

We have made the choice not to include the mixed derivatives in the definition
of $W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$ to simplify
some arguments. However the following result holds in a range of parameters
which is sufficient for the study of the operator $\mathcal{L}$.

###### Proposition 4.1

If $\alpha_{2}-\alpha_{1}<2$ and $\alpha_{1}^{-}<\frac{m+1}{p}$ then there
exists $C>0$ such that

$\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|u\|_{W^{2,p}_{\cal N}(\alpha_{1},\alpha_{2},m)}$

for every $u\in W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$.

Proof. This follows from [11, Theorem 7.1], choosing $c$ sufficiently large
therein, so that $\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}$.

###### Remark 4.2

With obvious changes we may consider also the analogous Sobolev spaces on
$\mathbb{R}_{+}$, $W^{2,p}(\alpha_{2},m)$ and $W^{2,p}_{\cal
N}(\alpha_{2},m)$. For example we have

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

For brevity sake, we consider in what follows, only the Sobolev spaces on
$\mathbb{R}^{N+1}_{+}$ but all the results of this section will be valid also
in $\mathbb{R}_{+}$ changing the condition $\alpha_{1}^{-}<\frac{m+1}{p}$
(which appears in Sections 4.2, 5) to $0<\frac{m+1}{p}$.

We clarify in which sense the condition $y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$
is a Neumann boundary condition.

###### Proposition 4.3

The following assertions hold.

  * (i)

If $\frac{m+1}{p}>1-\alpha_{2}$, then
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}(\alpha_{1},\alpha_{2},m)$.

  * (ii)

If $\frac{m+1}{p}<1-\alpha_{2}$, then

$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):\lim_{y\to 0}D_{y}u(x,y)=0\ {\rm for\ a.e.\
x\in\mathbb{R}^{N}}\\}.$

In both cases (i) and (ii), the norm of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ is equivalent to that of
$W^{2,p}(\alpha_{1},\alpha_{2},m)$.

Proof.  If $\frac{m+1}{p}>1-\alpha_{2}$ and $u\in
W^{2,p}(\alpha_{1},\alpha_{2},m)$, we apply Proposition 3.2 (ii) to $D_{y}u$
and obtain that $\lim_{y\to\infty}D_{y}u(x,y)=g(x)$ exists. At the points
where $g(x)\neq 0$, $u(x,\cdot)$ has at least a linear growth with respect to
$y$ and hence $\int_{0}^{\infty}|u(x,y)|^{p}y^{m}\,dy=\infty$ (since
$(m+1)/p>1-\alpha_{2}>-1$). Then $g=0$ a.e. and Proposition 3.2(ii) again
gives $\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}\leq
C\|y^{\alpha_{2}}D_{yy}u\|_{L^{p}_{m}}$.

If $\frac{m+1}{p}<1-\alpha_{2}$ we apply Proposition 3.2 (i) to $D_{y}u$ to
deduce that $\lim_{y\to 0}D_{y}u(x,y)=h(x)$ exists. If $h=0$, then Hardy
inequality yields $y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$. On the other hand,
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$ implies $h=0$, since
$y^{p(\alpha_{2}-1)}$ is not integrable with respect to the weight $y^{m}$.

###  4.1 An alternative description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$

We show an alternative description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$, adapted to the operator
$D_{yy}+cy^{-1}D_{y}$.

###### Lemma 4.4

Let $c\in\mathbb{R}$ and let us suppose that $\frac{m+1}{p}<c+1-\alpha_{2}$.
If $u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+})$ and $u,\
y^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in L^{p}_{m}$, then the
following properties hold.

  * (i)

The function $v=y^{c}D_{y}u$ satisfies $v,D_{y}v\in
L^{1}_{loc}\left(\mathbb{R}^{N}\times[0,\infty)\right)$ and therefore has a
trace $v_{0}(x):=\lim_{y\to 0}y^{c}D_{y}u(x,y)\in L^{1}_{loc}(\mathbb{R}^{N})$
at $y=0$.

  * (ii)

$v_{0}=0$ if and only $y^{\alpha_{2}-1}D_{y}u\in
L^{p}_{m}(\mathbb{R}^{N}\times[0,1])$. In this case

$\displaystyle\left\|y^{\alpha_{2}-1}D_{y}u\right\|_{L^{p}_{m}}\leq
C\left\|y^{\alpha_{2}}\left(D_{yy}u+cy^{-1}D_{y}u\right)\right\|_{L^{p}_{m}}$

with $C=\left(c+1-\alpha_{2}-\frac{m+1}{p}\right)^{-1}>0$.

  * (iii)

If the stronger assumption $0<\frac{m+1}{p}\leq c-1$ holds then $v_{0}=0$ and
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}(\mathbb{R}^{N}\times[0,1])$.

Proof. Let $v:=y^{c}D_{y}u$ and

$f:=y^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)=y^{\alpha_{2}-c}D_{y}v\in
L^{p}_{m}.$

Claim (i) is then a consequence of Proposition 3.2 (i) with
$\beta=\alpha_{2}-c$.

To prove (ii) we set $v_{0}(x)=\left(y^{c}D_{y}u\right)(x,0)$. Then one has
$g:=y^{\alpha_{2}-c-1}(v-v_{0})\in L^{p}_{m}$ by Proposition 3.2 (ii) again.
Then

$y^{\alpha_{2}-1}D_{y}u=g+y^{\alpha_{2}-1-c}v_{0}$

is $L^{p}_{m}$-integrable near $y=0$ if and only if $v_{0}=0$, since
$\frac{m+1}{p}<c+1-\alpha_{2}$.

