# Kipriyanov’s Fractional Calculus Prehistory and Legacy

Maksim V. Kukushkin

Moscow State University of Civil Engineering, 129337, Moscow, Russia
Russian Academy of Science, Institute of Applied Mathematics and Automation,
360051, Nalchik, Russia
<EMAIL_ADDRESS>
###### Abstract

This paper is partly a historical survey of various approaches and methods in
the fractional calculus, partly a description of the Kipriyanov extraordinary
theory in comparison with the classical one. The significance and outstanding
methods in constructing the independent Kipriyanov fractional calculus theory
are convexly stressed, also we represent modern results involving the
Kipriyanov operator and corresponding generalization under the semigroup
theory point of view.

Keywords: Fractional power of the m-accretive operator; infinitesimal
generator of a semigroup; Kipriyanov fractional differential operator;
strictly accretive operator; abstract evolution equations.

MSC 47B28; 47A10; 47B12; 47B10; 47B25; 20M05; 20M99; 26A33.

##  1 Brief historical review

1\. Birth of Kipriyanov’s fractional calculus

Ivan Alexandrovich Kipriyanov was born on the Urals in the Chelyabinsk region.
In 1945, Kipriyanov I.A. entered the Faculty of Physics and Mathematics of the
Chelyabinsk Pedagogical Institute. In 1949 he was accepted to graduate school
Steklov V.A. Mathematical Institute of the USSR Academy of Sciences, his
supervisor was Keldysh M.V. In 1954 Kupriyanov I.A. defended his Ph. D thesis
”On summation of Fourier series and interpolation processes for functions of
two variables”. In this work, Kupriyanov I.A. found a class of functions of
two variables, to which it is possible to completely transfer the results of
Lozinsky S.M. that, in a sense, every theorem on convergence and summation of
a one-dimensional Fourier series can be transferred to convergence and
summation of the corresponding trigonometric interpolation process with
equally spaced nodes. Moreover, these results remain valid for more general
approximating processes. In his review of the dissertation, Prof. Lozinsky
S.M. noted the successful choice of a class of functions of two variables.
This choice, as Lozinsky S.M. wrote, required analytical insight and provided
the success of the work. In his review of the abstract of the dissertation,
Prof. Kantorovich L.V. noted that, having familiarized himself with the work
of Kipriyanov I.A. on the abstract and two reports in Leningrad in November
1953 at the seminar on the theory of functions and functional analysis of
Leningrad University, he formed a very positive opinion about it. The work
made a very favorable impression on the other participants of the seminar, as
its discussion showed. It should be noted that the seminar was attended by
Professors Kantorovich L.V., Lozinsky S.M., Natanson I.P., etc. In the late
50s and early 60s of our century, active work was carried out on the study of
various kinds of functional spaces, important in themselves and also playing
an important role in the modern theory of partial differential equations and
probability theory. In 1958, Kipriyanov’s first work on fractional order
derivatives appeared, in which the concept of fractional order partial
derivatives for functions given in a cube was introduced, starting from the
fractional integral in the sense of Marchaud and derivatives of the integer
order in the sense of Sobolev S.L., here we should note that the implemented
approach was unique and distinctive in comparison with the regular fractional
calculus theory of the time. The definition of the Kipriyanov fractional
derivative is based upon some integral identity that relates to the Marchaud
fractional derivative. The corresponding integral representation is given, the
definition of two functional spaces is given, and embedding theorems and space
completeness theorems are proved for them. One of these theorems in the one-
dimensional case significantly complements the well-known theorem of Hardy and
Littlewood on fractional integrals. In the next cycle of works from 1959 to
1961, Kupriyanov I.A. studied the fractional derivatives in the direction
introduced by him [12],[13],[14], [15]. In 1960 Kupriyanov I.A. published a
paper on the operator of fractional differentiation [13], which is a
fractional order operator with respect to a second-order elliptic operator
with smooth coefficients. It is remarkable that this operator allows us to
study boundary value problems for differential equations containing, in
addition to partial derivatives, fractional derivatives. Here, we should
interrupt the description of Kipriyanov’s other achievements and focus on the
specific questions regarding his most significant scientific contribution -
fractional calculus as an independently constructed axiomatic theory.

2\. Branches in generalizations of the Riemann-Liouville an Marchaud operators

The central point of fractional calculus is a concept of fractional
differentiation. In this regard, we should admit that the Riemann-Liouville
operator of fractional differentiation is at the origin of the concept and
plays a special role in the science. Such operators as Caputo and Marchaud
certainly are worth mentioned in the context, the first one is not interesting
for us since it is more like a reduction of the Riemann-Liouville operator on
smooth functions disappearing at the initial point (if we consider the matter
from the point of view that is of functional analysis), but the second one
does completely reflect a true mathematical nature of fractional derivative as
a notion, since it has a representation in terms of infinitesimal generator of
the corresponding semigroup [23]. It is clear that considering such an
approach we are forced to deal with more general notions of the operator
theory and in this way the understanding of the notion of fractional
derivative as a fractional power of infinitesimal generator is harmoniously
completed, on the one hand.

On the other hand, for a harmony of the narrative, we should referee an
extract of the paper [25] appealing to another generalization, if we interpret
the fractional differential Riemann-Liouville operator as a particular case of
the derivative of the convolution operator for which the so called Sonin
condition holds [57]. We should note that the second direction in
understanding the matter was developed by mathematicians such as Rubin B.S.
[48],[51],[50], Vakulov B.G. [58], Samko S.G. [52],[54], Karapetyants N.K.
[10],[11]. Let us remind that the so called mapping theorem for the Riemann-
Liouville operator (the particular case of the Sonin operator) were firstly
studied by H. Hardy and Littlewood [6] and nowadays is known as the Hardy-
Littlewood theorem with limit index. However there was an attempt to extend
this theorem on some class of weighted Lebesgue spaces defined as functional
spaces endowed with the following norm
$\|f\|_{L_{p}(I\\!,\,\beta,\gamma)}:=\|f\|_{L_{p}(I\\!,\mu)},\,\mu(x)=\omega^{\,\beta,\gamma}(x):=(x-a)^{\beta}(b-x)^{\gamma},\,\beta,\gamma\in\mathbb{R},\,I:=(a,b).$
In this direction the mathematicians such as Rubin B.S., Karapetyants N.K.
[10] had success, the following problem was considered
$I^{\alpha}_{a+}:L_{p}(I\\!,\,\beta,\gamma)\longrightarrow\,?$ However the
converse theorem was not! All these create the prerequisite to invent another
approach for studying mapping properties of the Riemann-Liouville operator or,
more generally, integral operators. Thus, trying to solve (at least in
particular) more general problem, in the paper [20] we deal with mapping
theorems for operators acting on Banach spaces in order to obtain afterwards
the desired results applicable to integral operators. In this regard the
following papers are worth noticing [18], [21], where in additional, a special
technique based on the properties of the Jacobi polynomials was introduced.
Based on this approach, in the paper [25] we offer a method of studying the
Sonin operator [57], which is defined as a convolution operator
${}_{s}I^{\varrho}_{a+}\varphi:=\varrho\ast_{a}\\!\varphi$ under some
conditions (the so called Sonin conditions) imposed on the kernel $\varrho,$
i.e. there exists a function $\vartheta$ such that $\varrho\ast\vartheta=1.$
The particular case of the Sonin kernel is a kernel of the fractional integral
Riemman-Liouville operator, many other examples can be found in papers [55],
[49], the first one gives us a survey considering various types of kernels
such as the Bessel-type function, the power-exponential function, the
incomplete gamma function e.t.c., the main concept of the second one is to
construct a widest class of functions being a Sonin kernel. Here, we can
partly close the matter at this point having noted that there was a successful
attempt to establish a criterion of the solvability of the Sonin-Abel Equation
in the Weighted Lebesgue Space [25].

