11institutetext: European Southern Observatory, Karl-Schwarzschild-Straße 2,
D-85748 Garching bei München, Germany 11email<EMAIL_ADDRESS>22institutetext: INAF – Osservatorio di Astrofisica e Scienza dello Spazio,
via Piero Gobetti 93/3, 40129 Bologna, Italy 33institutetext: Technische
Universität München, TUM School of Natural Sciences, Physics Department,
James-Frank-Straße 1, D-85748 Garching bei München, Germany 44institutetext:
Simon Fraser University, Department of Physics, 8888 University Dr, Burnaby,
Canada V5A 1S6

# The neutrino background from non-jetted active galactic nuclei

P. Padovani 11 R. Gilli 22 E. Resconi 33 C. Bellenghi 33 F. Henningsen 44

(Received 19 March 2024; accepted 5 April 2024)

###### Abstract

Aims. We calculate the contribution to the neutrino background from the non-
jetted active galactic nuclei (AGN) population following the recent IceCube
association of TeV neutrinos with NGC 1068.

Methods. We exploit our robust knowledge of the AGN X-ray luminosity function
and evolution and convert it to the neutrino band by using NGC 1068 as a
benchmark and a theoretically motivated neutrino spectrum.

Results. The resulting neutrino background up to redshift 5 does not violate
either the IceCube diffuse flux or the upper bounds for non-jetted AGN,
although barely so. This is consistent with a scenario where the latter class
makes a substantial contribution mostly below 1 PeV, while jetted AGN, i.e.
blazars, dominate above this energy, in intriguing agreement with the dip in
the neutrino data at $\sim 300\,$TeV. More and better IceCube data on Seyfert
galaxies will allow us to constrain the fraction of neutrino emitters among
non-jetted AGN.

###### Key Words.:

Galaxies: active, Seyfert – quasars: general – Neutrinos – Radiation
mechanisms: non-thermal – X-rays: diffuse background

##  1 Introduction

IceCube has recently detected an excess of TeV neutrinos with a 4.2 $\sigma$
significance from the direction of NGC 1068, the prototype Seyfert II galaxy
(IceCube Collaboration et al., 2022). This event was somewhat unexpected. Non-
jetted AGN111We follow Padovani (2017) and define as jetted AGN with strong
relativistic jets, which is not the case for NGC 1068, as discussed in
Padovani et al. (2024). such as NGC 1068, in fact, are usually characterised
by thermal emission, in contrast to jetted AGN, known for their predominantly
non-thermal radiation (e.g. Padovani et al., 2017). Only the latter class was
deemed to have the capability to accelerate protons to the energies necessary
for neutrino production, as discussed, for example, in the review by Giommi &
Padovani (2021) (and references therein; but see Berezinsky 1977; Eichler
1979; Silberberg & Shapiro 1979).

Being so close, NGC 1068 can be spatially resolved into a number of
components, all possibly relevant to neutrino production (Padovani et al.,
2024). These include: 1. a starburst region in the spiral arms of its host
galaxy; 2. a $\lesssim$ kpc jet; 3. a sub-kpc molecular outflow; 4. and the
supermassive black hole (SMBH) vicinity. By first using simple order-of-
magnitude arguments and then applying specific theoretical models, Padovani et
al. (2024) (and references therein) have come to the conclusion that only the
region close to the accretion disc around the SMBH, most likely the X-ray
emitting corona, fulfils the conditions to have both the right density of
photons needed to provide the targets for protons to sustain neutrino
production and to absorb the expected but unobserved $\gamma$ rays.

How unique is NGC 1068? At present the extent to which the process that
produces neutrinos in NGC 1068 generalise to the wider AGN population is
unknown. Assuming a tight connection between neutrino emission and the plasma
in AGN coronae, theoretical studies of Seyfert galaxies that host X-ray bright
AGN have been carried out to single out possible new neutrino sources (e.g.
Kheirandish et al., 2021).

The purpose of this Letter is to approach this question from an observational
and population-wide perspective. We leverage the well-established knowledge of
X-ray luminosity functions (XLF) and evolution of AGN, along with the
constraints provided by the cosmic X-ray background (XRB; Gilli et al. 2007).
We then translate these findings to the neutrino band by normalizing them to
NGC 1068. In short, our aim is to utilize the limited neutrino data, both on
the observational and modelling side, to investigate the implications of
extrapolating from a single non-jetted AGN to the entire population. We assume
a distance to this source of 10.1 Mpc, following Padovani et al. (2024).

##  2 Population synthesis of the X-ray background

The spectrum of the XRB records the integrated emission of AGN across all
cosmic times (Setti & Woltjer, 1989). Several papers (e.g. Comastri et al.,
1995; Treister & Urry, 2006; Akylas et al., 2012; Ueda et al., 2014; Ananna et
al., 2019) demonstrated that the peak around $20-30$ keV in the XRB spectrum
cannot be reproduced by the emission of unobscured and moderately-obscured AGN
alone, but that a large population of heavily obscured, Compton-thick
AGN222These are defined by a column density along our line-of-sight $N_{\rm
H}>1/\sigma_{T}\sim 10^{24}$cm-2, where $\sigma_{T}$ is the Thomson cross-
section. is required. To date, population synthesis models of the XRB provide
the best inference about the overall abundance of accreting SMBHs. We adopt
here the model of Gilli et al. (2007) which, besides accurately fitting the
XRB spectrum, is in very good agreement with the most recent estimates of the
AGN XLFs (Ueda et al. 2014) and number counts in different redshift intervals
and X-ray bands (Luo et al., 2017; Nanni et al., 2020; Marchesi et al., 2020).
Starting from the $0.5-2$ keV XLF of unobscured AGN described in Hasinger et
al. (2005), and assuming a distribution of AGN obscuring column densities in
agreement with observational constraints (Risaliti et al., 1999; Tozzi et al.,
2006), Gilli et al. (2007) found that moderately obscured, Compton-thin AGN
($N_{\rm H}=10^{22-24}$cm-2) need to outnumber unobscured AGN (by a factor
decreasing from 4 to 1 with increasing luminosity) to reproduce the measured
$2-10$ keV AGN XLF (Ueda et al., 2003; La Franca et al., 2005). Furthermore, a
luminosity-dependent space density of Compton-thick AGN equal to that of
moderately obscured Compton-thin AGN was required to fit the XRB spectral peak
(assuming all populations follow the same cosmological evolution). We remark
that the AGN XLF used in the Gilli et al. (2007) are largely ($\gtrsim 90\%$)
dominated by non-jetted AGN. As discussed below, our computation will
specifically refer to the contribution to the neutrino background from the
non-jetted AGN population only.

##  3 Predicting the neutrino background

To compute the expected neutrino background using CXB population synthesis
models we first extended the AGN X-ray spectra considered by Gilli et al.
(2007) to the neutrino domain.

We first assumed that the $10^{-4}-10^{3}$ TeV neutrino spectrum of each non-
jetted AGN follows the “minimal $pp$ scenario” model of Murase (2022) (as
shown in their Fig. 3, left and middle panels), which provides an X-ray
corona-based theoretical fit to the NGC 1068 IceCube data. Such a spectrum
peaks at $E\sim 1$ TeV and has a slope in the $\sim 2-15$ TeV energy range
consistent with what IceCube has observed [$\gamma=3.2\pm 0.2$, where
$n(E)\propto E^{-\gamma}$ and $n(E)$ is the neutrino particle spectrum;
IceCube Collaboration et al. 2022].

As for the relative normalization between neutrino and X-ray AGN spectra, we
resorted to the observed X-ray and neutrino fluxes (or luminosities) of
NGC1068, the only non-jetted AGN for which these two crucial parameters are
available. We remark that what matters here is the $intrinsic$, i.e. corrected
for absorption, AGN X-ray flux (or luminosity), since the total background is
obtained by integrating $intrinsic$ AGN luminosity functions. The best
estimate of the intrinsic X-ray flux of NGC1068 is arguably the one presented
by Marinucci et al. (2016), who, by means of NuSTAR observations, obtained
spectra of the target up to $\sim 60$ keV at different epochs and discovered a
transient decrease of the column density along the line of sight, from
$\gtrsim 10^{25}$ to $7\times 10^{24}$ cm-2, which was sufficient to reveal
its direct nuclear radiation. The intrinsic flux density at 1 keV was found to
be $f_{1\;keV}=1.44\times 10^{-9}$ erg cm-2 s-1 keV-1. We then derived the
all-flavour neutrino flux density at 4 TeV, where it is likely best
constrained, by considering the muon-type neutrino spectrum measured by
IceCube, $EF(E)=5\times 10^{-11}(E/1{\rm TeV})^{-1.2}$ TeV cm-2 s-1, and
multiplying it by 3 (which assumes vacuum neutrino mixing). This gives a
neutrino flux density of $f_{4\;TeV}=1.14\times 10^{-20}$ erg cm-2 s-1 keV-1,
and hence an X-ray to neutrino flux ratio $\nu f_{\nu}|_{1\;keV}/\nu
f_{\nu}|_{4\;TeV}=31.5$. The uncertainty on this value, taking into account
the errors on the X-ray and neutrino fluxes and those on the photon indices,
is $\sim 0.5$ dex. We used this ratio as the relative normalization between
the neutrino and X-ray AGN spectra and computed the expected neutrino
background spectrum as:

$F(E)=\frac{1}{4\pi}\int_{0}^{z_{max}}\frac{1+z}{4\pi{d_{L}}^{2}}\;\frac{dV}{dz}\int_{L_{min}}^{L_{max}}f[E(1+z)]\Phi[L,z]L\,dL\;dz\;,$
(1)

where $d_{L}$ is the luminosity distance, $\frac{dV}{dz}$ is the comoving
volume element, $\Phi[L,z]$ is the total, intrinsic comoving AGN XLF (obtained
by summing both obscured and unobscured AGN) as per the CXB model of Gilli et
al. (2007) and $f[E(1+z)]$ is the X-ray normalised neutrino flux density at
the energy $E(1+z)$. We integrated the AGN XLF in the luminosity range
$L_{0.5-2keV}=10^{42}-10^{48}$ erg s-1 and in the redshift interval $z=0-5$.
We note that the contribution of AGN at $z>5$ to the total background flux is
negligible and that integrating the AGN XLF down to $L_{0.5-2keV}=10^{41}$ erg
s-1 to include the contribution of low-luminosity AGN would increase the total
background flux by at most $\sim$16%. For comparison, we also computed the
neutrino diffuse background expected from very local AGN, i.e. within the same
distance to NGC1068.