Finally, when $v_{0}=0$, $y^{\alpha_{2}-1}=g=y^{\alpha_{2}-c-1}v$ and we can
use Proposition 3.2 (ii).

Let us prove (iii). Note that $c-1<c+1-\alpha_{2}$, since $\alpha_{2}<2$. At
the points where $v_{0}(x)\neq 0$, we have for $0<y\leq\delta(x)$,
$|D_{y}u(x,y)|\geq\frac{1}{2}|v_{0}(x)|y^{-c}$ which implies
$|u(x,y)|\geq\frac{1}{4}|v_{0}(x)|y^{-c+1}$ for $0<y\leq\delta^{\prime}(x)$,
since $c>1$. This yields $\int_{0}^{\infty}|u(x,y)|^{p}y^{m}\,dy=\infty$,
since $(m+1)/p\leq c-1$, and then $v_{0}=0$ a.e.

To provide an equivalent description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ we need the following simple
lemma.

###### Lemma 4.5

Assume that $u\in L^{p}(\mathbb{R}^{N})\cap W^{2,1}_{loc}(\mathbb{R}^{N})$ for
some $1<p<\infty$ and that $\Delta u\in L^{p}(\mathbb{R}^{N})$. Then $u\in
W^{2,p}(\mathbb{R}^{N})$.

Proof. Let $v\in W^{2,p}(\mathbb{R}^{N})$ be such that $v-\Delta v=u-\Delta u$
and consider $w=u-v\in L^{p}(\mathbb{R}^{N})\cap
W^{2,1}_{loc}(\mathbb{R}^{N})$. If $\phi\in C_{c}^{\infty}(\mathbb{R}^{N})$,
then

$0=\int_{\mathbb{R}^{N}}(w-\Delta
w)\phi=\int_{\mathbb{R}^{N}}w(\phi-\Delta\phi).$

Since $w\in L^{p}(\mathbb{R}^{N})$ the above identity extends by density to
all $\phi\in W^{2,p^{\prime}}(\mathbb{R}^{N})$ and then, since $I-\Delta$ is
invertible from $W^{2,p^{\prime}}(\mathbb{R}^{N})$ to
$L^{p^{\prime}}(\mathbb{R}^{N})$, we have $\int_{\mathbb{R}^{N}}wg=0$ for
every $g\in L^{p^{\prime}}(\mathbb{R}^{N})$, so that $w=0$ and $u=v\in
W^{2,p}(\mathbb{R}^{N})$.

We can now show an equivalent description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$, adapted to the degenerate
operator $D_{yy}+cy^{-1}D_{y}$.

###### Proposition 4.6

Let $c\in\mathbb{R}$ and $\frac{m+1}{p}<c+1-\alpha_{2}$. Then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m}\right.$ $\displaystyle\left.\hskip
43.05542pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;\;and\;\;}\lim_{y\to 0}y^{c}D_{y}u=0\right\\}$

and the norms $\|u\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}$ and

$\|u\|_{L^{p}_{m}}+\|y^{\alpha_{1}}\Delta_{x}u\|_{L^{p}_{m}}+\|y^{\alpha_{2}}(D_{yy}u+cy^{-1}D_{y}u)\|_{L^{p}_{m}}$

are equivalent on $W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

Finally, when $0<\frac{m+1}{p}\leq c-1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\right\\}.$

Proof. Let $\mathcal{G}$ be the space on the right hand side with the
canonical norm indicated above. By Lemma 4.4
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)\subset\mathcal{G}$ and the
embedding is clearly continuous.

Conversely, let $u\in\mathcal{G}$. The estimate for $y^{\alpha_{2}-1}D_{y}u$
follows from Lemma 4.4(ii) and yields, by difference, also that for
$y^{\alpha_{2}}D_{yy}u$. Since for $y\leq 1$ one has
$y^{\frac{\alpha_{2}}{2}}\leq y^{\alpha_{2}-1}$ it follows that
$y^{\frac{\alpha_{2}}{2}}D_{y}u\in L^{p}_{m}(\mathbb{R}^{N}\times[0,1])$ and
$y^{\frac{\alpha_{2}}{2}}D_{y}u\in L^{p}_{m}(\mathbb{R}^{N}\times[1,\infty])$
by Lemma 3.5.

Finally, we prove the inequality

$\|y^{\frac{\alpha_{1}}{2}}D_{x}u\|_{L^{p}_{m}}+\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}\leq
C\left(\|u\|_{L^{p}_{m}}+\|y^{\alpha_{1}}\Delta_{x}u\|_{L^{p}_{m}}\right).$

Since $u(\cdot,y)\in L^{p}(\mathbb{R}^{N})\cap W_{loc}^{2,p}(\mathbb{R}^{N})$
for a.e. $y>0$, the lemma above and the Calderon-Zygmund inequality yield

$\displaystyle\int_{\mathbb{R}^{N}}|D_{x_{i}x_{j}}(x,y)|^{p}\,dx\leq
C\int_{\mathbb{R}^{N}}|\Delta_{x}(x,y)|^{p}\,dx.$

Multiplying by $y^{p\alpha_{1}+m}$ and integrating over $\mathbb{R}_{+}$ we
obtain $\sum_{i,j=1}^{n}\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}\leq
C\|y^{\alpha_{1}}\Delta_{x}u\|_{L^{p}_{m}}$. The estimate

$\|y^{\frac{\alpha_{1}}{2}}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\left(\|y^{\alpha_{1}}\Delta_{x}u\|_{L^{p}_{m}}+\|u\|_{L^{p}_{m}}\right)$

can be obtained similarly using the interpolative inequality

$\|\nabla_{x}u(\cdot,y)\|_{L^{p}(\mathbb{R}^{n})}\leq\epsilon\|\Delta_{x}u(\cdot,y)\|_{L^{p}(\mathbb{R}^{n})}+\frac{C(N,p)}{\epsilon}\|u(\cdot,y)\|_{L^{p}(\mathbb{R}^{n})}$

with $\epsilon=y^{\frac{\alpha_{1}}{2}}$.