However, let us be back to the first understanding of the matter which is
closely connected with the notion of the Kipriyanov operator.

3\. The semigroup approach and the spectral theory

The idea discussed in this paragraph relates to a model that gives us a
representation of a composition of fractional differential operators in terms
of the semigroup theory. For instance we can represent a second order
differential operator as a some kind of a transform of the infinitesimal
generator of a shift semigroup. Continuing this line of reasonings we
generalized a differential operator with a fractional integro-differential
composition in final terms to some transform of the corresponding
infinitesimal generator and introduced a class of transforms of m-accretive
operators. Further, we used methods obtained in the papers [22],[19] to study
spectral properties of non-selfadjoint operators acting in a complex separable
Hilbert space, these methods alow us to obtain an asymptotic equivalence
between the real component of the resolvent and the resolvent of the real
component of an operator. Due to such an approach we obtain relevant results
since an asymptotic formula for the operator real component can be established
in many cases (see [2], [45]). Thus, a classification in accordance with
resolvent belonging to the Schatten-von Neumann class was obtained, a
sufficient condition of completeness of the root vectors system was
formulated.

The latter approach allows to construct an abstract model of a differential
operator with a fractional Kipriyanov integro-differential operator
composition in final terms, where modeling is understood as an interpretation
of concrete differential operators in terms of the infinitesimal generator of
a corresponding semigroup. Moreover, we can consider an approach in
contracting the space of fractionally-differentiable functions which
originates from the analog created by Kipriyanov and goes further up to the
semigroup theory generalizations. In this paper we deal with a more general
operator — a differential operator with a fractional integro-differential
operator composition in final terms, which covers the corresponding one-
dimensional operator. Various types of fractional integro-differential
operator compositions were studied by such mathematicians as Prabhakar T.R.
[43], Love E.R. [34], Erdelyi A. [4], McBride A. [39], Dimovski I.H.,
Kiryakova V.S. [3], Nakhushev A.M. [41]. In particular the aim of this paper
is to represent a description of the previously obtained results under a
specific point of view related with Kipriyanov’s fractional calculus.

3\. Evolution equations with the operator function in the second term

Having created a direction of the spectral theory of non-selfadjoint
operators, we can consider abstract theoretical results as a base for further
research studying such mathematical objects as a Cauchy problem for evolution
equation of fractional order in the abstract Hilbert space. We consider in the
second term an operator function defined on a special operator class covering
a generator transform considered in [23] and discussed in the previous
paragraph, where a corresponding semigroup is supposed to be a $C_{0}$
semigroup of contractions. In its own turn the transform reduces to a linear
composition of differential operators of real order in various senses such as
the Riemann-Liouville fractional differential operator, the Kipriyanov
operator, the Riesz potential, the difference operator [13],[23],[53].
Moreover, in the paper [27] we broadened the class of differential operators
having considered the artificially constructed normal operator that cannot be
covered by the Lidskii results [33]. It should be noted that the Kipriyanov
operator is very useful in theoretical constructions as well as in
applications since it covers Euclidean spaces and can be considered as a term
in a perturbation of a differential operator of an arbitrary odd order acting
in n-dimensional Euclidean space. This fact is based upon the the brilliant
idea of Kipriyanov to consider directional coordinates in the n-dimensional
Euclidean space, the latter approach is independent on dimension what is an
enormous advantage for we can consider compositions of operators having
various nature.

The application part of the theory involving fractional integro-differential
constructions appeals to the results and problems which can be considered as
particular cases of the abstract ones, the following papers a worth noting
within the context [38],[37],[44]. At the same time, we should admit that
abstract methods can be ”clumsy” for some peculiarities can be considered only
by a unique technique what forms a main contribution of the specialists
dealing with concrete differential equations. Here, we should add that the
relevance of the abstract problems can be expressed convexly by virtue of the
application of the fractional integro-differential compositions with the
Kipriyanov operator in physics and engineering sciences.

Apparently, in the paper [30] we realized the idea to broaden the class of
fractional integro-differential compositions having considered a notion of
operator function applicably to a Cauchy problem for an abstract fractional
evolution equation with an operator function in the second term not containing
the time variable, where the derivative in the first term is supposed to be of
fractional order. Here, we should note that regarding to functional spaces we
have that an operator function generates a variety of operators acting in a
corresponding space. In this regard, even a power function gives us an
interesting result [27]. In the context of the existence and uniqueness
theorems, a significant refinement that is worth highlighting is the obtained
formula for the solution represented by a series on the root vectors. In the
absence of the norm convergence of the root vector series, we need to consider
a notion of convergence in weaker Bari, Riesz, Abel-Lidskii senses
[33],[1],[5].

In spite of the claimed rather applied objectives from the operator theory
point of view, we admit that the problem of the root vectors expansion for a
non-selfadjoint unbounded operator still remains relevant in the context of
the paper. It is remarkable that the problem origins nearly from the first
half of the last century [9], [36], [33], [32],[35],[1],[56], [40],[22], [19]
[23], [26]. However, we have a particular interest when an operator is
represented by a linear combination of operators where a so-called senior term
is non-selfadjoint for a case corresponding to a selfadjoint operator was
thoroughly studied in the papers [9],[32],[35],[36],[40], [56]. In this regard
the linear combination of the second order differential operator and the
Kipriyanov operator represents a relevant model class for which the obtained
spectral theory results [19] created a prerequisite for further abstract
generalizations [22],[23].