Figure 1: Computed all-flavour neutrino background derived from an X-ray AGN
population synthesis. Dark blue curves show the computed neutrino backgrounds
for source populations integrated up to the distance of NGC1068 and redshift
$z=5$ (dash-dotted and dotted, respectively). A high-energy extrapolation up
to $10^{7}\,$GeV is added to the integrated spectrum for $z=5$ and combined
with the blazar neutrino background model by Padovani et al. (2015) (dash-
dotted grey curve) to highlight the structure of the combined AGN neutrino
background flux (“double-humped” red solid curve). The estimated uncertainty
on the integrated neutrino component from X-ray AGN is assumed to be
$0.5\,$dex (dark blue band). Also shown are the current best-fit astrophysical
diffuse neutrino flux and the segmented neutrino flux fit assuming an $E^{-2}$
energy spectrum in each bin (green area and black points: Naab et al. 2023),
IceCube upper limits from stacking analyses for non-blazar AGN (grey solid
line: Privon et al. 2023), and the point-source neutrino flux of NGC1068 (blue
area: Abbasi et al. 2022).

##  4 Main results

Fig. 1 shows the resulting all-flavour neutrino background integrated up to
redshift $z=5$ (dotted blue curve). This can be compared with the current
best-fit for the IceCube astrophysical diffuse neutrino flux as obtained by
Naab et al. (2023), derived using a segmented neutrino flux fit with
individual energy bins assuming an $E^{-2}$ energy spectrum in each bin (black
points) and a single power-law fit $\propto E^{-2.52\pm 0.04}$ (green area).
It can be seen that our integrated non-jetted AGN contribution is consistent
with the IceCube diffuse component down to $\approx 10$ TeV. In other words,
the assumption that all non-jetted AGN behave like NGC 1068 in terms of their
neutrino properties is not inconsistent with present IceCube data above this
energy.

Most likely, different astrophysical populations contribute to the IceCube
diffuse and hence this comparison might be not be very constraining. In Fig.
1, therefore, we also show upper limits derived by IceCube from a stacking
analysis carried out on non-blazar (and therefore mostly non-jetted) AGN (grey
solid line: Privon et al. 2023). The analysis tested the hypothesis of
neutrino emission from hard X-ray AGN in the BASS catalogue (Ricci et al.,
2017), assuming that the neutrino flux correlates with the de-absorbed X-ray
one. After finding no significant results, Privon et al. (2023) placed 90%
C.L. upper limits on the neutrino emission from all non-jetted hard X-ray AGN
assuming an energy spectrum $\propto E_{\nu}^{-3}$. These limits were
corrected for catalogue incompleteness by accounting for the missing neutrino
signal from all non-detected non-jetted X-ray AGN in the Universe using the
XLF given in Ueda et al. (2014). Our results turn out to be not inconsistent
even with the estimated maximum emission from non-jetted AGN, although barely
so.

Fig. 1 shows also the diffuse neutrino background expected from all AGN within
10.1 Mpc, that is the most reliable distance to NGC1068 (Section 1: dash-
dotted blue curve). At first glance this is somewhat smaller than the neutrino
flux from NGC1068 alone. We note however that the 0.5 dex uncertainty in the
X-ray to neutrino flux ratio assumed for NGC1068 (Section 3) alleviates
significantly this tension. Moreover, the model predictions may suffer from
significant limitations when tiny redshift (distance) intervals are
considered, primarily because of statistical fluctuations. The shape and
evolution of the AGN XLFs are in fact derived from samples of thousands of
objects distributed up to large cosmological distances ($z\lesssim 5$). By
contrast, only three AGN with intrinsic $L_{0.5-2keV}>10^{42}$ erg s-1 (the
luminosity limit used in our integrations), including NGC 1068, fall within
10.1 Mpc, as derived from the BASS DR2 catalogue (Koss et al., 2022). The
number of sources predicted by the assumed XLFs within this small volume is
then bound to be somewhat inaccurate, whereas it becomes more precise, up to a
few percent level, when integrating over the full redshift range (dotted blue
curve in Fig.1).

Our results are crucially model-independent. This means that any model capable
of reproducing the IceCube data for NGC 1068 would yield very similar curves
in the energy range covered by current neutrino data. Consequently, our
conclusions hold regardless of the specific mechanism driving neutrino
emission in this source.

To try to get the bigger picture we added a high-energy extrapolation up to
$10^{7}\,$GeV to the whole AGN integrated spectrum and then combined it with
the blazar neutrino background model by Padovani et al. (2015)333As explained
in Padovani et al. (2022), the blazar curve is scaled down by a factor $\sim
6.2$ as compared to the original one not to violate the upper limits of
Aartsen et al. (2016). (dash-dotted grey curve). Our overall results are shown
by the “double-humped” red solid curve and present a possible scenario where
non-jetted AGN contribute mostly to the low-energy ($\lesssim$ 1 PeV) IceCube
diffuse whereas blazars dominate the high-energy part. This would be in
tantalising agreement with the dip in the data at $\sim 300\,$TeV, which might
then be related to the fall of the non-jetted AGN contribution and the rise of
the blazar one.

Can we improve on our predictions? Preliminary results indicate that the
IceCube data associated with the selection of Seyfert galaxies in the Northern
Sky, in particular NGC 4151 and CGCG 420-015, are inconsistent with the
neutrino background at the $2.7\,\sigma$ level of significance (Abbasi et al.,
2023). Moreover, the IceCube search for high-energy neutrino emission from
hard X-ray AGN has reported NGC 4151 at a significance level of $2.9\,\sigma$
level (Privon et al., 2023). Using the IceCube fluxes and spectral slopes for
these two sources (Qinrui, L. and IceCube Collaboration, 2023) and typical
X-ray data (powers and spectra) from Wang et al. (2010) and Tanimoto et al.
(2022) respectively, we derive X-ray to neutrino flux ratios $\nu
f_{\nu}|_{1\;keV}/\nu f_{\nu}|_{4\;TeV}\sim 5.4$ and $\sim 0.5$, i.e. $\sim 6$
and $\sim 60$ times lower than for NGC 1068. At face value this would then
mean that NGC 1068 has a lower than average neutrino to X-ray ratio, implying
that the dotted blue curve in Fig. 1 should be shifted upwards by $\sim 1-2$
orders of magnitude. As this would violate by the same amount the AGN IceCube
upper limits from Privon et al. (2023), it would then follow that only $\sim
1-10\%$ of non-jetted AGN are neutrino emitters. Clearly, this is an issue for
which more and better IceCube data are of paramount importance, both on the
source and population side.

In summary, a population synthesis model, which accounts for the X-ray
background from non-jetted AGN, predicts a neutrino background not violating
current IceCube data and upper limits, when using NGC 1068 to convert from one
band to the other. Preliminary results from two more Seyferts appear instead
to imply an over-prediction. Further IceCube observations are obviously needed
to sort out this issue.

###### Acknowledgements.

We thank Stefano Gabici and Björn Eichmann for useful comments. This work is
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through grant SFB 1258 “Neutrinos and Dark Matter in Astro- and
Particle Physics”.

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295310

# Elliptic boundary-value problems
in Hörmander spaces

Anna Anop  Institute of Mathematics, National Academy of Sciences of Ukraine,
3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine; Chernihiv National Pedagogical
University, 53 Het’mana Polubotka, Chernihiv, 14013, Ukraine<EMAIL_ADDRESS>Tetiana Kasirenko  Institute of Mathematics, National Academy of Sciences of
Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine<EMAIL_ADDRESS>
(Date: 19/09/2016)

###### Abstract.

We investigate general elliptic boundary-value problems in Hörmander inner
product spaces that form the extended Sobolev scale. The latter consists of
all Hilbert spaces that are interpolation spaces with respect to the Sobolev
Hilbert scale. We prove that the operator corresponding to an arbitrary
elliptic problem is Fredholm in appropriate couples of the Hörmander spaces
and induces a collection of isomorphisms on the extended Sobolev scale. We
obtain a local a priory estimate for generalized solutions to this problem and
prove a theorem on their local regularity in the Hörmander spaces. We find new
sufficient conditions under which generalized derivatives (of a given order)
of the solutions are continuous.

###### Key words and phrases:

Elliptic problem, Hörmander space, extended Sobolev scale, RO-varying
function, Fredholm property, a priori estimate, local regularity.

###### 2010 Mathematics Subject Classification:

Primary 35J40, 46E35

##  1\. Introduction

A fundamental result of the theory of general elliptic boundary-value problems
consists in that these problems are Fredholm on appropriate couples of Sobolev
spaces and induce isomorphisms between their subspaces of finite co-dimension
(see monographs [8, 13, 15, 18, 40, 36] and survey [1]). This result has
important applications to the investigation of regularity of solutions to
elliptic equations, to the spectral theory of elliptic differential operators
and others, the most profound results being obtained for the Hilbert scale of
Sobolev spaces. Nevertheless, the Sobolev scale is not finely calibrated
enough for various problems appearing in the theory of partial differential
equations and mathematical analysis (see monographs [13, 14, 16, 28, 31, 33,
41]).

In this connection, Hörmander [13, 14] introduced and investigated a broad
class of normed function spaces for which the index of regularity is a
sufficiently arbitrary function depending on frequency variables. This
function parameter allows one to characterize the regularity more finely than
it is possible within the Sobolev space. Hörmander gave applications of these
spaces to the investigation of regularity of solutions to partial differential
equations. But Hörmander spaces have not found applications to boundary-value
problems within decades. There was only one paper [39] by Slenzak devoted to
the study of regular elliptic problems in some Hörmander inner product spaces.

Recently Mikhailets and Murach [20, 21, 22, 23, 24, 26, 28] have built a
theory of solvability of general elliptic boundary-value problems in a scale
of Hörmander inner product spaces whose regularity index is an arbitrary
radial function varying regularly at infinity in the sense of Karamata. This
refined Sobolev scale is obtained by the method of interpolation with a
function parameter between Sobolev inner product spaces. This method plays a
key role in the mentioned theory.

Therefore of definite interest is the class of all Hilbert spaces that are
interpolation spaces between inner product Sobolev spaces. This class is
constructively described by Mikhailets and Murach [25, 27, 29] with the help
of Ovchinnikov’s theorem [32, Section 11.4] and is called the extended Sobolev
scale. This scale is formed by Hörmander inner product spaces whose regularity
index is an arbitrary radial function RO-varying at infinity in the sense of
Avakumović. This scale contains the refined Sobolev scale and allows us to
characterize the regularity of distributions more finely than the latter scale
does. So, an RO-varying function need not have a number order of varying at
infinity.

The purpose of the present paper is to investigate the character of
solvability of general elliptic boundary-value problems and the properties of
their generalized solutions on the extended Sobolev scale. Some results of the
paper are announced in [4]. Various classes of elliptic problems were
considered earlier in [2, 5, 6, 3] under stronger assumptions about the
regularity index than the assumption made in the present paper. Note that
elliptic systems given on $\mathbb{R}^{n}$ or a closed smooth manifold are
investigated on this scale by Murach and Zinchenko [30, 43, 44].