The equality for $0<\frac{m+1}{p}\leq c-1$ follows from Lemma 4.4(iii).

We provide now another equivalent description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ which involves a Dirichlet,
rather than Neumann, boundary condition, in a certain range of parameters.

###### Proposition 4.7

Let $c\geq 1$ and $\frac{m+1}{p}<c+1-\alpha_{2}$. The following properties
hold.

  * (i)

If $c>1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m},\right.$ $\displaystyle\left.\hskip
47.36096pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}y^{c-1}u=0\right\\}.$

  * (ii)

If $c=1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m},\right.$ $\displaystyle\left.\hskip
47.36096pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}u(x,y)\in\mathbb{C}\right\\}.$

Proof.  Let us prove (i). By Proposition 4.6 it is sufficient to show that the
conditions $\lim_{y\to 0}y^{c}D_{y}u=0$ and $\lim_{y\to 0}y^{c-1}u=0$ are
equivalent. We proceed as in Lemma 4.4 setting $v:=y^{c}D_{y}u$ and

$f:=y^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)=y^{\alpha_{2}-c}D_{y}v\in
L^{p}_{m}.$

If $v_{0}(x)=\left(y^{c}D_{y}u\right)(x,0)$, then
$g:=y^{\alpha_{2}-c-1}(v-v_{0})\in L^{p}_{m}$ by Proposition 3.2 (ii), and

$D_{y}u=y^{1-\alpha_{2}}g+y^{-c}v_{0}.$ (5)

Then, since $c>1$, we can write for $0<y<1$

$\displaystyle
u(x,1)-u(x,y)=\frac{1}{c-1}v_{0}(x)(y^{1-c}-1)+\int_{y}^{1}s^{1-\alpha_{2}}g(x,s)\,ds$
(6)

and

$\int_{y}^{1}s^{1-\alpha_{2}}|g(x,s)|\,ds\leq\|g\|_{L^{p}_{m}}\left(\int_{y}^{1}s^{(1-\alpha_{2}-\frac{m}{p})p^{\prime}}\right)^{\frac{1}{p^{\prime}}}\leq
C(1+y^{\gamma})$ (7)

with $\gamma=2-\alpha_{2}-(m+1)/p>1-c$ (when $\gamma=0$ the term $y^{\gamma}$
is substituted by $|\log y|^{\frac{1}{p^{\prime}}}$). Since $c>1$, it follows
that

$\displaystyle\lim_{y\to 0}y^{c-1}u(x,y)=\frac{v_{0}(x)}{1-c}$

and therefore $\displaystyle\lim_{y\to 0}y^{c-1}u(x,y)=0$ if and only if
$v_{0}(x)=0$ or, by Lemma 4.4(ii), if $\displaystyle\lim_{y\to
0}y^{c}D_{y}u(x,y)=0$.

To prove (ii) we proceed similarly. From (5) with $c=1$ we obtain

$\displaystyle u(x,1)-u(x,y)=-v_{0}(x)|\log
y|+\int_{y}^{1}s^{1-\alpha_{2}}g(x,s)\,ds,\qquad 0<y<1.$

The parameter $\gamma$ is positive, since $(m+1)/p<2-\alpha_{2}$ and the
integral on the right hand side of (6) converges. Therefore
$\displaystyle\lim_{y\to 0}u(x,y)\in\mathbb{C}$ if and only if $v_{0}(x)=0$.

###### Remark 4.8

We point out that the function $v=y^{c-1}u$ above satisfies $D_{y}v\in
L^{1}\left(Q\times[0,1]\right)$ for every cube $Q$. In particular $D_{y}u\in
L^{1}\left(Q\times[0,1]\right)$, if $\frac{m+1}{p}<2-\alpha_{2}$, by choosing
$c=1$.

Indeed, if $c>1$, using (6), (7) with $v_{0}=0$ one has $y^{c-2}u\in
L^{1}\left(Q\times[0,1]\right)$. Then the equality

$D_{y}v=y^{c-1}D_{y}u+(c-1)y^{c-2}u=y^{c-\alpha_{2}}g+(c-1)y^{c-2}u$

and $g\in L^{p}_{m}$ and Hölder inequality yield $y^{c-\alpha_{2}}g\in
L^{1}(Q\times[0,1])$.

When $c=1$ then $v=u$ and we use (5) with $v_{0}=0$ and then (7), since
$\gamma>0$, as observed in the above proof.

###  4.2 Approximation with smooth functions

The main result of this section is a density property of smooth functions in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$. We introduce the set

$\mathcal{C}:=\left\\{u\in
C_{c}^{\infty}\left(\mathbb{R}^{N}\times[0,\infty)\right),\ D_{y}u(x,y)=0\
{\rm for}\ y\leq\delta\ {\rm and\ some\ }\delta>0\right\\}$ (8)

and its one dimensional version

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)),\ D_{y}u(y)=0\ {\rm for}\
y\leq\delta\ {\rm and\ some\ }\delta>0\right\\}.$ (9)

Let

$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}=\left\\{u(x,y)=\sum_{i}u_{i}(x)v_{i}(y),\
u_{i}\in C_{c}^{\infty}(\mathbb{R}^{N}),\ v_{i}\in\cal D\right\\}$

(finite sums). Clearly
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subset\mathcal{C}$.