##  2 Abstract method

1\. Preliminaries

Let $C,C_{i},\;i\in\mathbb{N}_{0}$ be positive constants. We assume that a
value of $C$ can be different in various formulas and parts of formulas but
values of $C_{i}$ are certain. Denote by $\mathrm{Fr}\,M$ the set of boundary
points of the set $M.$ Everywhere further, if the contrary is not stated, we
consider linear densely defined operators acting on a separable complex
Hilbert space $\mathfrak{H}$. Denote by $\mathcal{B}(\mathfrak{H})$ the set of
linear bounded operators on $\mathfrak{H}.$ Denote by $\tilde{L}$ the closure
of an operator $L.$ We establish the following agreement on using symbols
$\tilde{L}^{i}:=(\tilde{L})^{i},$ where $i$ is an arbitrary symbol. Denote by
$\mathrm{D}(L),\,\mathrm{R}(L),\,\mathrm{N}(L)$ the domain of definition, the
range, and the kernel or null space of an operator $L,$ respectively. The
deficiency (codimension) of $\mathrm{R}(L),$ dimension of $\mathrm{N}(L)$ are
denoted by $\mathrm{def}\,L,\;\mathrm{nul}\,L$ respectively. Assume that $L$
is a closed operator acting on $\mathfrak{H},\,\mathrm{N}(L)=0,$ let us define
a Hilbert space
$\mathfrak{H}_{L}:=\big{\\{}f,g\in\mathrm{D}(L),\,(f,g)_{\mathfrak{H}_{L}}=(Lf,Lg)_{\mathfrak{H}}\big{\\}}.$
Consider a pair of complex Hilbert spaces $\mathfrak{H},\mathfrak{H}_{+},$ the
notation $\mathfrak{H}_{+}\subset\subset\mathfrak{H}$ means that
$\mathfrak{H}_{+}$ is dense in $\mathfrak{H}$ as a set of elements and we have
a bounded embedding provided by the inequality $\|f\|_{\mathfrak{H}}\leq
C_{0}\|f\|_{\mathfrak{H}_{+}},\,C_{0}>0,\;f\in\mathfrak{H}_{+},$ moreover any
bounded set with respect to the norm $\mathfrak{H}_{+}$ is compact with
respect to the norm $\mathfrak{H}.$ Let $L$ be a closed operator, for any
closable operator $S$ such that $\tilde{S}=L,$ its domain $\mathrm{D}(S)$ will
be called a core of $L.$ Denote by $\mathrm{D}_{0}(L)$ a core of a closeable
operator $L.$ Let $\mathrm{P}(L)$ be the resolvent set of an operator $L$ and
$R_{L}(\zeta),\,\zeta\in\mathrm{P}(L),\,[R_{L}:=R_{L}(0)]$ denotes the
resolvent of an operator $L.$ Denote by $\lambda_{i}(L),\,i\in\mathbb{N}$ the
eigenvalues of an operator $L.$ Suppose $L$ is a compact operator and
$N:=(L^{\ast}L)^{1/2},\,r(N):={\rm dim}\,\mathrm{R}(N);$ then the eigenvalues
of the operator $N$ are called the singular numbers (s-numbers) of the
operator $L$ and are denoted by $s_{i}(L),\,i=1,\,2,...\,,r(N).$ If
$r(N)<\infty,$ then we put by definition $s_{i}=0,\,i=r(N)+1,2,...\,.$ Let
$\nu(L)$ denotes the sum of all algebraic multiplicities of an operator $L.$
Denote by $n(r)$ a function equals to the quantity of the elements of the
sequence $\\{a_{n}\\}_{1}^{\infty},\,|a_{n}|\uparrow\infty$ within the circle
$|z|<r.$ Let $A$ be a compact operator, denote by $n_{A}(r),$ counting
function a function $n(r)$ corresponding to the sequence
$\\{s^{-1}_{i}(A)\\}_{1}^{\infty}.$ Let
$\mathfrak{S}_{p}(\mathfrak{H}),\,0<p<\infty$ be a Schatten-von Neumann class
and $\mathfrak{S}_{\infty}(\mathfrak{H})$ be the set of compact operators.
Denote by $\tilde{\mathfrak{S}}_{\rho}(\mathfrak{H})$ the class of the
operators such that
$A\in\tilde{\mathfrak{S}}_{\rho}(\mathfrak{H})\Rightarrow\\{A\in\mathfrak{S}_{\rho+\varepsilon},\,A\overline{\in}\,\mathfrak{S}_{\rho-\varepsilon},\,\forall\varepsilon>0\\}.$
In accordance with [26], we will call it Schatten-von Neumann class of the
convergence exponent. Suppose $L$ is an operator with a compact resolvent and
$s_{n}(R_{L})\leq C\,n^{-\mu},\,n\in\mathbb{N},\,0\leq\mu<\infty;$ then we
denote by $\mu(L)$ order of the operator $L$ (see [56]). Denote by
$\mathfrak{Re}L:=\left(L+L^{*}\right)/2,\,\mathfrak{Im}L:=\left(L-L^{*}\right)/2i$
the real and imaginary Hermitian components of an operator $L$ respectively.
In accordance with the terminology of the monograph [8], the set
$\Theta(L):=\\{z\in\mathbb{C}:z=(Lf,f)_{\mathfrak{H}},\,f\in\mathrm{D}(L),\,\|f\|_{\mathfrak{H}}=1\\}$
is called the numerical range of an operator $L.$ An operator $L$ is called
sectorial if its numerical range belongs to a closed sector
$\mathfrak{L}_{\iota}(\theta):=\\{\zeta:\,|\arg(\zeta-\iota)|\leq\theta<\pi/2\\},$
where $\iota$ is the vertex and $\theta$ is the semi-angle of the sector
$\mathfrak{L}_{\iota}(\theta).$ If we want to stress the correspondence
between $\iota$ and $\theta,$ then we will write $\theta_{\iota}.$ An operator
$L$ is called bounded from below if the following relation holds
$\mathrm{Re}(Lf,f)_{\mathfrak{H}}\geq\gamma_{L}\|f\|^{2}_{\mathfrak{H}},\,f\in\mathrm{D}(L),\,\gamma_{L}\in\mathbb{R},$
where $\gamma_{L}$ is called a lower bound of $L.$ An operator $L$ is called
accretive if $\gamma_{L}=0.$ An operator $L$ is called strictly accretive if
$\gamma_{L}>0.$ An operator $L$ is called m-accretive if the next relation
holds
$(A+\zeta)^{-1}\in\mathcal{B}(\mathfrak{H}),\,\|(A+\zeta)^{-1}\|\leq(\mathrm{Re}\zeta)^{-1},\,\mathrm{Re}\zeta>0.$
An operator $L$ is called symmetric if one is densely defined and the
following equality holds
$(Lf,g)_{\mathfrak{H}}=(f,Lg)_{\mathfrak{H}},\,f,g\in\mathrm{D}(L).$ Consider
a sesquilinear form $s[\cdot,\cdot]$ (see [8] ) defined on a linear manifold
of the Hilbert space $\mathfrak{H}.$ Let
$\mathfrak{h}=(s+s^{\ast})/2,\,\mathfrak{k}=(s-s^{\ast})/2i$ be a real and
imaginary component of the form $s$ respectively, where
$s^{\ast}[u,v]=s\overline{[v,u]},\;\mathrm{D}(s^{\ast})=\mathrm{D}(s).$ Denote
by $s[\cdot]$ the quadratic form corresponding to the sesquilinear form
$s[\cdot,\cdot].$ According to these definitions, we have
$\mathfrak{h}[\cdot]=\mathrm{Re}\,s[\cdot],\,\mathfrak{k}[\cdot]=\mathrm{Im}\,s[\cdot].$
Denote by $\tilde{s}$ the closure of a form $s.$ The range of a quadratic form
$s[f],\,f\in\mathrm{D}(s),\,\|f\|_{\mathfrak{H}}=1$ is called the range of the
sesquilinear form $s$ and is denoted by $\Theta(s).$ A form $s$ is called
sectorial if its range belongs to a sector having a vertex $\iota$ situated at
the real axis and a semi-angle $0\leq\theta<\pi/2.$ Due to Theorem 2.7 [8,
p.323] there exist unique m-sectorial operators $T_{s},T_{\mathfrak{h}}$
associated with the closed sectorial forms $s,\mathfrak{h}$ respectively. The
operator $T_{\mathfrak{h}}$ is called a real part of the operator $T_{s}$ and
is denoted by $\mathrm{Re}\,T_{s}.$

Assume that $T_{t},\,(0\leq t<\infty)$ is a semigroup of bounded linear
operators on $\mathfrak{H},$ by definition put

$Af=-\lim\limits_{t\rightarrow+0}\left(\frac{T_{t}-I}{t}\right)f,$

where $\mathrm{D}(A)$ is a set of elements for which the last limit exists in
the sense of the norm $\mathfrak{H}.$ In accordance with definition [42, p.1]
the operator $-A$ is called the infinitesimal generator of the semigroup
$T_{t}.$

Let $f_{t}:I\rightarrow\mathfrak{H},\,t\in I:=[a,b],\,-\infty<a<b<\infty.$ The
following integral is understood in the Riemann sense as a limit of partial
sums