The paper consists of six sections. Section 1 is Introduction. In Section 2,
we state a general elliptic boundary-value problem and consider a formally
adjoint problem with respect to the special Green formula. In section 3, we
give a definition of Hörmander spaces which form the extended Sobolev scale
and discuss some of their properties. Section 4 contains our main results
concerning the properties of this problem on the extended Sobolev scale. These
results are theorems on the Fredholm property of the problem and on
isomorphisms induced by the problem, a local a priory estimate for its
generalized solutions, and a theorem on their local regularity. As an
application of Hörmander spaces, we obtain new sufficient conditions under
which generalized derivatives (of a prescribed order) of these solutions are
continuous; specifically we give conditions for the generalized solutions to
be classical. Section 5 is devoted to the method of interpolation with a
function parameter between Hilbert spaces. This method plays a key role in our
investigation. The main results of the paper are proved in the last Section 6.

##  2\. Statement of the problem

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with an infinitely
smooth boundary $\Gamma$. (So, $\Gamma$ is a closed $C^{\infty}$-manifold of
dimension $n-1$.) We suppose that the integer $n\geq 2$. As usual,
$\overline{\Omega}=\Omega\cup\Gamma$. Let $\nu(x)$ denote the unit vector of
the inward normal to the boundary $\Gamma$ at a point $x\in\Gamma$.

We consider the following boundary-value problem:

$\displaystyle Au=f\quad\mbox{in}\quad\Omega,$ (1) $\displaystyle
B_{j}u=g_{j}\quad\mbox{on}\quad\Gamma,\quad j=1,\ldots,q.$ (2)

Here,

$A:=A(x,D):=\sum_{|\mu|\leq 2q}a_{\mu}(x)D^{\mu}\ $

is a linear differential operator on $\overline{\Omega}$ of an even order
$2q\geq 2$. Besides, every

$B_{j}:=B_{j}(x,D)=\sum_{|\mu|\leq m_{j}}b_{j,\mu}(x)D^{\mu}\ $

is a linear boundary differential operator on $\Gamma$ of order $m_{j}\leq
2q-1$. The coefficients of these operators are complex-valued infinitely
smooth functions given on $\overline{\Omega}$ and $\Gamma$, respectively.

Here, we use the following standard designations:
$\mu:=(\mu_{1},\ldots,\mu_{n})$ is a multi-index with nonnegative integer
components, $|\mu|:=\mu_{1}+\cdots+\mu_{n}$, $D^{\mu}:=D_{1}^{\mu_{1}}\cdots
D_{n}^{\mu_{n}}$, $D_{k}:=i\partial/\partial x_{k}$, $k=1,\ldots,n$, where $i$
is imaginary unit and $x=(x_{1},\ldots,x_{n})$ is an arbitrary point in
$\mathbb{R}^{n}$. We also put $D_{\nu}:=i\partial/\partial\nu(x)$.

In the paper, we suppose that the boundary-value problem (1), (2) is elliptic
in the domain $\Omega$. This means that the differential operator $A$ is
properly elliptic on $\overline{\Omega}$ and that the system of boundary
differential operators $B:=(B_{1},\ldots,B_{q})$ satisfies Lopatynsky
condition with respect to $A$ on $\Gamma$ (see, e.g., survey [1, Section
1.2]).

With the problem (1), (2), we associate the linear mapping

$\begin{gathered}u\mapsto(Au,Bu)=(Au,B_{1}u,\ldots,B_{q}u),\quad\mbox{with}\quad
u\in C^{\infty}(\overline{\Omega}).\end{gathered}$ (3)

We investigate properties of the extension (by continuity) of this mapping in
appropriate couples of Hörmander inner product spaces which form the extended
Sobolev scale.

In order to describe the range of this extension, we need the following
special Green formula [17, Theorem 3.1.1]:

$(Au,v)_{\Omega}+\sum_{j=1}^{q}(B_{j}u,h_{j})_{\Gamma}=(u,A^{+}v)_{\Omega}+\sum_{j=1}^{2q}\biggl{(}D_{\nu}^{j-1}u,K_{j}v+\sum_{k=1}^{q}Q_{k,j}^{+}h_{k}\biggr{)}_{\Gamma},$

where $u,v\in C^{\infty}(\overline{\Omega})$, $h_{1},\ldots,h_{q}\in
C^{\infty}(\Gamma)$, and, besides, $(\cdot,\cdot)_{\Omega}$ and
$(\cdot,\cdot)_{\Gamma}$ denote the inner products in the Hilbert spaces
$L_{2}(\Omega)$ and $L_{2}(\Gamma)$ of square integrable functions over
$\Omega$ and $\Gamma$ respectively and later denote the extension of these
inner products by continuity. As usual, the differential operator $A^{+}$ is
formally adjoint to $A$; namely,

$(A^{+}v)(x):=\sum_{|\mu|\leq
2q}D^{\mu}\bigl{(}\overline{a_{\mu}(x)}v(x)\bigr{)}.$

Besides, each tangent differential operator $Q_{k,j}^{+}$ is adjoint to the
tangent differential operator $Q_{k,j}:=Q_{j,k}(x,D_{\tau})$ appearing in the
representation

$B_{j}(x,D)=\sum_{k=1}^{2q}Q_{j,k}(x,D_{\tau})D_{\nu}^{k-1}.$

(Of course, if $k>m_{j}+1$, then $Q_{j,k}:=0$.) Furthermore, each
$K_{j}:=K_{j}(x,D)$ is a certain linear boundary differential operator of
order $2q-j$ with coefficients from $C^{\infty}(\Gamma)$.

Note that we do not suppose the system $B$ of boundary operators to be normal.
Therefore the classical Green formula [1, Section 4.2] is not generally valid
for the problem (1), (2).

In view of the special Green formula, we consider the following boundary-value
problem in $\Omega$ with $q$ additional unknown functions on $\Gamma$:

$\displaystyle A^{+}v=\omega\quad\mbox{in}\quad\Omega,$ (4) $\displaystyle
K_{j}v+\sum_{k=1}^{q}Q_{k,j}^{+}h_{k}=\zeta_{j}\quad\mbox{on}\quad\Gamma,\quad
j=1,\ldots,2q.$ (5)

This problem is formally adjoint to the problem (1), (2) with respect to the
given Green formula. The boundary-value problem (1), (2) is elliptic if and
only if the formally adjoint problem (4), (5) is elliptic in the relevant
sense [17, Theorem 3.1.2].

##  3\. The extended Sobolev scale

As we have noted, this scale was selected and investigated by Mikhailets and
Murach [25, 27, 28, 29]. The scale consists of isotropic Hörmander inner
product spaces $H^{\alpha}$ for which the index $\alpha$ of regularity runs
through the function class $\mathrm{RO}$.

By definition, the class $\mathrm{RO}$ consists of all Borel measurable
functions $\alpha:\nobreak[1,\infty)\rightarrow(0,\infty)$ for which there
exist numbers $b>1$ and $c\geq 1$ such that

$c^{-1}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq c\quad\mbox{for every}\quad
t\geq 1\quad\mbox{and}\quad\lambda\in[1,b]$

($b$ and $c$ may depend on $\alpha$). Such functions are called RO-varying (or
OR-varying) at infinity. The class $\mathrm{RO}$ was introduced by Avakumović
[7] in 1936 and is sufficiently investigated (see, e.g., monographs [10,
Sections 2.0–2.2] and [38, Appendix 1]).

This class admits a simple description [38, Theorem A.1], namely

$\alpha\in\mathrm{RO}\;\;\Leftrightarrow\;\;\alpha(t)=\exp\Biggl{(}\beta(t)+\int_{1}^{\>t}\frac{\gamma(\tau)}{\tau}\;d\tau\Biggr{)}\quad\mbox{for}\quad
t\geq 1,$

where the real-valued functions $\beta$ and $\gamma$ are Borel measurable and
bounded on $[1,\infty)$.

We need the following property of the class $\mathrm{RO}$ [38, Theorem A.2]:
for every function $\alpha\in\mathrm{RO}$ there exist numbers
$s_{0},s_{1}\in\mathbb{R}$, $s_{0}\leq s_{1}$, and $c_{0},c_{1}>0$ such that

$c_{0}\lambda^{s_{0}}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq
c_{1}\lambda^{s_{1}}\quad\mbox{for all}\quad t\geq 1,\quad\lambda\geq 1.$ (6)

We put

$\displaystyle\sigma_{0}(\alpha):=\sup\,\\{s_{0}\in\mathbb{R}:\,\mbox{the
left-hand inequality in \eqref{Ax=b3} holds}\\},$
$\displaystyle\sigma_{1}(\alpha):=\inf\,\\{s_{1}\in\mathbb{R}:\,\mbox{the
right-hand inequality in \eqref{Ax=b3} holds}\\}.$

The numbers $\sigma_{0}(\alpha)$ and $\sigma_{1}(\alpha)$ are the lower and
upper Matuszewska indices of the function $\alpha\in\mathrm{RO}$ (see [19] and
[10, Section 2.1.2]). Of course,
$-\infty<\sigma_{0}(\alpha)\leq\sigma_{1}(\alpha)<\infty$.

We consider some examples of RO-varying functions.

######  Example 1.

Let a continuous function $\alpha:[1,\infty)\rightarrow(0,\infty)$ be such
that

$\alpha(t):=t^{s}(\log t)^{r_{1}}(\log\log
t)^{r_{2}}\ldots(\underbrace{\log\ldots\log}_{k\;\mathrm{times}}t)^{r_{k}}\quad\mbox{for}\quad
t\gg 1.$

Here, the integer $k\geq 1$ and real numbers $s,r_{1},\ldots,r_{k}$ are
arbitrarily chosen. Then $\alpha\in\mathrm{RO}$ and
$\sigma_{0}(\alpha)=\sigma_{1}(\alpha)=s$.

######  Example 2.

Let $\theta\in\mathbb{R}$, $\delta>0$ and $r\in(0,1]$. We put

$\alpha(t):=\left\\{\begin{array}[]{ll}t^{\theta+\delta\sin(\log\log
t)^{r}}&\hbox{for}\;\;t>e,\\\ t^{\theta}&\hbox{for}\;\;1\leq t\leq
e.\end{array}\right.$

Then $\alpha\in\mathrm{RO}$, with $\sigma_{0}(\alpha)=\theta-\delta$ and
$\sigma_{1}(\alpha)=\theta+\delta$.

Let $\alpha\in\mathrm{RO}$. We will first introduce the Hörmander space
$H^{\alpha}$ over $\mathbb{R}^{n}$, where $1\leq n\in\mathbb{Z}$, and then
over $\Omega$ and $\Gamma$.

By definition, the complex linear space $H^{\alpha}(\mathbb{R}^{n})$ consists
of all tempered distributions in $\mathbb{R}^{n}$ such that their Fourier
transform $\widehat{w}$ is locally Lebesgue integrable over $\mathbb{R}^{n}$
and satisfies the condition

$\|w\|_{H^{\alpha}(\mathbb{R}^{n})}^{2}:=\int_{\mathbb{R}^{n}}\alpha^{2}((1+|\xi|^{2})^{1/2})\,|\widehat{w}(\xi)|^{2}\,d\xi<\infty.$

This space is Hilbert and separable with respect to the norm
$\|\cdot\|_{H^{\alpha}(\mathbb{R}^{n})}$. In the paper we interpret
distributions as _anti_ linear functionals given on the relevant spaces of
test functions.