###### Theorem 4.9

If $\frac{m+1}{p}>\alpha_{1}^{-}$ then
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is dense in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

Note that the condition $(m+1)/p>\alpha_{1}^{-}$ or $m+1>0$ and
$(m+1)/p+\alpha_{1}>0$ is necessary for the inclusion
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subset
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

For technical reason we start from the case $\alpha_{2}=0$ and write $\alpha$
for $\alpha_{1}$. Then

$W^{2,p}_{\mathcal{N}}(\alpha,0,m)=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):\ u,\,y^{\alpha}D_{x_{i}x_{j}}u,\
y^{\frac{\alpha}{2}}D_{x_{i}}u,\ D_{y}u\ D_{yy}u,\ \frac{D_{y}u}{y}\in
L^{p}_{m}\right\\}.$

We need some preliminary results which show the density of smooth functions
with compact support in $W^{2,p}_{\mathcal{N}}(\alpha,0,m)$. In the first no
restriction on $\alpha$ is needed.

###### Lemma 4.10

The functions in $W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ having support in
$\mathbb{R}^{N}\times[0,b[$ for some $b>0$ are dense in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$.

Proof. Let $0\leq\phi\leq 1$ be a smooth function depending only on the $y$
variable which is equal to $1$ in $(0,1)$ and to $0$ for $y\geq 2$. Set
$\phi_{n}(y)=\phi\left(\frac{y}{n}\right)$ and $u_{n}(x,y)=\phi_{n}(y)u(x,y)$.
Then $u_{n}\in W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ and has compact support in
$\mathbb{R}^{N}\times[0,2n]$. By dominated convergence $u_{n}\to u$ in
$L^{p}_{m}$. Since $D_{x_{i}x_{j}}u_{n}=\phi_{n}D_{x_{i}x_{j}}u$,
$D_{x_{i}}u_{n}=\phi_{n}D_{x_{i}}u$ we have $y^{\alpha}D_{x_{i}x_{j}}u_{n}\to
y^{\alpha}D_{x_{i}x_{j}}u$, $y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}\to
y^{\frac{\alpha}{2}}D_{x_{i}}u$ , by dominated convergence again.

For the convergence of the $y$-derivatives, we observe that
$|D_{y}\phi_{n}|\leq\frac{C}{n}\chi_{[n,2n]}$,
$|D_{yy}\phi_{n}|\leq\frac{C}{n^{2}}\chi_{[n,2n]}$. Since
$D_{y}u_{n}=\phi_{n}D_{y}u+D_{y}\phi_{n}u$ and
$D_{yy}u_{n}=\phi_{n}D_{yy}u+2D_{y}\phi_{n}D_{y}u+uD_{yy}\phi_{n}$, we have
also $D_{y}u_{n}\to D_{y}u$, $D_{yy}u_{n}\to D_{yy}u$ and
$\frac{D_{y}u_{n}}{y}\to\frac{D_{y}u}{y}$ in $L^{p}_{m}$.

###### Lemma 4.11

Assume that $\frac{m+1}{p}<2$ and $\frac{m+1}{p}+\alpha>0$. Then the functions
in $W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ with compact support are dense in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$.

Proof. Let $u\in W^{2,p}_{\mathcal{N}}(\alpha,0,m)$. By Lemma 4.10, we may
assume that $u$ has support in $\mathbb{R}^{N}\times[0,b[$ for some $b>0$. Let
$0\leq\phi\leq 1$ be a smooth function depending only on the $x$ variable
which is equal to $1$ if $|x|\leq 1$ and to $0$ for $|x|\geq 2$. Set
$\phi_{n}(x)=\phi\left(\frac{x}{n}\right)$ and $u_{n}(x,y)=\phi_{n}(x)u(x,y)$.
Then $u_{n}\in W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ and has compact support. By
dominated convergence $u_{n}\to u$ in $L^{p}_{m}$. Moreover, since
$D_{y}u_{n}=\phi_{n}D_{y}u$ and $D_{yy}u_{n}=\phi_{n}D_{yy}u$, we have
immediately $D_{yy}u_{n}\to D_{yy}u$,
$\frac{D_{y}u_{n}}{y}\to\frac{D_{y}u}{y}$ in $L^{p}_{m}$.

Concerning the derivatives with respect to the $x$ variable, we have
$|D_{x_{i}}\phi_{n}(x)|\leq\frac{C}{n}\chi_{[n,2n]}(|x|)$,
$|D_{x_{i}x_{j}}\phi_{n}(x)|\leq\frac{C}{n^{2}}\chi_{[n,2n]}(|x|)$ and

$\displaystyle D_{x_{i}}u_{n}$
$\displaystyle=\phi_{n}D_{x_{i}}u+uD_{x_{i}}\phi_{n},$ $\displaystyle
D_{x_{j}x_{i}}u_{n}$
$\displaystyle=\phi_{n}D_{x_{i}x_{j}}u+D_{x_{j}}\phi_{n}D_{x_{i}}u+D_{x_{j}}uD_{x_{i}}\phi_{n}+uD_{x_{i}x_{j}}\phi_{n}.$
(10)

Let us show that $y^{\alpha}u,\ y^{\frac{\alpha}{2}}u\in L^{p}_{m}$. Since $u$
has support in $\mathbb{R}^{N}\times[0,b[$ this is trivial for $\alpha\geq 0$.
When $\alpha<0$ let $f(x,y)=\frac{u(x,y)-u(x,0)}{y^{2}}$ so that