$\sum\limits_{i=0}^{n}f_{\xi_{i}}\Delta
t_{i}\stackrel{{\scriptstyle\mathfrak{H}}}{{\longrightarrow}}\int\limits_{I}f_{t}dt,\,\lambda\rightarrow
0,$ (1)

where $(a=t_{0}<t_{1}<...<t_{n}=b)$ is an arbitrary splitting of the segment
$I,\;\lambda:=\max\limits_{i}(t_{i+1}-t_{i}),\;\xi_{i}$ is an arbitrary point
belonging to $[t_{i},t_{i+1}].$ The sufficient condition of the last integral
existence is a continuous property (see[31, p.248]) i.e.
$f_{t}\stackrel{{\scriptstyle\mathfrak{H}}}{{\longrightarrow}}f_{t_{0}},\,t\rightarrow
t_{0},\;\forall t_{0}\in I.$ The improper integral is understood as a limit

$\int\limits_{a}^{b}f_{t}dt\stackrel{{\scriptstyle\mathfrak{H}}}{{\longrightarrow}}\int\limits_{a}^{c}f_{t}dt,\,b\rightarrow
c,\,c\in[-\infty,\infty].$ (2)

Using notations of the paper [12], we assume that $\Omega$ is a convex domain
of the $n$ \- dimensional Euclidean space $\mathbb{E}^{n}$, $P$ is a fixed
point of the boundary $\partial\Omega,$ $Q(r,\mathbf{e})$ is an arbitrary
point of $\Omega.$ Let $\mathrm{d}:=\mathrm{diam}\Omega,$ we denote by
$\mathbf{e}$ a unit vector having a direction from $P$ to $Q,$ denote by
$r=|P-Q|$ the Euclidean distance between the points $P,Q,$ and use the
shorthand notation $T:=P+\mathbf{e}t,\,t\in\mathbb{R}.$ We consider the
Lebesgue classes $L_{p}(\Omega),\;1\leq p<\infty$ of complex valued functions.
For the function $f\in L_{p}(\Omega),$ we have

$\int\limits_{\Omega}|f(Q)|^{p}dQ=\int\limits_{\omega}d\chi\int\limits_{0}^{d(\mathbf{e})}|f(Q)|^{p}r^{n-1}dr<\infty,$
(3)

where $d\chi$ is an element of solid angle of the unit sphere surface (the
unit sphere belongs to $\mathbb{E}^{n}$) and $\omega$ is a surface of this
sphere, $d:=d(\mathbf{e})$ is the length of the segment of the ray going from
the point $P$ in the direction $\mathbf{e}$ within the domain $\Omega.$
Without loss of generality, we consider only those directions of $\mathbf{e}$
for which the inner integral on the right-hand side of equality (3) exists and
is finite. It is the well-known fact that these are almost all directions. We
use a shorthand notation $P\cdot Q=P^{i}Q_{i}=\sum^{n}_{i=1}P_{i}Q_{i}$ for
the inner product of the points
$P=(P_{1},P_{2},...,P_{n}),\,Q=(Q_{1},Q_{2},...,Q_{n})$ which belong to
$\mathbb{E}^{n}.$ Denote by $D_{i}f$ a weak partial derivative of the function
$f$ with respect to a coordinate variable with index $1\leq i\leq n.$ We
assume that all functions have a zero extension outside of $\bar{\Omega}.$
Everywhere further, unless otherwise stated, we use notations of the papers
[5], [8], [12], [13], [53].

Bellow, we represent the conditions of Theorem 1 [23] that gives us a
description of spectral properties, in terms of the real part order, of a non-
selfadjoint operator $L$ acting in $\mathfrak{H}.$

$(\mathrm{H}1)$ There exists a Hilbert space
$\mathfrak{H}_{+}\subset\subset\mathfrak{H}$ and a linear manifold
$\mathfrak{M}$ that is dense in $\mathfrak{H}_{+}.$ The operator $L$ is
defined on $\mathfrak{M}.$

$(\mathrm{H2})\,\left|(Lf,g)_{\mathfrak{H}}\right|\\!\leq\\!C_{1}\|f\|_{\mathfrak{H}_{+}}\|g\|_{\mathfrak{H}_{+}},\,\,\mathrm{Re}(Lf,f)_{\mathfrak{H}}\\!\geq\\!C_{2}\|f\|^{2}_{\mathfrak{H}_{+}},\,f,g\in\mathfrak{M},\;C_{1},C_{2}>0.$

Here, we should remark that since there is no general statement claiming that
the intersection of the domain of definitions of an operator and its adjoint
is a dense set, then we cannot restrict the reasonings considering Hermitian
real component but compelled to involve the notion of the operator real part.
This is why it is rather reasonable to suggest the the issue should be
undergone to a comprehensive analysis.

Consider a condition $\mathfrak{M}\subset\mathrm{D}(W^{\ast}),$ in this case
the real Hermitian component $\mathcal{H}:=\mathfrak{Re}\,W$ of the operator
is defined on $\mathfrak{M},$ the fact is that $\tilde{\mathcal{H}}$ is
selfadjoint, bounded from bellow (see Lemma 3 [22]), where $H=ReW.$ Hence a
corresponding sesquilinear form (denote this form by $h$) is symmetric and
bounded from bellow also (see Theorem 2.6 [8, p.323]). It can be easily shown
that $h\subset\mathfrak{h},$ but using this fact we cannot claim in general
that $\tilde{\mathcal{H}}\subset H$ (see [8, p.330] ). We just have an
inclusion $\tilde{\mathcal{H}}^{1/2}\subset H^{1/2}$ (see [8, p.332]). Note
that the fact $\tilde{\mathcal{H}}\subset H$ follows from a condition
$\mathrm{D}_{0}(\mathfrak{h})\subset\mathrm{D}(h)$ (see Corollary 2.4 [8,
p.323]). However, it is proved (see proof of Theorem 4 [22]) that relation H2
guaranties that $\tilde{\mathcal{H}}=H.$ Note that the last relation is very
useful in applications, since in most concrete cases we can find a concrete
form of the operator $\mathcal{H}.$

2\. Intrinsic properties of the Kipriyanov operator

Here, we study a case $\alpha\in(0,1).$ Assume that
$\Omega\subset\mathbb{E}^{n}$ is a convex domain, with a sufficient smooth
boundary ($C^{3}$ class) of the n-dimensional Euclidian space. For the sake of
the simplicity we consider that $\Omega$ is bounded, but the results can be
extended to some type of unbounded domains. In accordance with the definition
given in the paper [17], we consider the directional fractional integrals. By
definition, put

$(\mathfrak{I}^{\alpha}_{0+}f)(Q):=\frac{1}{\Gamma(\alpha)}\int\limits^{r}_{0}\frac{f(P+t\mathbf{e})}{(r-t)^{1-\alpha}}\left(\frac{t}{r}\right)^{n-1}\\!\\!\\!\\!dt,\,(\mathfrak{I}^{\alpha}_{d-}f)(Q):=\frac{1}{\Gamma(\alpha)}\int\limits_{r}^{d}\frac{f(P+t\mathbf{e})}{(t-r)^{1-\alpha}}\,dt,$
$\;f\in L_{p}(\Omega),\;1\leq p\leq\infty.$

The properties of these operators are described in detail in the papers
[17],[16]. Similarly to the monograph [53] we consider left-side and right-
side cases. For instance, $\mathfrak{I}^{\alpha}_{0+}$ is called a left-side
directional fractional integral. We suppose $\mathfrak{I}^{0}_{0+}=I.$
Nevertheless, this fact can be easily proved dy virtue of the reasonings
corresponding to the one-dimensional case and given in [53]. We also consider
integral operators with a weighted factor (see [53, p.175]) defined by the
following formal construction

$\left(\mathfrak{I}^{\alpha}_{0+}\mu
f\right)(Q):=\frac{1}{\Gamma(\alpha)}\int\limits^{r}_{0}\frac{(\mu
f)(P+t\mathbf{e})}{(r-t)^{1-\alpha}}\left(\frac{t}{r}\right)^{n-1}\\!\\!\\!\\!dt,$

where $\mu$ is a real-valued function. We introduce the classes of functions
representable by the directional fractional integrals.