The space $H^{\alpha}(\mathbb{R}^{n})$ is a Hilbert and isotropic case of the
spaces $B_{p,k}$ introduced and investigated by Hörmander (see his monographs
[13, Sec. 2.2] and [14, Sec. 10.1]). Namely,
$H^{\alpha}(\mathbb{R}^{n})=B_{p,k}$ if $p=2$ and
$k(\xi)=\alpha((1+|\xi|^{2})^{1/2})$ for all $\xi\in\mathbb{R}^{n}$. Remark
that, in the Hilbert case of $p=2$, the Hörmander spaces coincide with the
spaces introduced and investigated by Volevich and Paneah [42, Sec. 2].

If $\alpha(t)\equiv t^{s}$ for some $s\in\mathbb{R}$, then
$H^{\alpha}(\mathbb{R}^{n})$ becomes the inner product Sobolev space
$H^{(s)}(\mathbb{R}^{n})$ of order $s$. Generally,

$(s_{0}<\sigma_{0}(\alpha),\;\;\sigma_{1}(\alpha)<s_{1})\;\Rightarrow\;H^{(s_{1})}(\mathbb{R}^{n})\hookrightarrow
H^{\alpha}(\mathbb{R}^{n})\hookrightarrow H^{(s_{0})}(\mathbb{R}^{n}),$ (7)

with embeddings being continuous and dense.

Consider the class of Hilbert function spaces
$\\{H^{\alpha}(\mathbb{R}^{n}):\alpha\in\mathrm{RO}\bigr{\\}}$. Following
Mikhailets and Murach [27, 28], we call it the extended Sobolev scale over
$\mathbb{R}^{n}$. Its analogs for the Euclidean domain $\Omega$ and the closed
infinitely smooth manifold $\Gamma$ are introduced in the standard way; see
[29, Sec. 2] and [28, Sec. 2.4.2] respectively). Let us give the necessary
definitions.

By definition, the linear space $H^{\alpha}(\Omega)$ consists of the
restrictions of all distributions $w\in H^{\alpha}(\mathbb{R}^{n})$ to the
domain $\Omega$. The norm in $H^{\alpha}(\Omega)$ is defined by the formula

$\|v\|_{H^{\alpha}(\Omega)}:=\inf\bigl{\\{}\,\|w\|_{H^{\alpha}(\mathbb{R}^{n})}:\,w\in
H^{\alpha}(\mathbb{R}^{n}),\ w=v\;\,\mbox{in}\;\,\Omega\,\bigr{\\}},$

with $v\in H^{\alpha}(\Omega)$. The space $H^{\alpha}(\Omega)$ is Hilbert and
separable with respect to this norm; the set $C^{\infty}(\overline{\Omega})$
is dense in $H^{\alpha}(\Omega)$.

In short, the linear space $H^{\alpha}(\Gamma)$ consists of all distributions
on $\Gamma$ that belong to $H^{\alpha}(\mathbb{R}^{n-1})$ in local
coordinates. Let us give the detailed definition. We arbitrarily choose a
finite atlas from $C^{\infty}$-structure on $\Gamma$ formed by local charts
$\pi_{j}:\mathbb{R}^{n-1}\leftrightarrow\Gamma_{j}$, where
$j=1,\ldots,\varkappa$. Here the open sets
$\\{\Gamma_{1},\ldots,\Gamma_{\varkappa}\\}$ form a covering of the manifold
$\Gamma$. Let functions $\chi_{j}\in C^{\infty}(\Gamma)$, with
$j=1,\ldots,\varkappa$, form a partition of unity on $\Gamma$ that satisfies
the condition $\mathrm{supp}\,\chi_{j}\subset\Gamma_{j}$ for each
$j\in\\{1,\ldots,\varkappa\\}$.

By definition, the complex linear space $H^{\alpha}(\Gamma)$ consists of all
distributions $h$ on $\Gamma$ such that $(\chi_{j}h)\circ\pi_{j}\in
H^{\alpha}(\mathbb{R}^{n-1})$ for all $j\in\\{1,\ldots,\varkappa\\}$. Here,
$(\chi_{j}h)\circ\pi_{j}$ is the representation of the distribution
$\chi_{j}h$ in the local chart $\pi_{j}$. The space $H^{\alpha}(\Gamma)$ is
endowed with the norm

$\|h\|_{H^{\alpha}(\Gamma)}:=\biggl{(}\,\sum_{j=1}^{\varkappa}\,\|(\chi_{j}h)\circ\pi_{j}\|_{H^{\alpha}(\mathbb{R}^{n-1})}^{2}\biggr{)}^{1/2}.$

This space is Hilbert and separable with respect to this norm and does not
depend (up to equivalence of norms) on our choice of the atlas and the
partition of unity [28, Theorem 2.21]. The set $C^{\infty}(\Gamma)$ is dense
in $H^{\alpha}(\Gamma)$.

The above-defined function spaces form the extended Sobolev scales

$\\{H^{\alpha}(\Omega):\alpha\in\mathrm{RO}\\}\quad\mbox{and}\quad\\{H^{\alpha}(\Gamma):\alpha\in\mathrm{RO}\\}$

over $\Omega$ and $\Gamma$ respectively. They contain the Hilbert scale of
Sobolev space: if $\alpha(t)\equiv t^{s}$ then
$H^{\alpha}(\Omega)=:H^{(s)}(\Omega)$ and
$H^{\alpha}(\Gamma)=:H^{(s)}(\Gamma)$ are the inner product Sobolev spaces
over $\Omega$ or $\Gamma$ of order $s\in\mathbb{R}$.

Dealing with the extended Sobolev scale, we will use some of its properties
relating to embeddings of spaces. Let $\alpha,\eta\in\mathrm{RO}$ and
$G\in\\{\Omega,\Gamma\\}$. The function $\alpha/\eta$ is bounded in a
neighborhood of infinity if and only if $H^{\eta}(G)\hookrightarrow
H^{\alpha}(G)$. This embedding is continuous and dense. It is compact if and
only if $\alpha(t)/\eta(t)\rightarrow 0$ as $t\rightarrow\infty$. This
property is a consequence of Theorems 2.2.2 and 2.2.3 from Hörmander’s
monograph [13]. Specifically, property (7) remains true if we replace
$\mathbb{R}^{n}$ with $\Omega$ or $\Gamma$. Then the embeddings are compact
and dense.

We also mention Hörmander’s result [13, Theorem 2.2.7] on the embeddings of
his spaces in spaces of continuously differentiable functions. For the
extended Sobolev scale over $\Omega$, this result is formulated as follows:

$\int_{1}^{\infty}t^{2p+n-1}\alpha^{-2}(t)\,dt<\infty\;\;\Leftrightarrow\;\;H^{\alpha}(\Omega)\hookrightarrow
C^{p}(\overline{\Omega});$ (8)

here, $0\leq p\in\mathbb{Z}$ and $\alpha\in\mathrm{RO}$. This embedding is
continuous (see [28, Proposition 2.6(vi)] and [43, Lemma 2]). Note that, in
the Sobolev case where $\alpha(t)\equiv t^{s}$ for some $s\in\mathbb{R}$,
property (8) becomes the well-know Sobolev embedding theorem according to
which the inequality $s>p+n/2$ is equivalent to the embedding
$H^{(s)}(\Omega)\hookrightarrow C^{p}(\overline{\Omega})$.

##  4\. Main results

In this section we will formulate our main results concerning the properties
of the elliptic boundary-value problem (1), (2) on the extended Sobolev scale.

Let us previously agree about the following. We will use the Hörmander spaces
$H^{\alpha}$ for which the order of regularity is given in the form
$\alpha(t)\equiv\varphi(t)t^{s}$ for some $\varphi\in\mathrm{RO}$ and
$s\in\mathbb{R}$. To not write the argument $t$ in superscripts, we resort to
the function parameter $\varrho(t):=t$ of $t\geq 1$. Then $H^{\alpha}$ can be
written as $H^{\varphi\varrho^{s}}$. Note that if $\varphi\in\mathrm{RO}$ and
$s\in\mathbb{R}$, then $\varphi\varrho^{s}\in\mathrm{RO}$ and
$\sigma_{j}(\varrho^{s}\varphi)=s+\sigma_{j}(\varphi)$ for each
$j\in\\{0,1\\}$.

We associate two finite-dimensional spaces with the boundary-value problem
(1), (2) and its adjoint problem (4), (5). Namely, let $N$ denote the linear
space of all solutions $u\in C^{\infty}(\overline{\Omega})$ to the problem
(1), (2) in the case where $f=0$ on $\Omega$ and each $g_{j}=0$ on $\Gamma$.
Besides, we let $N_{*}$ denote the linear space of all solutions

$(v,h_{1},\ldots,h_{q})\in
C^{\infty}(\overline{\Omega})\times(C^{\infty}(\Gamma))^{q}$

to the adjoint problem (4), (5) in the case where $\omega=0$ on $\Omega$ and
each $\zeta_{j}=0$ on $\Gamma$. We put
$m:=\mathrm{max}\\{m_{1},\ldots,m_{q}\\}$; recall that each
$m_{j}:=\mathrm{ord}\,B_{j}$. Since these problems are elliptic in $\Omega$,
the spaces $N$ and $N_{*}$ are finite-dimensional (see, e.g., [17, Lemma
3.4.2]).

###### Theorem 1

Let $\varphi\in\mathrm{RO}$ and $\sigma_{0}(\varphi)>m+1/2$. Then the mapping
(3) extends uniquely (by continuity) to a bounded operator

$(A,B):H^{\varphi}(\Omega)\rightarrow
H^{\varphi\varrho^{-2q}}(\Omega)\oplus\bigoplus_{j=1}^{q}H^{\varphi\varrho^{-m_{j}-1/2}}(\Gamma)=:\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma).$
(9)

This operator is Fredholm. Its kernel is equal to $N$, and its range consists
of all vectors
$(f,g_{1},\ldots,g_{q})\in\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$
such that

$(f,v)_{\Omega}+\sum_{j=1}^{q}(g_{j},h_{j})_{\Gamma}=0\quad\mbox{for
all}\quad(v,h_{1},\ldots,h_{q})\in N_{*}.$ (10)

The index of the operator (9) is equal to $\dim N-\dim N_{*}$ and does not
depend on $\varphi$.

In view of this theorem, it is worthwhile to recall that a linear bounded
operator $T:E_{1}\rightarrow E_{2}$, where $E_{1}$ and $E_{2}$ are Banach
spaces, is called Fredholm if its kernel $\ker T$ and co-kernel
$E_{2}/T(E_{1})$ are both finite-dimensional. If this operator is Fredholm,
then its range is closed in $E_{2}$ (see, e.g., [15, Lemma 19.1.1]), and its
index $\mathrm{ind}\,T:=\dim\ker T-\dim(E_{2}/T(E_{1}))$ is well defined and
finite.