$\displaystyle y^{\alpha}u=y^{\alpha+2}f+y^{\alpha}u(\cdot,0).$

By Proposition 3.2, $f\in L^{p}_{m}$ and $u(\cdot,0)\in
L^{p}(\mathbb{R}^{N})$. Since $u$ and $f$ have support in
$\mathbb{R}^{N}\times[0,b[$, the assumption $-\alpha<\frac{m+1}{p}<2$ then
implies that $y^{\alpha}u\in L^{p}_{m}$ and also $y^{\frac{\alpha}{2}}u\in
L^{p}_{m}$. Using the classic interpolative inequality

$\|\nabla_{x}u(\cdot,y)\|_{L^{p}(\mathbb{R}^{N})}\leq\epsilon\|\Delta_{x}u(\cdot,y)\|_{L^{p}(\mathbb{R}^{N})}+\frac{C(N,p)}{\epsilon}\|u(\cdot,y)\|_{L^{p}(\mathbb{R}^{N})}$

with $\epsilon=1$ we easily get (after raising to the power $p$, multiplying
by $y^{\alpha}$ and integrating in $y$), $y^{\alpha}\nabla_{x}u\in L^{p}_{m}$.
Using (4.2) and the fact that $y^{\alpha}u,\ y^{\frac{\alpha}{2}}u,\
y^{\alpha}\nabla_{x}u\in L^{p}_{m}$ we deduce using dominated convergence that
$y^{\alpha}D_{x_{i}x_{j}}u_{n}\to y^{\alpha}D_{x_{i}x_{j}}u$,
$y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}\to y^{\frac{\alpha}{2}}D_{x_{i}}u$ in
$L^{p}_{m}$.

In the next lemma we add regularity with respect to the $x$-variable.

###### Lemma 4.12

Let $\frac{m+1}{p}<2$, $\frac{m+1}{p}+\alpha>0$ and $u\in
W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ with compact support. Then there exist
$(u_{n})_{n\in\mathbb{N}}\subset W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ with
compact support such that $u_{n}$ converges to $u$ in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ and, for every $y\geq 0$, $u_{n}(\cdot,y)$
belongs to $C^{\infty}(\mathbb{R}^{N})$ and has bounded $x$-derivatives of any
order.

Proof. Let $u\in W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ as above and let us fix a
standard sequence of mollifiers in $\mathbb{R}^{N}$ $\rho_{n}=n^{N}\rho(nx)$
where $0\leq\rho\in C_{c}^{\infty}(\mathbb{R}^{N})$,
$\int_{\mathbb{R}^{N}}\rho(x)\ dx=1$. Let us set
$u_{n}(x,y)=\left(\rho_{n}\ast u(\cdot,y)\right)(x)$ where $\ast$ means
convolution with respect to the $x$ variable.

By Lemma 3.1 and Proposition 3.2, $u(\cdot,y)\in L^{p}(\mathbb{R}^{N})$ for
every $y\geq 0$ and therefore, by standard properties, $u_{n}$ has a compact
support and $u_{n}(\cdot,y)\in C^{\infty}_{b}(\mathbb{R}^{N})$ for every
$y\geq 0$. By Young’s inequality

$\displaystyle\|u_{n}(\cdot,y)\|_{L^{p}(\mathbb{R}^{N})}\leq\|u(\cdot,y)\|_{L^{p}(\mathbb{R}^{N})},\qquad
u_{n}(\cdot,y)\to u(\cdot,y)\quad\text{in}\quad
L^{p}(\mathbb{R}^{N}),\qquad\forall y\geq 0.$

Raising to the power $p$, multiplying by $y^{m}$ and by integrating with
respect to $y$, we get

$\displaystyle\|u_{n}\|_{L^{p}_{m}}\leq\|u\|_{L^{p}_{m}}$

which, using dominated convergence with respect to $y$, implies $u_{n}\to u$
in $L^{p}_{m}$.

Using the equalities

$\displaystyle y^{\alpha}D_{x_{i}x_{j}}u_{n}$
$\displaystyle=\rho_{n}\ast(y^{\alpha}D_{x_{i}x_{j}}u),\qquad
y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}=\rho_{n}\ast(y^{\frac{\alpha}{2}}D_{x_{i}}u),$
$\displaystyle D_{yy}u_{n}$ $\displaystyle=\rho_{n}\ast D_{yy}u,\hskip
62.43036pty^{\gamma}D_{y}u_{n}=\rho_{n}\ast(y^{\gamma}D_{y}u),$

$\gamma=0,1$ a similar argument as before yields $u_{n}\to u$ in
$W^{2,p}_{n}(\alpha,0,m)$.

We can now prove a weaker version of Theorem 4.9 when $\alpha_{2}=0$.

###### Proposition 4.13

If $\frac{m+1}{p}>\alpha^{-}$ then $\mathcal{C}$, defined in (8), is dense in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$.

Proof. (i) We first consider the case $\frac{m+1}{p}>2$. Let $u\in
W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ which, by Lemma 4.10, we may assume to have
the support in $\mathbb{R}^{N}\times[0,b]$. Let $\phi$ be a smooth function
depending only on $y$, equal to $0$ in $(0,1)$ and to $1$ for $y\geq 2$. Let
$\phi_{n}(y)=\phi(ny)$ and $u_{n}(x,y)=\phi_{n}(y)u(x,y)$. Then

$\displaystyle D_{x_{i}x_{j}}u_{n}$
$\displaystyle=\phi_{n}D_{x_{i}x_{j}}u,\hskip
51.6665ptD_{x_{i}}u_{n}=\phi_{n}D_{x_{i}}u,$ $\displaystyle D_{y}u_{n}$
$\displaystyle=\phi_{n}D_{y}u+D_{y}\phi_{n}u,\qquad
D_{yy}u_{n}=\phi_{n}D_{yy}u+2D_{y}\phi_{n}D_{y}u+uD_{yy}\phi_{n}.$