$\mathfrak{I}^{\alpha}_{0+}(L_{p}):=\left\\{u:\,u(Q)=(\mathfrak{I}^{\alpha}_{0+}g)(Q)\right\\},\;\mathfrak{I}^{\alpha}_{d-}(L_{p})=\left\\{u:\,u(Q)=(\mathfrak{I}^{\alpha}_{d-}g)(Q)\right\\},\,g\in
L_{p}(\Omega),\,1\leq p\leq\infty.$

Define the following auxiliary operators acting in $L_{p}(\Omega)$ and
depended on the parameter $\varepsilon>0.$ In the left-side case

$(\psi^{+}_{\varepsilon}f)(Q)=\left\\{\begin{aligned}
\int\limits_{0}^{r-\varepsilon}\frac{f(Q)r^{n-1}-f(T)t^{n-1}}{(r-t)^{\alpha+1}r^{n-1}}dt,\;\varepsilon\leq
r\leq d,\\\
\frac{f(Q)}{\alpha}\left(\frac{1}{\varepsilon^{\alpha}}-\frac{1}{r^{\alpha}}\right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\leq
r<\varepsilon.\\\ \end{aligned}\right.$ (4)

In the right-side case

$(\psi^{-}_{\varepsilon}f)(Q)=\left\\{\begin{aligned}
\int\limits_{r+\varepsilon}^{d}\frac{f(Q)-f(T)}{(t-r)^{\alpha+1}}dt,\;0\leq
r\leq d-\varepsilon,\\\
\frac{f(Q)}{\alpha}\left(\frac{1}{\varepsilon^{\alpha}}-\frac{1}{(d-r)^{\alpha}}\right),\;\;\;d-\varepsilon<r\leq
d.\\\ \end{aligned}\right.$

Using the definitions of the monograph [53, p.181] we consider the following
operators. In the left-side case

$(\mathfrak{D}^{\alpha}_{0+\\!,\,\varepsilon}f)(Q)=\frac{1}{\Gamma(1-\alpha)}f(Q)r^{-\alpha}+\frac{\alpha}{\Gamma(1-\alpha)}(\psi^{+}_{\varepsilon}f)(Q).$
(5)

In the right-side case

$(\mathfrak{D}^{\alpha}_{d-\\!,\,\varepsilon}f)(Q)=\frac{1}{\Gamma(1-\alpha)}f(Q)(d-r)^{-\alpha}+\frac{\alpha}{\Gamma(1-\alpha)}(\psi^{-}_{\varepsilon}f)(Q).$

The left-side and right-side fractional derivatives are understood
respectively as the following limits

$\mathfrak{D}^{\alpha}_{0+}f=\lim\limits_{\stackrel{{\scriptstyle\varepsilon\rightarrow
0}}{{(L_{p})}}}\mathfrak{D}^{\alpha}_{0+\\!,\,\varepsilon}f,\;\mathfrak{D}^{\alpha}_{d-}f=\lim\limits_{\stackrel{{\scriptstyle\varepsilon\rightarrow
0}}{{(L_{p})}}}\mathfrak{D}^{\alpha}_{d-\\!,\,\varepsilon}f,\,1\leq p<\infty.$
(6)

Consider the Kipriyanov fractional differential operator defined in the paper
[13] by the formal expression

$\mathfrak{D}^{\alpha}(Q)=\frac{\alpha}{\Gamma(1-\alpha)}\int\limits_{0}^{r}\frac{[f(Q)-f(T)]}{(r-t)^{\alpha+1}}\left(\frac{t}{r}\right)^{n-1}dt+C^{(\alpha)}_{n}f(Q)r^{-\alpha}\\!,\,P\in\partial\Omega,$

where $C^{(\alpha)}_{n}=(n-1)!/\Gamma(n-\alpha).$ It is remarkable that
Theorem 2 [13] establishes the mapping properties of the Kipriyanov operator,
here we represent its statement in the explicit form: under the assumptions

$lp\leq n,\;0<\alpha<l-\frac{n}{p}+\frac{n}{q},\,q>p,$ (7)

we have that for sufficiently small $\delta>0$ the following inequality holds

$\|\mathfrak{D}^{\alpha}f\|_{L_{q}(\Omega)}\leq\frac{K}{\delta^{\nu}}\|f\|_{L_{p}(\Omega)}+\delta^{1-\nu}\|f\|_{L^{l}_{p}(\Omega)},\,f\in\dot{W}_{p}^{\,l}(\Omega),$
(8)

where

$\nu=\frac{n}{l}\left(\frac{1}{p}-\frac{1}{q}\right)+\frac{\alpha+\beta}{l}.$
(9)

The constant $K$ does not depend on $\delta,\,f;$ the point
$P\in\partial\Omega;\;\beta$ is an arbitrarily small fixed positive number. It
is remarkable that Lemma 2.5 [17] establishes the connection between the
fractional differential operators, more precisely it establishes the following
relation

$(\mathfrak{D}^{\alpha}f)(Q)=\left(\mathfrak{D}^{\alpha}_{0+}f\right)(Q),\,f\in\dot{W}_{p}^{l}(\Omega),$

what leads us to the inclusion
$\mathfrak{D}^{\alpha}\subset\mathfrak{D}^{\alpha}_{0+}.$

The following theorem [17] establishes the mapping properties of directional
fractional integral operators.

######  Theorem 1.

The following estimates hold

$\|\mathfrak{I}^{\alpha}_{0+}u\|_{L_{p}(\Omega)}\leq
C_{\alpha,\mathrm{d}}\|u\|_{L_{p}(\Omega)},\;\|\mathfrak{I}^{\alpha}_{d-}u\|_{L_{p}(\Omega)}\leq
C_{\alpha,\mathrm{d}}\|u\|_{L_{p}(\Omega)},\;C_{\alpha,\mathrm{d}}=\mathrm{d}^{\alpha}/\Gamma(\alpha+1),\,1\leq
p<\infty.$

The proof of the following so-called representation theorem given in [17]
implements the scheme of the proof corresponding to the one-dimensional case
invented by Rubin B.S. [46], [47]. The author’s own merit is a creation of the
adopted version applicable to the Kipriyanov operator, we represent it in the
expanded form since it may be treated as the intersection of the classical
fractional calculus with the theory invented by Kipriyanov I.A.

######  Theorem 2.

Suppose $f\in L_{p}(\Omega),$ there exists
$\lim\limits_{\varepsilon\rightarrow 0}\psi^{+}_{\varepsilon}f$ or
$\lim\limits_{\varepsilon\rightarrow 0}\psi^{-}_{\varepsilon}f$ with respect
to the norm $L_{p}(\Omega),\,1\leq p<\infty,$ then
$f\in\mathfrak{I}^{\alpha}_{0+}(L_{p})$ or
$f\in\mathfrak{I}^{\alpha}_{d-}(L_{p})$ respectively.