Note that the assumption $\sigma_{0}(\varphi)>m+1/2$ made in Theorem 1 cannot
be weakened. Specifically, if $\varphi(t)\equiv t^{s}$ for some real $s\leq
m+1/2$, then the mapping $u\mapsto Bu$, with $u\in
C^{\infty}(\overline{\Omega})$, cannot be extended to a continuous linear
operator from $H^{(s)}(\Omega)$ to $(\mathcal{D}^{\prime}(\Gamma))^{q}$. Here,
as usual, $\mathcal{D}^{\prime}(\Gamma)$ denotes the linear topological space
of all distribution on $\Gamma$. Anop [2, 5, 6, 3] proved Theorem 1 for
various classes of elliptic boundary-value problems under stronger
restrictions on $\sigma_{0}(\varphi)$ than $\sigma_{0}(\varphi)>m+1/2$. At
least, it was supposed that $\sigma_{0}(\varphi)>2q-1/2$.

Perhaps, the expression $(f,v)_{\Omega}$ in (10) needs commenting. If
$\sigma_{0}(\varphi)>2q$, then $f\in H^{\varphi\varrho^{-2q}}(\Omega)\subset
L_{2}(\Omega)$ and, hence, $(f,v)_{\Omega}$ is the inner product in
$L_{2}(\Omega)$. If $m+1/2<\sigma_{0}(\varphi)\leq 2q$, then $(f,v)_{\Omega}$
is well defined by the passage to the limit; namely,
$(f,v)_{\Omega}:=\lim_{k\to\infty}(f_{k},v)_{\Omega}$ where
$C^{\infty}(\overline{\Omega})\ni f_{k}\to f$ in
$H^{\varphi\varrho^{-2q}}(\Omega)$ as $k\to\infty$. Here, the condition
$(v,h_{1},\ldots,h_{q})\in N_{*}$ is essential. So, if
$\sigma_{0}(\varphi)\leq 2q-1/2$ and $v$ is an arbitrarily chosen function
from $C^{\infty}(\overline{\Omega})$, then the mapping
$f\mapsto(f,v)_{\Omega}$, with $f\in C^{\infty}(\overline{\Omega})$, cannot be
extended to a continuous linear functional on
$H^{\varphi\varrho^{-2q}}(\Omega)$. Specifically, for Sobolev spaces, this
follows from [40, Theorems 4.8.2(c) and 4.3.2/1(c)].

In the case where $N=\\{0\\}$ and $N_{*}=\\{0\\}$, the operator (9) is an
isomorphism between spaces $H^{\varphi}(\Omega)$ and
$\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$. This follows from Theorem
1 and the Banach theorem on inverse operator. In the general case, the
operator (9) induced an isomorphism between their certain subspaces of finite
co-dimension. These subspaces and projectors onto them can be build in the
following way.

Let $\varphi\in\mathrm{RO}$ and $\sigma_{0}(\varphi)>m+1/2$. We consider the
decompositions of the spaces in the direct sums of their subspaces:

$\displaystyle H^{\varphi}(\Omega)=N\dotplus\bigl{\\{}u\in
H^{\varphi}(\Omega):\,(u,v)_{\Omega}=0\;\;\mbox{for all}\;\;v\in N\bigr{\\}},$
(11)
$\displaystyle\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)=N_{*}\dotplus\bigl{\\{}(f,g_{1},\ldots,g_{q})\in\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma):\eqref{1f9}\;\mbox{is
true}\bigr{\\}}.$ (12)

The representation (11) exists since it is a restriction of the decomposition
of the space $L_{2}(\Omega)$ into the orthogonal sum of the subspace $N$ and
its complement. The equality (12) holds because the intersection of the
subspaces on its right-hand side is zero space and because the finite
dimension of the first subspace is equal to the co-dimension of the second due
to Theorem 1. Let $P$ and $P_{*}$ respectively denote the oblique projectors
of the spaces $H^{\varphi}(\Omega)$ and
$\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$ onto the second summands
in (11) and (12) parallel to the first. Evidently, these projectors do not
depend on $\varphi$.

###### Theorem 2

For arbitrary $\varphi\in\mathrm{RO}$ with $\sigma_{0}(\varphi)>m+1/2$, the
restriction of the mapping (9) to the subspace $P(H^{\varphi}(\Omega))$ is an
isomorphism

$(A,B):\,P(H^{\varphi}(\Omega))\leftrightarrow
P_{*}\bigl{(}\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)\bigr{)}.$ (13)

Let us now consider properties of the generalized solutions to the elliptic
boundary-value problem (1), (2) on the extended Sobolev scale. We recall the
definition of these solutions. We put

$H^{m+1/2+}(\Omega):=\bigcup_{\begin{subarray}{c}\alpha\in\mathrm{RO}:\\\
\sigma_{0}(\alpha)>m+1/2\end{subarray}}H^{\alpha}(\Omega)=\bigcup_{s>m+1/2}H^{(s)}(\Omega)$

The latter equality is valid in view of (7). Owing to Theorem 1, for every
function $u\in H^{m+1/2+}(\Omega)$ we can reasonably define the vector

$(f,g):=(f,g_{1},\ldots,g_{q}):=(A,B)u\in\mathcal{D}^{\prime}(\Omega)\times(\mathcal{D}^{\prime}(\Gamma))^{q}.$

Here, as usual, $\mathcal{D}^{\prime}(\Omega)$ and
$\mathcal{D}^{\prime}(\Gamma)$ stand respectively for the linear topological
spaces of all distributions on the domain $\Omega$ or on the manifold
$\Gamma$. The function $u$ is called a generalized (strong) solution of the
boundary-value problem (1), (2) with the right-hand side $(f,g)$.

###### Theorem 3

Let $\varphi\in\mathrm{RO}$ with $\sigma_{0}(\varphi)>m+1/2$, and let
functions $\chi,\eta\in C^{\infty}(\overline{\Omega})$ satisfy the condition
$\eta=1$ in a neighborhood of $\mathrm{supp}\,\chi$. Then there exists a
number $c=c(\varphi,\chi,\eta)>0$ such that an arbitrary function $u\in
H^{\varphi}(\Omega)$ satisfies the estimate

$\|\chi u\|_{H^{\varphi}(\Omega)}\leq
c\,\bigl{(}\|\chi(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+\|\eta
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}\bigr{)}.$ (14)

Here, $c$ does not depend on $u$.

In the special case where $\chi=\eta=1$, inequality (14) is a global a priori
estimate of the generalized solution $u$ to the elliptic boundary-value
problem (1), (2). Generally, this inequality is a local a priory estimate of
the solution. Indeed, for every nonempty open subset of $\overline{\Omega}$,
we can choose the functions $\chi,\eta$ so that they satisfy the condition of
Theorem 3 and that their supports lie in this subset.

Let us discuss the local regularity of the generalized solutions. Let $V$ be
an open subset of $\mathbb{R}^{n}$ that has a nonempty intersection with
$\Omega$. We put $\Omega_{0}:=\Omega\cap V$ and $\Gamma_{0}:=\Gamma\cap V$
(the $\Gamma_{0}=\varnothing$ case is possible). For arbitrary parameter
$\alpha\in\mathrm{RO}$, we introduce a local analogs of the spaces
$H^{\alpha}(\Omega)$ and $H^{\alpha}(\Gamma)$ in the following standard way.
By definition, the linear space
$H^{\alpha}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ consists of all
distributions $u\in\mathcal{D}^{\prime}(\Omega)$ such that $\chi u\in
H^{\alpha}(\Omega)$ for arbitrary function $\chi\in
C^{\infty}(\overline{\Omega})$ with
$\mathrm{supp}\,\chi\subset\Omega_{0}\cup\Gamma_{0}$. The topology in the
linear space $H^{\alpha}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ is induced by
the seminorms $u\mapsto\|\chi u\|_{H^{\alpha}(\Omega)}$, where $\chi$ is an
arbitrary function from the definition of this space. Similarly, the linear
space $H^{\alpha}_{\mathrm{loc}}(\Gamma_{0})$ consists of all distributions
$h\in\nobreak\mathcal{D}^{\prime}(\Gamma)$ such that $\chi h\in
H^{\alpha}(\Gamma)$ for arbitrary function $\chi\in C^{\infty}(\Gamma)$ with
$\mathrm{supp}\,\chi\subset\Gamma_{0}$. The topology in the linear space
$H^{\alpha}_{\mathrm{loc}}(\Gamma_{0})$ is induced by the seminorms
$h\mapsto\|\chi h\|_{H^{\alpha}(\Omega)}$, where $\chi$ is an arbitrary
function from the definition of this space.

###### Theorem 4

Let a function $u\in H^{m+1/2+}(\Omega)$ be a generalized solution to the
elliptic boundary-value problem (1), (2) in which

$(f,g_{1},\ldots,g_{q})\in
H^{\varphi\varrho^{-2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})\oplus\bigoplus_{j=1}^{q}H^{\varphi\varrho^{-m_{j}-1/2}}_{\mathrm{loc}}(\Gamma_{0})=:\mathcal{H}^{\varphi\varrho^{-2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$

for a certain parameter $\varphi\in\mathrm{RO}$ such that
$\sigma_{0}(\varphi)>m+1/2$. Then $u\in
H^{\varphi}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$.

It is worthwhile to mention two special cases of Theorem 4. In the case where
$\Omega_{0}=\Omega$ and $\Gamma_{0}=\Gamma$, the spaces
$H^{\varphi}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ and
$\mathcal{H}^{\varphi\varrho^{-2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$
coincide respectively with $H^{\varphi}(\Omega)$ and
$\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$. Therefore Theorem 4
asserts in this case that the regularity of $u$ increases globally, i.e. in
the whole domain $\Omega$ up to its boundary $\Gamma$. The case where
$\Omega_{0}=\Omega$ and $\Gamma_{0}=\varnothing$ leads us to the assertion
that the smoothness of $u$ increases in neighborhoods of all internal points
of $\overline{\Omega}$.

As an application of the extended Sobolev scale, we give sufficient conditions
under which the generalized derivatives (of a given order) of the solution $u$
are continuous.

###### Theorem 5

Let $0\leq p\in\mathbb{Z}$ and suppose that a function $u\in
H^{m+1/2+}(\Omega)$ satisfies the condition of Theorem 4 for a certain
parameter $\varphi\in\mathrm{RO}$ such that $\sigma_{0}(\varphi)>m+1/2$ and

$\int_{1}^{\infty}t^{2p+n-1}\varphi^{-2}(t)dt<\infty.$ (15)

Then $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$.

######  Remark 1.

Condition (15) is sharp in Theorem 5. Namely, let $0\leq p\in\mathbb{Z}$ and
$\varphi\in\mathrm{RO}$ with $\sigma_{0}(\varphi)>m+1/2$. Then it follows from
the implication

$\bigl{(}u\in
H^{m+1/2+}(\Omega)\quad\mbox{and}\quad(A,B)u\in\mathcal{H}^{\varphi\varrho^{-2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})\bigr{)}\;\Rightarrow\;u\in
C^{p}(\Omega_{0}\cup\Gamma_{0})$ (16)

that $\varphi$ satisfies condition (15).