By dominated convergence $u_{n}\to u$, $y^{\alpha}D_{x_{i}x_{j}}u_{n}\to
y^{\alpha}D_{x_{i}x_{j}}u$, $y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}\to
y^{\frac{\alpha}{2}}D_{x_{i}}u$ in $L^{p}_{m}$. Let us consider now the terms
containing the $y$ derivatives and observe that

$\displaystyle|D_{y}\phi_{n}|\leq
Cn\chi_{[\frac{1}{n},\frac{2}{n}]}\leq\frac{2C}{y}\chi_{[\frac{1}{n},\frac{2}{n}]},\qquad|D_{yy}\phi_{n}|\leq
Cn^{2}\chi_{[\frac{1}{n},\frac{2}{n}]}\leq\frac{4C}{y^{2}}\chi_{[\frac{1}{n},\frac{2}{n}]}.$
(11)

Using these estimates and since $y^{-2}u\in L^{p}_{m}$ by Proposition 3.2

$\frac{D_{y}u_{n}}{y}=\phi_{n}\frac{D_{y}u}{y}+\frac{u}{y}(D_{y}\phi_{n})\to\frac{D_{y}u}{y}$

in $L^{p}_{m}$, by dominated convergence.

In a similar way one shows that $D_{y}u_{n}\to D_{y}u$ and $D_{yy}u_{n}\to
D_{yy}u$ in $L^{p}_{m}$ and hence functions with compact support in
$\mathbb{R}^{n}\times]0,\infty[$ are dense in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$. At this point, a standard smoothing by
convolutions shows the density of
$C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$ in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$.

(ii) Let $\frac{m+1}{p}=2$. We proceed similarly to (i) and fix $u\in
W^{2,p}_{\mathcal{N}}(\alpha,0,m)$ with support in
$\mathbb{R}^{N}\times[0,b]$. Let $\phi$ be a smooth function which is equal to
$0$ in $\left(0,\frac{1}{4}\right)$ and to $1$ for $y\geq\frac{1}{2}$. Let
$\phi_{n}(y)=\phi\left(y^{\frac{1}{n}}\right)$ and $u_{n}=\phi_{n}u$. By
dominated convergence it is immediate to see that $u_{n}\to u$,
$y^{\alpha}D_{x_{i}x_{j}}u_{n}\to y^{\alpha}D_{x_{i}x_{j}}u$,
$y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}\to y^{\frac{\alpha}{2}}D_{x_{i}}u$ in
$L^{p}_{m}$. To treat the terms concerning the $y$ derivatives we observe that

$\displaystyle|\phi_{n}^{\prime}|$
$\displaystyle=\left|\frac{1}{n}\phi^{\prime}\left(y^{\frac{1}{n}}\right)y^{\frac{1}{n}-1}\right|\leq\frac{C}{ny}\chi_{[(\frac{1}{4})^{n},(\frac{1}{2})^{n}]}$
$\displaystyle|\phi_{n}^{\prime\prime}|$
$\displaystyle=\left|\frac{1}{n^{2}}\phi^{\prime\prime}\left(y^{\frac{1}{n}}\right)y^{\frac{2}{n}-2}+\frac{1}{n}(\frac{1}{n}-1)\phi^{\prime}\left(y^{\frac{1}{n}}\right)y^{\frac{1}{n}-2}\right|\leq\frac{C}{ny^{2}}\chi_{[(\frac{1}{4})^{n},(\frac{1}{2})^{n}]}.$
(12)

Moreover,

$\displaystyle D_{y}u_{n}=\phi_{n}D_{y}u+\phi_{n}^{\prime}u,\qquad
D_{yy}u_{n}=\phi_{n}D_{yy}u+2\phi_{n}^{\prime}D_{y}u+\phi_{n}^{\prime\prime}u.$

Then $\frac{1}{y}D_{y}u_{n}\to\frac{1}{y}D_{y}u$ in $L^{p}_{m}$ since
$\phi_{n}\frac{D_{y}u}{y}\to\frac{1}{y}D_{y}u$ by dominated convergence and
$\phi_{n}^{\prime}\frac{u}{y}\to 0$. In fact, using (4.2) and (3) of Lemma 3.3
we have

$\displaystyle\left\|\phi_{n}^{\prime}\frac{u}{y}\right\|^{p}_{L^{p}_{m}}\leq\frac{C}{n^{p}}\int_{(\frac{1}{4})^{n}}^{(\frac{1}{2})^{n}}|\log
y|^{p-1}y^{m-2p}\,dy=\frac{C}{n^{2p}}\int_{\frac{1}{4}}^{\frac{1}{2}}|\log
s|^{p-1}s^{-1}\,dy$

which tends to $0$ as $n\to\infty$.

Concerning the second order derivative we have $D_{yy}u_{n}\to D_{yy}u$ since
$\phi_{n}D_{yy}u\to D_{yy}u$ by dominated convergence and the other terms tend
to $0$. Indeed proceeding as before we have $|\phi_{n}^{\prime}D_{y}u|\leq
C\frac{C}{n}\chi_{[(\frac{1}{4})^{n},(\frac{1}{2})^{n}]}\frac{|D_{y}u|}{y}$
which tends to $0$ by dominated convergence. Finally,

$\displaystyle\|\phi_{n}^{\prime\prime}u\|^{p}_{L^{p}_{m}}\leq\frac{C}{n^{p}}\int_{(\frac{1}{4})^{n}}^{(\frac{1}{2})^{n}}|\log
y|^{p-1}y^{m-2p}\,dy=\frac{C}{n^{2p}}\int_{\frac{1}{4}}^{\frac{1}{2}}|\log
s|^{p-1}s^{-1}\,dy$

which tends to $0$ as $n\to\infty$.