###### Proof.

Let $f\in L_{p}(\Omega)$ and
$\lim\limits_{\stackrel{{\scriptstyle\varepsilon\rightarrow
0}}{{(L_{p})}}}\psi^{+}_{\varepsilon}f=\psi.$ Consider the function

$(\varphi^{+}_{\varepsilon}f)(Q)=\frac{1}{\Gamma(1-\alpha)}\left\\{\frac{f(Q)}{r^{\alpha}}+\alpha(\psi^{+}_{\varepsilon}f)(Q)\right\\}.$

Taking into account (4), we can easily prove that
$\varphi^{+}_{\varepsilon}f\in L_{p}(\Omega).$ Obviously, there exists the
limit $\varphi^{+}_{\varepsilon}f\rightarrow\varphi\in
L_{p}(\Omega),\,\varepsilon\rightarrow 0.$ Taking into account Theorem 1, we
can complete the proof, if we show that

$\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f\stackrel{{\scriptstyle
L_{p}}}{{\longrightarrow}}f,\,\varepsilon\rightarrow 0.$ (10)

In the case $\varepsilon\leq r\leq d,$ we have

$(\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f)(Q)\cdot\frac{\pi
r^{n-1}}{\sin\alpha\pi}=\int\limits_{\varepsilon}^{r}\frac{f(P+y\mathbf{e})y^{n-1-\alpha}}{(r-y)^{1-\alpha}}dy+\alpha\int\limits_{\varepsilon}^{r}(r-y)^{\alpha-1}dy\int\limits_{0}^{y-\varepsilon}\frac{f(P+y\mathbf{e})y^{n-1}-f(T)t^{n-1}}{(y-t)^{\alpha+1}}dt+$
$+\frac{1}{\varepsilon^{\alpha}}\int\limits_{0}^{\varepsilon}f(P+y\mathbf{e})(r-y)^{\alpha-1}y^{n-1}dy=I.$

By direct calculation, we obtain

$I=\frac{1}{\varepsilon^{\alpha}}\int\limits_{0}^{r}f(P+y\mathbf{e})(r-y)^{\alpha-1}y^{n-1}dy-\alpha\int\limits_{\varepsilon}^{r}(r-y)^{\alpha-1}dy\int\limits_{0}^{y-\varepsilon}\frac{f(T)}{(y-t)^{\alpha+1}}t^{n-1}dt.$
(11)

Changing the variable in the second integral, we have

$\alpha\int\limits_{\varepsilon}^{r}(r-y)^{\alpha-1}dy\int\limits_{0}^{y-\varepsilon}\frac{f(T)}{(y-t)^{\alpha+1}}t^{n-1}dt=\alpha\int\limits_{0}^{r-\varepsilon}(r-y-\varepsilon)^{\alpha-1}dy\int\limits_{0}^{y}\frac{f(T)}{(y+\varepsilon-t)^{\alpha+1}}t^{n-1}dt=$
$=\alpha\int\limits_{0}^{r-\varepsilon}f(T)t^{n-1}dt\int\limits_{t}^{r-\varepsilon}\frac{(r-y-\varepsilon)^{\alpha-1}}{(y+\varepsilon-t)^{\alpha+1}}dy=\alpha\int\limits_{0}^{r-\varepsilon}f(T)t^{n-1}dt\int\limits_{t+\varepsilon}^{r}(r-y)^{\alpha-1}(y-t)^{-\alpha-1}dy.$
(12)

Applying formula (13.18) [53, p.184], we get

$\int\limits_{t+\varepsilon}^{r}(r-y)^{\alpha-1}(y-t)^{-\alpha-1}dy=\frac{1}{\alpha\varepsilon^{\alpha}}\cdot\frac{(r-t-\varepsilon)^{\alpha}}{r-t}.$
(13)

Combining relations (11),(12),(13), using the change of the variable
$t=r-\varepsilon\tau,$ we get

$(\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f)(Q)\cdot\frac{\pi
r^{n-1}}{\sin\alpha\pi}=\frac{1}{\varepsilon^{\alpha}}\left\\{\int\limits_{0}^{r}f(P+y\mathbf{e})(r-y)^{\alpha-1}y^{n-1}dy-\int\limits_{0}^{r-\varepsilon}\frac{f(T)(r-t-\varepsilon)^{\alpha}}{r-t}t^{n-1}dt\right\\}=$
$=\frac{1}{\varepsilon^{\alpha}}\int\limits_{0}^{r}\frac{f(T)\left[(r-t)^{\alpha}-(r-t-\varepsilon)_{+}^{\alpha}\right]}{r-t}t^{n-1}dt=\int\limits_{0}^{r/\varepsilon}\frac{\tau^{\alpha}-(\tau-1)_{+}^{\alpha}}{\tau}f(P+[r-\varepsilon\tau]\mathbf{e})(r-\varepsilon\tau)^{n-1}d\tau,$
$\tau_{+}=\left\\{\begin{array}[]{cc}\tau,\;\tau\geq 0;\\\\[7.11317pt]
0,\;\tau<0\,.\end{array}\right.$ (14)

Consider the auxiliary function $\mathcal{K}$ defined in the paper [53, p.105]

$\mathcal{K}(t)=\frac{\sin\alpha\pi}{\pi}\cdot\frac{t_{+}^{\alpha}-(t-1)_{+}^{\alpha}}{t},\;\int\limits_{0}^{\infty}\mathcal{K}(t)dt=1;\;\mathcal{K}(t)>0.$
(15)

Combining (2),(15) and taking into account that $f$ has the zero extension
outside of $\bar{\Omega},$ we obtain

$(\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f)(Q)-f(Q)=\int\limits_{0}^{\infty}\mathcal{K}(t)\left\\{f(P+[r-\varepsilon
t]\mathbf{e})(1-\varepsilon t/r)_{+}^{n-1}-f(P+r\mathbf{e})\right\\}dt.$ (16)

Consider the case $0\leq r<\varepsilon.$ Taking into account (4), we get

$(\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f)(Q)-f(Q)=\frac{\sin\alpha\pi}{\pi\varepsilon^{\alpha}}\int\limits_{0}^{r}\frac{f(T)}{(r-t)^{1-\alpha}}\left(\frac{t}{r}\right)^{n-1}dt-f(Q)=$
$=\frac{\sin\alpha\pi}{\pi\varepsilon^{\alpha}}\int\limits_{0}^{r}\frac{f(P+[r-t]\mathbf{e})}{t^{1-\alpha}}\left(\frac{r-t}{r}\right)^{n-1}dt-f(Q).$
(17)

Consider the domains

$\Omega_{\varepsilon}:=\\{Q\in\Omega,\,d(\mathbf{e})\geq\varepsilon\\},\;\tilde{\Omega}_{\varepsilon}=\Omega\setminus\Omega_{\varepsilon}.$
(18)

In accordance with this definition we can divide the surface $\omega$ into two
parts $\omega_{\varepsilon}$ and $\tilde{\omega}_{\varepsilon},$ where
$\omega_{\varepsilon}$ is the subset of $\omega$ such that
$d(\mathbf{e})\geq\varepsilon$ and $\tilde{\omega}_{\varepsilon}$ is the
subset of $\omega$ such that $d(\mathbf{e})<\varepsilon.$ Using (16),(2), we
get

$\|(\mathfrak{I}^{\alpha}_{0+}\varphi^{+}_{\varepsilon}f)-f\|^{p}_{L_{p}(\Omega)}=\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{\varepsilon}^{d}\left|\int\limits_{0}^{\infty}\mathcal{K}(t)[f(Q-\varepsilon
t\mathbf{e})(1-\varepsilon t/r)_{+}^{n-1}-f(Q)]dt\right|^{p}r^{n-1}dr+$
$+\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{0}^{\varepsilon}\left|\frac{\sin\alpha\pi}{\pi\varepsilon^{\alpha}}\int\limits_{0}^{r}\frac{f(P+[r-t]\mathbf{e})}{t^{1-\alpha}}\left(\frac{r-t}{r}\right)^{n-1}dt-f(Q)\right|^{p}r^{n-1}dr+$
$+\int\limits_{\tilde{\omega}_{\varepsilon}}d\chi\int\limits_{0}^{d}\left|\frac{\sin\alpha\pi}{\pi\varepsilon^{\alpha}}\int\limits_{0}^{r}\frac{f(P+[r-t]\mathbf{e})}{t^{1-\alpha}}\left(\frac{r-t}{r}\right)^{n-1}dt-f(Q)\right|^{p}r^{n-1}dr=I_{1}+I_{2}+I_{3}.$
(19)