Theorem 5 implies a sufficient condition for the generalized solution $u$ to
be classical.

###### Corollary 1

Let a function $u\in H^{m+1/2+}(\Omega)$ be a generalized solution to the
elliptic boundary-value problem (1), (2) in which

$\displaystyle f\in
H^{\varphi_{1}\varrho^{-2q}}_{\mathrm{loc}}(\Omega,\varnothing)\cap
H^{\varphi_{2}\varrho^{-2q}}(\Omega),$ (17) $\displaystyle g_{j}\in
H^{\varphi_{2}\varrho^{-m_{j}-1/2}}(\Gamma),\quad\mbox{with}\quad
j=1,\ldots,q,$ (18)

for some parameters $\varphi_{1},\varphi_{2}\in\mathrm{RO}$ that satisfy the
conditions $\sigma_{0}(\varphi_{1})>m+1/2$, $\sigma_{0}(\varphi_{2})>m+1/2$
and

$\displaystyle\int_{1}^{\infty}t^{n+2q-1}\varphi_{1}^{-2}(t)dt<\infty,$ (19)
$\displaystyle\int_{1}^{\infty}t^{2m+n-1}\varphi_{2}^{-2}(t)dt<\infty.$ (20)

Then $u$ is a classical solution, i.e. $u\in C^{2q}(\Omega)\cap
C^{m}(\overline{\Omega})$.

Note that if the solution $u$ is classical, then the left-hand sides of the
problem (1), (2) are calculated by means of classical derivatives and are
continuous on $\Omega$ or $\Gamma$ respectively.

##  5\. Interpolation with function parameter

An important property of the extended Sobolev scale consists in that this
scale can be obtained by the interpolation with a function parameter between
Sobolev inner product spaces. This interpolation will be a main tool in our
proof of the key Theorem 1. Therefore we recall the definition of this
interpolation method and formulate its properties being used in the paper.

Note that the method of interpolation with a function parameter between normed
spaces was introduced by Foiaş and Lions in paper [12], where the case of
Hilbert spaces was separately considered. As to Hilbert spaces, this method is
a natural generalization of the classical interpolation method by Lions and S.
Krein (see, e.g., [18, Chapter 1, Section 1 and 5]) to the case where a
general enough function serves as an interpolation parameter instead of a
number. Setting forth the interpolation with a function parameter between
Hilbert spaces, we follow the monograph [28, Section 1.1]. It is enough for
our purposes to restrict ourselves to the case of separable Hilbert spaces.

Let $X:=[X_{0},X_{1}]$ be an ordered couple of separable complex Hilbert
spaces that the continuous and dense embedding $X_{1}\hookrightarrow X_{0}$
holds. We say that this couple is admissible. As is known [18, Chapter 1,
Section 1], for $X$ there exists an isometric isomorphism
$J:X_{1}\leftrightarrow X_{0}$ such that $J$ is a self-adjoint positive
operator on $X_{0}$ with the domain $X_{1}$. The operator $J$ is called a
generating operator for the couple $X$. This operator is uniquely determined
by $X$.

By $\mathcal{B}$ we denote the set of all Borel measurable functions
$\psi:(0,\infty)\rightarrow(0,\infty)$ such that $\psi$ is bounded on each
compact interval $[a,b]$, with $0<a<b<\infty$, and that $1/\psi$ is bounded on
every set $[r,\infty)$, with $r>0$.

Let $\psi\in\mathcal{B}$. Consider the operator $\psi(J)$, which is defined
(and positive) in $X_{0}$ as the Borel function $\psi$ of the positive self-
adjoint operator $J$. Denote by $[X_{0},X_{1}]_{\psi}$ or, simply, by
$X_{\psi}$ the domain of the operator $\psi(J)$ endowed with the inner product
$(w_{1},w_{2})_{X_{\psi}}:=(\psi(J)w_{1},\psi(J)w_{2})_{X_{0}}$ and the
corresponding norm $\|w\|_{X_{\psi}}=(w,w)_{X_{\psi}}^{1/2}$. The space
$X_{\psi}$ is Hilbert and separable. The continuous and dense embedding
$X_{\psi}\hookrightarrow X_{0}$ holds true.

A function $\psi\in\mathcal{B}$ is called an interpolation parameter if the
following condition is fulfilled for all admissible couples $X=[X_{0},X_{1}]$
and $Y=[Y_{0},Y_{1}]$ of Hilbert spaces and for an arbitrary linear mapping
$T$ given on $X_{0}$: if the restriction of $T$ to $X_{j}$ is a bounded
operator $T:X_{j}\rightarrow Y_{j}$ for each $j\in\\{0,1\\}$, then the
restriction of $T$ to $X_{\psi}$ is also a bounded operator
$T:X_{\psi}\rightarrow Y_{\psi}$. If $\psi$ is an interpolation parameter,
then we say that the Hilbert space $X_{\psi}$ is obtained by the interpolation
of the couple $X=[X_{0},X_{1}]$ (or between $X_{0}$ and $X_{1}$) with the
function parameter $\psi$. In this case, the dense and continuous embeddings
$X_{1}\hookrightarrow X_{\psi}\hookrightarrow X_{0}$ hold.

Note that a function $\psi\in\mathcal{B}$ is an interpolation parameter if and
only if $\psi$ is pseudoconcave on a neighborhood of $+\infty$ (see [28, Sect.
1.1.9]). The latter condition means that there exists a concave function
$\psi_{1}:(b,\infty)\rightarrow(0,\infty)$, with $b\gg 1$, such that both
functions $\psi/\psi_{1}$ and $\psi_{1}/\psi$ are bounded on $(b,\infty)$.
This fundamental result follows from Peetre’s theorem [34, 35] about
description of all interpolation functions of positive order (see also [9,
Sect. 5.4]).

The above-mentioned interpolation property of the extended Sobolev is stated
in the following way.

###### Proposition 1

Assume that a function $\alpha\in\mathrm{RO}$ and numbers
$s_{0},s_{1}\in\mathbb{R}$ satisfy the conditions $s_{0}<\sigma_{0}(\alpha)$
and $s_{1}>\sigma_{1}(\alpha)$. Put

$\psi(t):=\begin{cases}\;t^{{-s_{0}}/{(s_{1}-s_{0})}}\,\alpha\bigl{(}t^{1/{(s_{1}-s_{0})}}\bigr{)}&\text{for}\quad
t\geq 1,\\\ \;\alpha(1)&\text{for}\quad 0<t<1.\end{cases}$ (21)

Then $\psi\in\mathcal{B}$ is an interpolation parameter, and

$[H^{(s_{0})}(G),H^{(s_{1})}(G)]_{\psi}=H^{\alpha}(G)$

with equality of norms if $G=\mathbb{R}^{n}$, and with equivalence of norms if
$G=\Omega$ or $G=\Gamma$.

This proposition is proved by Mikhailets and Murach [29, Theorem 5.1] for
$G=\Omega$ and in [28, Theorems 2.19 and 2.22] for
$G\in\\{\mathbb{R}^{n},\Gamma\\}$.

It is worthwhile to note that the extended Sobolev scale is closed with
respect to the interpolation with a function parameter and coincides (up to
equivalence of norms) with the class of all Hilbert spaces that are
interpolation spaces for the couples of the Sobolev spaces $H^{(s_{0})}(G)$
and $H^{(s_{1})}(G)$ with $s_{0},s_{1}\in\mathbb{R}$ and $s_{0}<s_{1}$. The
latter property follows from Ovchinnikov’s theorem [32, p. 511], which gives a
description of all Hilbert spaces that are interpolation spaces for an
arbitrarily chosen compatible couple of Hilbert spaces. Recall that the
property of a (Hilbert) space $H$ to be an interpolation space for an
admissible couple $X=[X_{0},X_{1}]$ means the following: a) the continuous and
dense embeddings $X_{1}\hookrightarrow H\hookrightarrow X_{0}$ hold; b) every
linear operator which is bounded on each of the spaces $X_{0}$ and $X_{1}$
should be bounded on $H$ as well.

Let us recall two general properties of the interpolation; they will be used
in our proof of Theorem 1.

###### Proposition 2

Let $X=[X_{0},X_{1}]$ and $Y=[Y_{0},Y_{1}]$ be two admissible couples of
Hilbert spaces. Suppose that a linear mapping $T$ is given on $X_{0}$ and
satisfies the following condition: the restrictions of $T$ to the spaces
$X_{0}$ and $X_{1}$ are bounded and Fredholm operators $T:X_{0}\rightarrow
Y_{0}$ and $T:X_{1}\rightarrow Y_{1}$ respectively, and these operators have
the same kernel and the same index. Then, for an arbitrary interpolation
parameter $\psi\in\mathcal{B}$, the restriction of $T$ to $X_{\psi}$ is also a
bounded and Fredholm operator $T:X_{\psi}\rightarrow Y_{\psi}$ with the same
kernel and the same index; the range of this operator is equal to
$Y_{\psi}\cap T(X_{0})$.

###### Proposition 3

Let $[X_{0}^{(k)},X_{1}^{(k)}]$, with $k=1,\ldots,p$, be a finite collection
of admissible couples of Hilbert spaces. Then for every function
$\psi\in\mathcal{B}$ we have

$\biggl{[}\,\bigoplus_{k=1}^{p}X_{0}^{(k)},\,\bigoplus_{k=1}^{p}X_{1}^{(k)}\biggr{]}_{\psi}=\,\bigoplus_{k=1}^{p}\bigl{[}X_{0}^{(k)},\,X_{1}^{(k)}\bigr{]}_{\psi}$

with equality of norms.

These properties of the interpolation are expounded, e.g., in monograph [28,
Sections 1.1.5 and 1.1.7].

As an application of this interpolation method, we obtain the following
property of linear differential operators on the extended Sobolev scale.

###### Proposition 4

$\mathrm{(i)}$ Let $L$ be a linear differential operator on
$\overline{\Omega}$ of order $\ell\geq 0$ with coefficients from
$C^{\infty}(\overline{\Omega})$. Then the mapping $u\mapsto Lu$, with $u\in
C^{\infty}(\overline{\Omega})$, extends uniquely (by continuity) to the
bounded linear operator

$L:\,H^{\alpha}(\Omega)\rightarrow H^{\alpha\varrho^{-\ell}}(\Omega)$

for every $\alpha\in\mathrm{RO}$.

$\mathrm{(ii)}$ Let $K$ be a boundary linear differential operator on $\Gamma$
of order $k\geq\nobreak 0$ with coefficients from $C^{\infty}(\Gamma)$. Then
the mapping $u\mapsto Ku$, with $u\in C^{\infty}(\overline{\Omega})$, extends
uniquely (by continuity) to the bounded linear operator

$K:\,H^{\alpha}(\Omega)\rightarrow H^{\alpha\varrho^{-k-1/2}}(\Gamma)$

for every function parameter $\alpha\in\mathrm{RO}$ such that
$\sigma_{0}(\alpha)>k+1/2$.