Now the proof is as for (i) and shows that
$C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$ is dense in
$W^{2,p}_{\mathcal{N}}(\alpha,0,m)$.

(iii) Let assume finally that $\frac{m+1}{p}<2$. By Lemmas 4.11, 4.12 we may
assume that $u$ has compact support and that for every $y\geq 0$,
$u(\cdot,y)\in C^{\infty}_{b}(\mathbb{R}^{N})$.

By Proposition 3.2, $\frac{u-u(\cdot,0)}{y^{2}}\in L^{p}_{m}$. Let $\phi$ be a
smooth function equal to $0$ in $(0,1)$ and to $1$ for $y\geq 2$ and
$\phi_{n}(y)=\phi(ny)$. Setting

$u_{n}(x,y)=(1-\phi_{n}(y))u(x,0)+\phi_{n}(y)u(x,y),$

then

$\displaystyle D_{x_{i}}u_{n}$
$\displaystyle=(1-\phi_{n})D_{x_{i}}u(\cdot,0)+\phi_{n}D_{x_{i}}u,$
$\displaystyle D_{x_{i}x_{j}}u_{n}$
$\displaystyle=(1-\phi_{n})D_{x_{i}x_{j}}u(\cdot,0)+\phi_{n}D_{x_{i}x_{j}}u,$
$\displaystyle D_{y}u_{n}$
$\displaystyle=\phi_{n}^{\prime}(u-u(\cdot,0))+\phi_{n}D_{y}u,$ $\displaystyle
D_{yy}u_{n}$
$\displaystyle=\phi_{n}^{\prime\prime}(u-u(\cdot,0))+2\phi_{n}^{\prime}D_{y}u+\phi_{n}D_{yy}u.$

It follows that $u_{n}\to u$, $y^{\alpha}D_{x_{i}x_{j}}u_{n}\to
y^{\alpha}D_{x_{i}x_{j}}u$, $y^{\frac{\alpha}{2}}D_{x_{i}}u_{n}\to
y^{\frac{\alpha}{2}}D_{x_{i}}u$ in $L^{p}_{m}$. Since the argument is always
the same, let us explain it for $u_{n}$. The term $\phi_{n}u$ converges to $u$
by dominated convergence and $(1-\phi_{n})u(\cdot,0)$ converges to zero since
$u(\cdot,0)$ is bounded with compact support.

Using (11) one has

$\frac{|\phi_{n}^{\prime}(u-u(\cdot,0))|}{y}\leq
C\chi_{[\frac{1}{n},\frac{2}{n}]}(y)\frac{|u-u(\cdot,0)|}{y^{2}}$

which tend to $0$ in $L^{p}_{m}$ by dominated convergence and then
$\frac{D_{y}u_{n}}{y}$ converges to $\frac{D_{y}u}{y}$ in $L^{p}_{m}$.
Similarly $D_{yy}u_{n}$ converges $D_{yy}u$ in $L^{p}_{m}$.

Each function $u_{n}$ has compact support, does not depend on $y$ for small
$y$ and is smooth with respect to the $x$ variable for any fixed $y$.
Smoothness with respect to $y$ is however not yet guaranteed. This last
property can be added by taking appropriate convolutions in $y$ with a compact
support mollifier.

We can now prove the general density result.

Proof of Theorem 4.9 The density of $\mathcal{C}$, defined in (8), in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ follows by Lemma 2.2 and
Proposition 4.13 since the isometry $T_{0,-\frac{\alpha_{2}}{2}}$
isometrically maps dense subsets of
$W^{2,p}_{\mathcal{N}}(\tilde{\alpha},0,\tilde{m})$ into dense subsets of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ and, since $\alpha_{2}<2$,
leaves invariant $\mathcal{C}$. Note also that the conditions
$(m+1)/p>\alpha_{1}^{-}$ and $(\tilde{m}+1)/p>\tilde{\alpha}^{-}$ are
equivalent, since $\alpha_{2}<2$, again.

In order to prove the density of
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$, we may therefore assume
that $u$ is in $\mathcal{C}$, that is $u\in
C_{c}^{\infty}(\mathbb{R}^{N}\times[0,\infty))$ and $D_{y}u(x,y)=0$ for
$y\leq\delta$ for some $\delta>0$. Let $\eta$ be a smooth function depending
only on the $y$ variable which is equal to $1$ in $[0,\frac{\delta}{2}]$ and
to $0$ for $y\geq\delta$. Then, since $D_{y}u(x,y)=0$ for $y\leq\delta$,

$u(x,y)=\eta(y)u(x,y)+(1-\eta(y))u(x,y)=\eta(y)w(x)+(1-\eta(y))u(x,y)=u_{1}(x,y)+u_{2}(x,y)$

with $u_{1}(x,y)=\eta(y)w(x)$, $w(x)$ depending only on the $x$ variable.
Observe now that $u_{2}(x,y)=(1-\eta(y))u(x,y)=0$ in $[0,\frac{\delta}{2}]$
and outside the support of $u$, therefore it belongs to
$C^{\infty}_{c}(\mathbb{R}^{N+1}_{+})$ and the approximation with respect to
the $W^{2,p}(\mathbb{R}^{N+1}_{+})$ norm by functions in
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes C_{c}^{\infty}(]0,\infty[)$ is standard
(just use a sequence of polynomials converging uniformly to $u_{2}$ with all
first and second order derivatives on a cube containing the support of $u_{2}$
and truncate outside the cube by a cut-off of the same type). This proves the
result.