Consider $I_{1},$ using the generalized Minkowski inequality, we get

$I^{\frac{1}{p}}_{1}\leq\int\limits_{0}^{\infty}\mathcal{K}(t)\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{\varepsilon}^{d}|f(Q-\varepsilon
t\mathbf{e})(1-\varepsilon
t/r)_{+}^{n-1}-f(Q)|^{p}r^{n-1}dr\right)^{\frac{1}{p}}dt.$

Let us define the function

$h(\varepsilon,t):=\mathcal{K}(t)\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{\varepsilon}^{d}|f(Q-\varepsilon
t\mathbf{e})(1-\varepsilon
t/r)_{+}^{n-1}-f(Q)|^{p}r^{n-1}dr\right)^{\frac{1}{p}}dt.$

It can easily be checked that the following inequalities hold

$|h(\varepsilon,t)|\leq
2\mathcal{K}(t)\|f\|_{L_{p}(\Omega)},\;\forall\varepsilon>0;$ (20)
$|h(\varepsilon,t)|\leq\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{\varepsilon}^{d}\left|(1-\varepsilon
t/r)_{+}^{n-1}[f(Q-\varepsilon
t\mathbf{e})-f(Q)]\right|^{p}r^{n-1}dr\right)^{\frac{1}{p}}dt+$
$+\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{0}^{d}\left|f(Q)[1-(1-\varepsilon
t/r)_{+}^{n-1}]\right|^{p}r^{n-1}dr\right)^{\frac{1}{p}}dt=I_{11}+I_{12}.$
(21)

By virtue of the average continuity property of the functions belonging to
$L_{p}(\Omega),$ we have $\forall t>0:\,I_{11}\rightarrow
0,\;\varepsilon\rightarrow 0.$ Consider $I_{12}$ and let us define the
function

$h_{1}(\varepsilon,t,r):=\left|f(Q)\right|\cdot\left|1-(1-\varepsilon
t/r)_{+}^{n-1}\right|.$

Apparently, the following relations hold almost everywhere in $\Omega$

$\forall
t>0,\,h_{1}(\varepsilon,t,r)\leq|f(Q)|,\;h_{1}(\varepsilon,t,r)\rightarrow
0,\;\varepsilon\rightarrow 0.$

Applying the Lebesgue dominated convergence theorem, we get $I_{12}\rightarrow
0,\;\varepsilon\rightarrow 0.$ It implies that

$\forall t>0,\,\lim\limits_{\varepsilon\rightarrow 0}h(\varepsilon,t)=0.$ (22)

Taking into account (20), (22) and applying the Lebesgue dominated convergence
theorem again, we obtain

$I_{1}\rightarrow 0,\;\;\varepsilon\rightarrow 0.$

Consider $I_{2},$ using the Minkowski inequality, we get

$I^{\frac{1}{p}}_{2}\leq\frac{\sin\alpha\pi}{\pi\varepsilon^{\alpha}}\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{0}^{\varepsilon}\left|\int\limits_{0}^{r}\frac{f(Q-t\mathbf{e})}{t^{1-\alpha}}\left(\frac{r-t}{r}\right)^{n-1}dt\right|^{p}r^{n-1}dr\right)^{\frac{1}{p}}+\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{0}^{\varepsilon}\left|f(Q)\right|^{p}r^{n-1}dr\right)^{\frac{1}{p}}=$
$=I_{21}+I_{22}.$

Applying the generalized Minkowski inequality, we obtain

$I_{21}\frac{\pi}{\sin\alpha\pi}=\varepsilon^{-\alpha}\left(\int\limits_{\omega_{\varepsilon}}d\chi\int\limits_{0}^{\varepsilon}\left|\int\limits_{0}^{r}\frac{f(Q-t\mathbf{e})}{t^{1-\alpha}}\left(\frac{r-t}{r}\right)^{n-1}\\!\\!\\!dt\right|^{p}r^{n-1}\\!dr\right)^{\frac{1}{p}}\leq$
$\leq\varepsilon^{-\alpha}\left\\{\int\limits_{\omega_{\varepsilon}}\\!\\!\left[\int\limits_{0}^{\varepsilon}\\!\\!t^{\alpha-1}\\!\\!\left(\int\limits_{t}^{\varepsilon}\\!\\!|f(Q-t\mathbf{e})|^{p}\\!\left(\frac{r-t}{r}\right)^{\\!\\!\\!(p-1)(n-1)}\\!\\!\\!(r-t)^{n-1}\\!dr\right)^{\frac{1}{p}}\\!\\!dt\right]^{p}\\!\\!d\chi\right\\}^{\frac{1}{p}}\leq$
$\leq\varepsilon^{-\alpha}\left\\{\int\limits_{\omega_{\varepsilon}}\left[\int\limits_{0}^{\varepsilon}t^{\alpha-1}\left(\int\limits_{t}^{\varepsilon}\left|f(P+[r-t]\mathbf{e})\right|^{p}(r-t)^{n-1}dr\right)^{\frac{1}{p}}\\!\\!dt\right]^{p}\\!\\!d\chi\right\\}^{\frac{1}{p}}\leq$
$\leq\varepsilon^{-\alpha}\left\\{\int\limits_{\omega_{\varepsilon}}\left[\int\limits_{0}^{\varepsilon}t^{\alpha-1}\left(\int\limits_{0}^{\varepsilon}|f(P+r\mathbf{e})|^{p}r^{n-1}dr\right)^{\frac{1}{p}}\\!\\!dt\right]^{p}\\!\\!d\chi\right\\}^{\frac{1}{p}}\\!\\!=\alpha^{-1}\|f\|_{L_{p}(\Delta_{\varepsilon})},$

where $\Delta_{\varepsilon}:=\\{Q\in\Omega_{\varepsilon},\,r<\varepsilon\\}.$
Note that ${\rm mess}\,\Delta_{\varepsilon}\rightarrow
0,\,\varepsilon\rightarrow 0,$ therefore $I_{21},I_{22}\rightarrow
0,\,\varepsilon\rightarrow 0.$ It follows that $I_{2}\rightarrow
0,\,\varepsilon\rightarrow 0.$ In the same way, we obtain $I_{3}\rightarrow
0,\,\varepsilon\rightarrow 0.$ Since we proved that
$I_{1},I_{2},I_{3}\rightarrow 0,\,\varepsilon\rightarrow 0,$ then relation
(10) holds. This completes the proof corresponding to the left-side case. The
proof corresponding to the right-side case is absolutely analogous. ∎

The following theorem proved in [17] establishes the strictly accretive
property (see [8]) of the Kipriyanov operator what gives us an opportunity to
establish the numerical range of values of the operator, the latter notion
plays a significant role in the spectral theory. Denote by ${\rm
Lip}\,\lambda,\;0<\lambda\leq 1$ the set of functions satisfying the Holder-
Lipschitz condition

${\rm Lip}\,\lambda:=\left\\{\rho(Q):\;|\rho(Q)-\rho(P)|\leq
Mr^{\lambda},\,P,Q\in\bar{\Omega}\right\\}.$