Proposition 4 is well-known in the case of Sobolev spaces. For the extended
Sobolev scale, this proposition follows directly from the Sobolev case due to
Proposition 1.

##  6\. Proofs

In this section, we will prove Theorems 1–5, Remark 1, and Corollary 1.

_Proof of Theorem 1._ It follows directly from Proposition 4 that the mapping
$C^{\infty}(\overline{\Omega})\ni u\mapsto(Au,Bu)$ extends uniquely to the
bounded linear operator (9). We first discuss Theorem 1 in the Sobolev case
and then deduce it in the general situation with the help of the interpolation
with a function parameter between Sobolev spaces.

Consider Theorem 1 in the Sobolev case where $\varphi(t)\equiv t^{s}$ and
$s>m+1/2$. The central and well-known fact of the theory of elliptic boundary-
value problems in Sobolev spaces consists in the following: the operator (9)
is Fredholm for every real $s\geq 2q$, its kernel coincides with $N$, and its
index does not depend on $s$ (see, e.g., survey [1, Section 2.4]). It is less
known that this fact holds true for arbitrary real $s>m+1/2$ (see, e.g., [11,
Chapter III, Section 2.2]). The proof of the remaining part of Theorem 1 –
concerning the range of (9) – is given in monographs [17, Theorem 3.4.1 and
Section 4.4.1] and [37, Theorem 2.4.1] for every real $s\geq 2q$. Therefore it
remains to prove this part in the case where $m+1/2<s<2q$.

Let $\Lambda_{s}$ denote the Fredholm bounded operator (9) in the Sobolev case
where $\varphi(t)\equiv t^{s}$ and $s>m+1/2$. As usual, $\Lambda_{s}^{*}$
stands for the adjoint operator to (9). If $m+1/2<s_{0}<s_{1}$, then
$\Lambda_{s_{0}}^{\ast}$ is a restriction of $\Lambda_{s_{1}}^{\ast}$ and
therefore $\ker\Lambda_{s_{0}}^{\ast}\subseteq\ker\Lambda_{s_{1}}^{\ast}$. But
$\ker\Lambda_{s_{0}}^{\ast}$ and $\ker\Lambda_{s_{1}}^{\ast}$ have the same
finite dimension. Indeed, the latter is equal to the co-dimension of the range
of $\Lambda_{s}$, and this codimension is finite and does not depend on $s$,
as we have mentioned. Thus,
$\ker\Lambda_{s_{0}}^{\ast}=\ker\Lambda_{s_{1}}^{\ast}$. According to [17,
Theorem 3.4.2], we have the equality $\ker\Lambda_{s}^{\ast}=N_{\ast}$ if
$s\geq 2q$. Here, every vector $(v,h_{1},\ldots,h_{q})\in N_{\ast}$ is
interpreted as the continuous linear functional

$\Upsilon:(f,g_{1},\ldots,g_{q})\mapsto(f,v)_{\Omega}+\sum_{j=1}^{q}(g_{j},h_{j})_{\Gamma}$

on the space

$\mathcal{H}^{(s-2q)}(\Omega,\Gamma):=H^{(s-2q)}(\Omega)\oplus\bigoplus_{j=1}^{q}H^{(s-m_{j}-1/2)}(\Gamma).$

Hence, $\ker\Lambda_{s}^{\ast}=N_{\ast}$ whenever $s>m+1/2$. Therefore,
$\Upsilon$ extends uniquely (by continuity) to a linear continuous functional
on $\mathcal{H}^{(s-2q)}(\Omega,\Gamma)$ in the case where $m+1/2<s<2q$.
Specifically, the form $(f,v)_{\Omega}$ is well defined for every $f\in
H^{(s-2q)}(\Omega)$ via the passage to the limit in this case. Now, since the
operator $\Lambda_{s}$ is Fredholm whenever $s>m+1/2$, then its range consists
of all vectors $(f,g_{1},\ldots,g_{q})\in\mathcal{H}^{(s-2q)}(\Omega,\Gamma)$
such that $\Upsilon(f,g_{1},\ldots,g_{q})=\nobreak 0$ for all
$\Upsilon\in\ker\Lambda_{s}^{\ast}$. This condition becomes (10) in view of
the equality $\ker\Lambda_{s}^{\ast}=N_{\ast}$. Besides, the index of
$\Lambda_{s}$ is $\dim N-\dim N^{\ast}$ because $\dim\ker\Lambda_{s}^{\ast}$
is equal to the co-dimension of the range of $\Lambda_{s}$.

Thus, Theorem 1 is justified in the Sobolev case. We now deduce this theorem
in the general situation from this case with the help of the interpolation.

We arbitrarily choose $\varphi\in\mathrm{RO}$ such that
$\sigma_{0}(\varphi)>m+1/2$. Let real numbers $l_{0}$ and $l_{1}$ satisfy the
conditions $m+1/2<l_{0}<\sigma_{0}(\varphi)$ and $\sigma_{1}(\varphi)<\nobreak
l_{1}$. We have the Fredholm bounded operators

$\Lambda_{l_{i}}=(A,B):\,H^{(l_{i})}(\Omega)\rightarrow\mathcal{H}^{(l_{i}-2q)}(\Omega,\Gamma),\quad
i\in\\{0,1\\}.$ (22)

They have the common kernel $N$ and the same index $\dim N-\dim N_{*}$.
Besides,

$\displaystyle(A,B)\bigl{(}H^{(l_{i})}(\Omega)\bigr{)}=\bigl{\\{}(f,g)\in\mathcal{H}^{(l_{i}-2q)}(\Omega,\Gamma):\,\mbox{\eqref{1f9}
is true}\bigr{\\}}.$ (23)

We define an interpolation parameter $\psi$ by formula (21) with
$\alpha:=\varphi$, $s_{0}:=l_{0}$, and $s_{1}:=\nobreak l_{1}$. Applying the
interpolation with the function parameter $\psi$ to (22), we conclude by
Proposition 2 that the restriction of $\Lambda_{l_{0}}$ to the interpolation
space $[H^{(l_{0})}(\Omega),H^{(l_{1})}(\Omega)]_{\psi}$ is a bounded and
Fredholm operator

$(A,B):\,\bigl{[}H^{(l_{0})}(\Omega),H^{(l_{1})}(\Omega)\bigr{]}_{\psi}\to\bigl{[}\mathcal{H}^{(l_{0}-2q)}(\Omega,\Gamma),\mathcal{H}^{(l_{1}-2q)}(\Omega,\Gamma)\bigr{]}_{\psi}.$
(24)

Since $C^{\infty}(\overline{\Omega})$ is dense in this space, operator (24) is
an extension by continuity of the mapping $C^{\infty}(\overline{\Omega})\ni
u\mapsto(Au,Bu)$. This operator coincides with (9) because

$\displaystyle\bigl{[}H^{(l_{0})}(\Omega),H^{(l_{1})}(\Omega)\bigr{]}_{\psi}=H^{\varphi}(\Omega),$
$\displaystyle\bigl{[}\mathcal{H}^{(l_{0}-2q)}(\Omega,\Gamma),\mathcal{H}^{(l_{1}-2q)}(\Omega,\Gamma)\bigr{]}_{\psi}=\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$

with equivalence of norms.

Indeed, the first formula is due to Proposition 1, in which $\alpha:=\varphi$,
$s_{0}:=l_{0}$, $s_{1}:=l_{1}$, and $G:=\Omega$. The second formula follows
from Propositions 3 and 1; namely,

$\displaystyle\bigl{[}\mathcal{H}^{(l_{0}-2q)}(\Omega,\Gamma),\mathcal{H}^{(l_{1}-2q)}(\Omega,\Gamma)\bigr{]}_{\psi}$
$\displaystyle\quad=\bigl{[}H^{(l_{0}-2q)}(\Omega),H^{(l_{1}-2q)}(\Omega)\bigr{]}_{\psi}\oplus\bigoplus_{j=1}^{q}\bigl{[}H^{(l_{0}-m_{j}-1/2)}(\Gamma),H^{(l_{1}-m_{j}-1/2)}(\Gamma)\bigr{]}_{\psi}$
$\displaystyle\quad=H^{\varphi\varrho^{-2q}}(\Omega)\oplus\bigoplus_{j=1}^{q}H^{\varphi\varrho^{-m_{j}-1/2}}(\Gamma)=\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma).$

Note that in Proposition 1 we have first taken

$\alpha:=\varphi\varrho^{-2q},\quad s_{0}:=l_{0}-2q,\quad
s_{1}:=l_{1}-2q,\quad G:=\Omega$

and then

$\alpha:=\varphi\varrho^{-m_{j}-1/2},\quad s_{0}:=l_{0}-m_{j}-1/2,\quad
s_{1}:=l_{1}-m_{j}-1/2,\quad G:=\Gamma.$

Formula (21) gives the same function $\psi$ in both of these cases.

According to Proposition 2, the kernel and index of the operator (9) coincide
with the common kernel $N$ and index $\dim N-\dim N_{*}$ of the operators
(22). Besides, the range of (9) equals

$\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)\cap(A,B)\bigl{(}H^{(l_{0})}(\Omega)\bigr{)}=\bigl{\\{}(f,g)\in\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma):\,\mbox{\eqref{1f9}
is true}\bigr{\\}}.$ (25)

Here, we use the equality (23) with $i:=0$ and the embedding

$\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)\hookrightarrow\mathcal{H}^{(l_{0}-2q)}(\Omega,\Gamma).$

Thus, the operator (9) has all the properties stated in Theorem 1.

This theorem is proved.

_Proof of Theorem 2._ According to Theorem 1, the bounded linear operator (13)
is a bijection. Hence it is an isomorphism due to the Banach theorem on
inverse operator. Theorem 2 is proved.

_Proof of Theorem 3._ At first we will prove the estimate (14) in the case
where $\chi=\eta=1$. According to (11), we write an arbitrary function $u\in
H^{\varphi}(\Omega)$ in the form $u=u_{0}+u_{1}$ with $u_{0}:=(1-P)u\in N$ and
$u_{1}:=Pu\in P(H^{\varphi}(\Omega))$. By virtue of Theorem 2, we have

$\|u_{1}\|_{H^{\varphi}(\Omega)}\leq
c_{1}\|(A,B)u_{1}\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}=c_{1}\|(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}.$
(26)

Here, $c_{1}$ is the norm of the inverse operator to the isomorphism (13)
(note also that $(A,B)u_{1}=(A,B)u$). Since the space $N$ is finite-
dimensional, all norms are equivalent on $N$, specifically, the norms in
$H^{\varphi}(\Omega)$ and $H^{\varphi\varrho^{-1}}(\Omega)$. Hence, in view of
$u_{0}\in N$ and (26), we obtain the following inequalities:

$\displaystyle\|u_{0}\|_{H^{\varphi}(\Omega)}$ $\displaystyle\leq
c_{0}\|u_{0}\|_{H^{\varphi\varrho^{-1}}(\Omega)}\leq
c_{0}\|u\|_{H^{\varphi\varrho^{-1}}(\Omega)}+c_{0}\|u_{1}\|_{H^{\varphi\varrho^{-1}}(\Omega)}$
$\displaystyle\leq
c_{0}\|u\|_{H^{\varphi\varrho^{-1}}(\Omega)}+c_{0}\|u_{1}\|_{H^{\varphi}(\Omega)}$
$\displaystyle\leq
c_{0}\|u\|_{H^{\varphi\varrho^{-1}}(\Omega)}+c_{0}c_{1}\|(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}.$

Here, $c_{0}$ is a certain positive number that does not depend on $u$. So,

$\|u_{0}\|_{H^{\varphi}(\Omega)}\leq
c_{0}c_{1}\|(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+c_{0}\|u\|_{H^{\varphi\varrho^{-1}}(\Omega)}.$
(27)

The inequalities (26) and (27) gives the estimate (14) in the case of
$\chi=\eta=1$; namely,

$\|u\|_{H^{\varphi}(\Omega)}\leq
c_{3}\bigl{(}\|(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+\|u\|_{H^{\varphi\varrho^{-1}}(\Omega)}\bigr{)}$
(28)

for arbitrary $u\in H^{\varphi}(\Omega)$. Here, the positive number
$c_{3}:=c_{0}+c_{1}+c_{0}c_{1}$ does not depend on $u$.