###### Remark 4.14

From the proofs of Proposition 4.13 and Theorem 4.9 it follows that if $u\in
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ has support in
$\mathbb{R}^{N}\times[0,b]$, then there exists a sequence
$\left(u_{n}\right)_{n\in\mathbb{N}}\in\mathcal{C}$ such that $\mbox{supp
}u_{n}\subseteq\mathbb{R}^{N}\times[0,b]$ and $u_{n}\to u$ in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

###### Corollary 4.15

Assume $\frac{m+1}{p}\geq 2-\alpha_{2}$ and $\frac{m+1}{p}>\alpha_{1}^{-}$.
Then $C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ and
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes C_{c}^{\infty}\left(]0,\infty[\right)$
are dense in $W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

Proof.  This follows from the the proofs of Proposition 4.13 and of Theorem
4.9.

Specializing Proposition 3.2 to
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ we get the following
corollary.

###### Corollary 4.16

Let $\frac{m+1}{p}>\alpha_{1}^{-}$. The following properties hold for any
$u\in W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

  * (i)

If $\frac{m+1}{p}>1-\frac{\alpha_{2}}{2}$ then

$\displaystyle\|y^{\frac{\alpha_{2}}{2}-1}u\|_{L^{p}_{m}}\leq
C\|y^{\frac{\alpha_{2}}{2}}D_{y}u\|_{L^{p}_{m}}.$

  * (ii)

If $\frac{m+1}{p}>2-\alpha_{2}$ then

$\displaystyle\|y^{\alpha_{2}-2}u\|_{L^{p}_{m}}\leq
C\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}.$

  * (iii)

If $\frac{m+1}{p}<2-\alpha_{2}$ then

$\displaystyle\|y^{\alpha_{2}-2}(u-u(\cdot,0))\|_{L^{p}_{m}}\leq
C\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}.$

  * (iv)

If $\alpha_{2}-\alpha_{1}<2$ and
$\frac{m+1}{p}>1-\frac{\alpha_{1}+\alpha_{2}}{2}$,
$\frac{m+1}{p}>\alpha_{1}^{-}$ then

$\displaystyle\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}-1}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{y}\nabla_{x}u\|_{L^{p}_{m}}.$

Proof. By density we may assume that $u\in
C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$. All points follow by
applying Proposition 3.2 to $u$ in the cases (i), (ii) and (iii) and to
$\nabla_{x}u$ in the case (iv), recalling Proposition 4.1.

##  5 The space $W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$

We consider an integral version of Dirichlet boundary conditions, namely a
weighted summability of $y^{-2}u$ and introduce for $m\in\mathbb{R}$,
$\alpha_{2}<2$

$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):y^{\alpha_{2}-2}u\in L^{p}_{m}\\}$ (13)

with the norm

$\|u\|_{W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-2}u\|_{L^{p}_{m}}.$

We remark that $W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ will be
considered for every $m\in\mathbb{R}$ whereas
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ only for $m+1>0$. The symbol
$\mathcal{R}$ stands for ”Rellich”, since Rellich inequalities concern with
the summability of $y^{-2}u$.

###### Proposition 5.1

The following properties hold.

  * (i)

if $u\in W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ then
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}.$

  * (ii)

If $\alpha_{2}-\alpha_{1}<2$ and $\frac{m+1}{p}>2-\alpha_{2}$, then
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)=W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}(\alpha_{1},\alpha_{2},m)$,
with equivalence of the corresponding norms. In particular,
$C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ is dense in
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$.

Proof.  The proof of (i) follows by integrating with respect to $x$ the
inequality of Lemma 3.4. The proof of (ii) follows from Proposition 4.3(i) and
Corollary 4.16(ii), after noticing that $\alpha_{2}-\alpha_{1}<2$ and
$\frac{m+1}{p}>2-\alpha_{2}$ yield $\frac{m+1}{p}>\alpha_{1}^{-}$. The density
of $C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ in
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ now follows from Corollary
4.15.

Finally, we investigate the action of the multiplication operator
$T_{k,0}:u\mapsto y^{k}u$. The following lemma is the companion of Lemma 2.2
which deals with the transformation $T_{0,\beta}$.

###### Lemma 5.2

Let $\alpha_{2}-\alpha_{1}<2$ and $\frac{m+1}{p}>2-\alpha_{2}$. For every
$k\in\mathbb{R}$

$\displaystyle T_{k,0}:W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)\to
W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$

is an isomorphism (we shall write
$y^{k}W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$).

Proof. Let $u=y^{k}v$ with $v\in
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$. Since all $x$-derivatives
commute with $T_{k,0}$ we deal only with the $y$-derivatives. We observe that

$\displaystyle D_{y}u=y^{k}(D_{y}v+k\frac{v}{y}),\qquad
D_{yy}u=y^{k}\left(D_{yy}v+2k\frac{D_{y}v}{y}+k(k-1)\frac{v}{y^{2}}\right).$

Corollary 4.16 yields

$\displaystyle\|y^{\alpha_{2}-2}v\|_{L^{p}_{m}}+\|y^{\frac{\alpha_{2}}{2}-1}v\|_{L^{p}_{m}}+\|y^{\alpha_{2}-1}D_{y}v\|_{L^{p}_{m}}\leq
c\|v\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}$

and then $u\in W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$. Conversely,
if $u\in W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$, then
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m-kp}$ by Proposition 5.1(i) and similar
formulas as above show that $y^{\alpha_{2}-1}D_{y}v,y^{\alpha_{2}}D_{yy}v\in
L^{p}_{m}$. Since $y^{\alpha_{2}/2-1}\leq 1+y^{\alpha_{2}-2}$, then
$y^{\alpha_{2}/2-1}u\in L^{p}_{m-kp}$ and $y^{\alpha/2}D_{y}v\in L^{p}_{m}$.

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