We will now deduce Theorem 3 in the general situation from (28). Choose a
function $u\in H^{\varphi}(\Omega)$ arbitrarily. In our reasoning, $c_{3}$,
$c_{4}$, and $c_{5}$ denote some positive numbers that are independent of $u$.

Consider the estimate (28) with $\chi u\in H^{\varphi}(\Omega)$ instead of
$u$, namely,

$\|\chi u\|_{H^{\varphi}(\Omega)}\leq c_{3}\,\bigl{(}\|(A,B)(\chi
u)\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+\|\chi
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}\bigr{)}.$ (29)

Rearranging the operator of multiplication by $\chi$ with the differential
operators $A$ and $B_{1},\ldots,B_{q}$, we write

$\displaystyle(A,B)(\chi u)$ $\displaystyle=(A,B)(\chi\eta u)=\chi(A,B)(\eta
u)+(A^{\prime},B^{\prime})(\eta u)$
$\displaystyle=\chi(A,B)u+(A^{\prime},B^{\prime})(\eta u).$

Here, $A^{\prime}$ is a linear differential operator on $\overline{\Omega}$ of
order $\mathrm{ord}\,A^{\prime}\leq 2q-1$, and
$B^{\prime}:=(B_{1}^{\prime},\ldots,B_{q}^{\prime})$ is a collection of
boundary linear differential operators on $\Gamma$ of order
$\mathrm{ord}\,B_{j}^{\prime}\leq m_{j}-1$ for each $j\in\\{1,\ldots,q\\}$.
All the coefficients of $A^{\prime}$ and $B_{j}^{\prime}$ belong respectively
to $C^{\infty}(\overline{\Omega})$ and $C^{\infty}(\Gamma)$. Thus,

$(A,B)(\chi u)=\chi(A,B)u+(A^{\prime},B^{\prime})(\eta u),$ (30)

where, by Proposition 4,

$\|(A^{\prime},B^{\prime})(\eta
u)\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}\leq c_{4}\|\eta
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}.$ (31)

Now, according to (29)–(31), we obtain

$\displaystyle\|\chi u\|_{H^{\varphi}(\Omega)}$ $\displaystyle\leq
c_{3}\,\bigl{(}\|\chi(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}$
(32) $\displaystyle\quad+\|(A^{\prime},B^{\prime})(\eta
u)\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+\|\chi
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}\bigr{)}$ $\displaystyle\leq
c_{3}\|\chi(A,B)u\|_{\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)}+c_{3}c_{4}\|\eta
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}+c_{3}\|\chi
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}.$

Note that, by Proposition 4,

$\|\chi u\|_{H^{\varphi\varrho^{-1}}(\Omega)}=\|\chi\eta
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}\leq c_{5}\|\eta
u\|_{H^{\varphi\varrho^{-1}}(\Omega)}.$ (33)

Formulas (32) and (33) immediately give the required estimate (14), in which
$c:=c_{3}(1+c_{4}+c_{5})$.

Theorem 3 is proved.

_Proof of Theorem 4._ We will first prove this theorem in the case where
$\Omega_{0}=\Omega$ and $\Gamma_{0}=\Gamma$. By the condition, $u\in
H^{(s)}(\Omega)$ for some real $s\in(m+1/2,\sigma_{0}(\varphi))$, and

$(f,g):=(f,g_{1},\ldots,g_{q})=(A,B)u\in\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma).$

Therefore, owing to Theorem 1 applied also in the Sobolev case, the vector
$(f,g)$ satisfies condition (10), and hence $(A,B)v=(f,g)$ for some function
$v\in H^{\varphi}(\Omega)$. Thus, $(A,B)(u-v)=0$, with $u-v\in
H^{(s)}(\Omega)$. Then $w:=u-v\in N\subset C^{\infty}(\overline{\Omega})$ due
to Theorem 1. Hence, $u=v+w\in H^{\varphi}(\Omega)$. Theorem 4 is proved in
the case considered.

Let us now prove this theorem in the general situation. We arbitrarily choose
an open set $V_{1}\subset\mathbb{R}^{n}$ such that $\overline{V_{1}}\subset V$
and $\Omega\cap V_{1}\neq\varnothing$. Put $\Omega_{1}:=\Omega\cap V_{1}$ and
$\Gamma_{1}:=\Gamma\cap V_{1}$. Let us prove that $u\in
H^{\varphi}_{\mathrm{loc}}(\Omega_{1},\Gamma_{1})$.

We choose functions $\chi,\eta\in C^{\infty}(\overline{\Omega})$ so that their
supports lie in $\Omega_{0}\cup\Gamma_{0}$, $\chi=1$ in a neighborhood of
$\Omega_{1}\cup\Gamma_{1}$, and $\eta=1$ on $\mathrm{supp}\,\chi$. By the
condition, $u\in H^{(s)}(\Omega)$ for some real
$s\in(m+1/2,\sigma_{0}(\varphi))$ and $(A,B)u=(f,g)$ with
$(f,g)\in\mathcal{H}^{\varphi\varrho^{-2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$.
Therefore

$(A,B)(\chi u)=\eta(A,B)(\chi u)=\eta(f,g)-\eta(A,B)((1-\chi)u).$

Using the projector $P_{\ast}$, we write

$(A,B)(\chi u)=P_{\ast}(\eta(f,g))+F,$

where

$\displaystyle P_{\ast}(\eta(f,g))\in
P_{\ast}\bigl{(}\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)\bigr{)},$
$\displaystyle F:=(1-P_{\ast})(\eta(f,g))-\eta(A,B)((1-\chi)u)\in
P_{\ast}\bigl{(}\mathcal{H}^{\varrho^{s-2q}}(\Omega,\Gamma)\bigr{)}.$

Hence, by Theorem 2, there exist functions $u_{1}\in H^{\varphi}(\Omega)$ and
$u_{2}\in H^{(s)}(\Omega)$ such that $(A,B)u_{1}=P_{\ast}(\eta(f,g))$ and
$(A,B)u_{2}=F$. Then $(A,B)(\chi u-u_{1}-u_{2})=0$, which implies

$w:=\chi u-u_{1}-u_{2}\in N\subset C^{\infty}(\overline{\Omega})$

in view of Theorem 1. We note that
$F\in\mathcal{H}^{\varrho^{l-2q}}_{\mathrm{loc}}(\Omega_{1},\Gamma_{1})$ for
every $l>\sigma_{1}(\varphi)$ because $(1-P_{\ast})(\eta(f,g))\in N_{\ast}$
and $\eta(A,B)((1-\chi)u)=0$ on $\Omega_{1}\cup\Gamma_{1}$. Therefore

$u_{2}\in H_{\mathrm{loc}}^{\varrho^{l}}(\Omega_{1},\Gamma_{1})\subset
H^{\varphi}_{\mathrm{loc}}(\Omega_{1},\Gamma_{1})$

according to the theory of elliptic boundary-value problems in Sobolev spaces
(see, e.g., [23, Theorem 5.2]). We conclude now that

$\chi u=u_{1}+u_{2}+w\in H^{\varphi}_{\mathrm{loc}}(\Omega_{1},\Gamma_{1}).$

Then $\zeta u=\zeta\chi u\in H^{\varphi}(\Omega)$ for an arbitrary function
$\zeta\in C^{\infty}(\overline{\Omega})$ with
$\mathrm{supp}\,\zeta\subset\Omega_{1}\cup\Gamma_{1}$; i.e., $u\in
H^{\varphi}_{\mathrm{loc}}(\Omega_{1},\Gamma_{1})$. The latter inclusion
implies that $u\in H^{\varphi}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ due to
our choice of $V_{1}$.

Theorem 4 is proved.

_Proof of Theorem 5._ According to Theorem 4, the inclusion $u\in
H^{\varphi}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ holds. We arbitrarily
choose a point $x\in\Omega_{0}\cup\Gamma_{0}$. Let a function $\chi\in
C^{\infty}(\overline{\Omega})$ satisfy the conditions
$\mathrm{supp}\,\chi\subset\Omega_{0}\cup\Gamma_{0}$ and $\chi=1$ in some
neighborhood $V(x)$ of $x$. Owing to (8), we get $\chi u\in
H^{\varphi}(\Omega)\subset C^{p}(\overline{\Omega})$. Then $u\in C^{p}(V(x))$.
Hence, $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$ due to the arbitrariness of $x$.
Theorem 5 is proved.

_Proof of Remark 1._ Suppose that the implication (16) holds true. Let $V$ be
an open ball such that $\overline{V}\subset\Omega_{0}$, and choose a function
$v\in H^{\varphi}(V)$ arbitrarily. Note that $v=u\\!\upharpoonright\\!V$ for
certain $u\in H^{\varphi}(\Omega)$. Besides,
$(A,B)u\in\mathcal{H}^{\varphi\varrho^{-2q}}(\Omega,\Gamma)$. Now, according
to (16), we conclude that $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$. Hence, $v\in
C^{p}(\overline{V})$. Thus, $H^{\varphi}(V)\subset C^{p}(\overline{V})$, which
implies condition (15) in view of (8). Remark 1 is proved.

_Proof of Corollary 1._ By virtue of Theorem 5 with $p:=2q$,
$\varphi:=\varphi_{1}$, $\Omega_{0}:=\Omega$, and $\Gamma_{0}:=\varnothing$,
it follows from conditions (17) and (19) that $u\in C^{2q}(\Omega)$. Besides,
by the same theorem with $p:=m$, $\varphi:=\varphi_{2}$, $\Omega_{0}:=\Omega$,
and $\Gamma_{0}:=\Gamma$, it follows from conditions (17), (18), and (20) that
$u\in C^{m}(\overline{\Omega})$. Thus, $u$ is a classical solution to the
boundary-value problem (1), (2). Corollary 1 is proved.

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