######  Theorem 3.

Suppose $\rho(Q)$ is a real non-negative function, $\rho\in{\rm
Lip}\,\lambda,\;\lambda>\alpha;$ then the following inequality holds

${\rm Re}(f,\mathfrak{D}^{\alpha}f)_{L_{2}(\Omega,\rho)}\geq
C_{\alpha,\rho}\|f\|^{2}_{L_{2}(\Omega,\rho)},\;f\in H^{1}_{0}(\Omega),$ (23)

where

$C_{\alpha,\rho}=\frac{1}{2\mathrm{d}^{\alpha}}\left\\{\frac{1}{\Gamma(1-\alpha)}+\frac{(n-1)!}{\Gamma(n-\alpha)}-\frac{\alpha
M\mathrm{d}^{\lambda}}{2\Gamma(1-\alpha)(\lambda-\alpha)\inf\rho}\right\\}.$

Moreover, if we have in additional that for every fixed direction $\mathbf{e}$
the function $\rho$ is monotonically non-increasing, then

$C_{\alpha,\rho}=\frac{1}{2\mathrm{d}^{\alpha}}\left\\{\frac{1}{\Gamma(1-\alpha)}+\frac{(n-1)!}{\Gamma(n-\alpha)}\right\\}.$

Consider a linear combination of the uniformly elliptic operator, which is
written in the divergence form, and a composition of the fractional integro-
differential operator, where the fractional differential operator is
understood as the adjoint operator regarding the Kipriyanov operator (see
[12],[13],[19])

$L:=-\mathcal{T}\,+\mathfrak{I}^{\sigma}_{0+}\rho\,\mathfrak{D}^{\alpha}_{d-},\;\sigma\in[0,1),$
$\mathrm{D}(L)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$ (24)

where $\,\mathcal{T}:=D_{j}(a^{ij}D_{i}\cdot),\,i,j=1,2,...,n,$ under the
following assumptions regarding coefficients

$a^{ij}(Q)\in
C^{2}(\bar{\Omega}),\,\mathrm{Re}a^{ij}\xi_{i}\xi_{j}\geq\gamma_{a}|\xi|^{2},\,\gamma_{a}>0,\,\mathrm{Im}a^{ij}=0\;(n\geq
2),\,\rho\in L_{\infty}(\Omega).$ (25)

Note that in the one-dimensional case the operator
$\mathfrak{I}^{\sigma}_{0+}\rho\,\mathfrak{D}^{\alpha}_{d-}$ is reduced to a
weighted fractional integro-differential operator composition, which was
studied properly by many researchers (see introduction, [53, p.175]).

2\. The semi-group model

Bellow, we explore a special operator class for which a number of spectral
theory theorems can be applied. Further we construct an abstract model of a
differential operator in terms of m-accretive operators and call it an
m-accretive operator transform, we find such conditions that being imposed
guaranty that the transform belongs to the class. As an application of the
obtained abstract results we study a differential operator with a fractional
integro-differential operator composition in final terms on a bounded domain
of the $n$ \- dimensional Euclidean space. One of the central points is a
relation connecting fractional powers of m-accretive operators and fractional
derivative in the most general sense. By virtue of such an approach we express
fractional derivatives in terms of infinitesimal generators, in this regard
the Kipriyanov operator is considered.

We represent propositions devoted to properties of accretive operators and
related questions. For the reader convenience, we would like to establish
well-known facts of the operator theory under an appropriate point of view.

######  Lemma 1.

Assume that $A$ is a closed densely defined operator, the following condition
holds

$\|(A+\lambda)^{-1}\|_{\mathrm{R}\rightarrow\mathfrak{H}}\leq\frac{1}{\lambda},\,\lambda>0,$
(26)

where a notation $\mathrm{R}:=\mathrm{R}(A+\lambda)$ is used. Then the
operators $A,A^{\ast}$ are m-accretive.

In accordance with the definition given in [31], we can define a positive and
negative fractional powers of a positive operator $A$ as follows

$A^{\alpha}:=\frac{\sin\alpha\pi}{\pi}\int\limits_{0}^{\infty}\lambda^{\alpha-1}(\lambda+A)^{-1}A\,d\lambda;\,\,A^{-\alpha}:=\frac{\sin\alpha\pi}{\pi}\int\limits_{0}^{\infty}\lambda^{-\alpha}(\lambda+A)^{-1}\,d\lambda,\,\alpha\in(0,1).$
(27)

This definition can be correctly extended on m-accretive operators, the
corresponding reasonings can be found in [8]. Thus, further we define positive
and negative fractional powers of m-accretive operators by formula (27). The
following lemma reflects the property of the fractional powers of m-accretive
operators what gives us the invaluable technique to deal with the
infinitesimal generators.

######  Lemma 2.

Assume that $\alpha\in(0,1),$ the operator $J$ is m-accretive, $J^{-1}$ is
bounded, then

$\|J^{-\alpha}f\|_{\mathfrak{H}}\leq
C_{1-\alpha}\|f\|_{\mathfrak{H}},\,C_{1-\alpha}=2(1-\alpha)^{-1}\|J^{-1}\|+\alpha^{-1},\,f\in\mathfrak{H}.$
(28)

Consider a transform of an m-accretive operator $J$ acting in $\mathfrak{H}$

$Z^{\alpha}_{G,F}(J):=J^{\ast}GJ+FJ^{\alpha},\,\alpha\in[0,1),$ (29)

where symbols $G,F$ denote operators acting in $\mathfrak{H}.$ Further, using
a relation $L=Z^{\alpha}_{G,F}(J)$ we mean that there exists an appropriate
representation for the operator $L.$ The following theorem gives us a tool to
describe spectral properties of transform (29), as it will be shown further it
has an important application in fractional calculus since allows to represent
fractional differential operators as a transform of the infinitesimal
generator of a semigroup.

######  Theorem 4.

Assume that the operator $J$ is m-accretive, $J^{-1}$ is compact, $G$ is
bounded, strictly accretive, with a lower bound
$\gamma_{G}>C_{\alpha}\|J^{-1}\|\cdot\|F\|,\;\mathrm{D}(G)\supset\mathrm{R}(J),$
$F\in\mathcal{B}(\mathfrak{H}),$ where $C_{\alpha}$ is a constant (28). Then
$Z^{\alpha}_{G,F}(J)$ satisfies conditions H1 - H2.

Consider the shift semigroup in a direction acting on $L_{2}(\Omega)$ and
defined as follows $T_{t}f(Q)=f(Q+\mathbf{e}t),$ where
$Q\in\Omega,\,Q=P+\mathbf{e}r.$ Bellow, we represent the complete proof of the
lemma proved in [23] to show the reader some techniques related to the shift
semigroup.

######  Lemma 3.

The semigroup $T_{t}$ is a $C_{0}$ semigroup of contractions.

###### Proof.

By virtue of the continuous in average property, we conclude that $T_{t}$ is a
strongly continuous semigroup. It can be easily established due to the
following reasonings, using the Minkowski inequality, we have

$\left\\{\int\limits_{\Omega}|f(Q+\mathbf{e}t)-f(Q)|^{2}dQ\right\\}^{\frac{1}{2}}\leq\left\\{\int\limits_{\Omega}|f(Q+\mathbf{e}t)-f_{m}(Q+\mathbf{e}t)|^{2}dQ\right\\}^{\frac{1}{2}}+$
$+\left\\{\int\limits_{\Omega}|f(Q)-f_{m}(Q)|^{2}dQ\right\\}^{\frac{1}{2}}+\left\\{\int\limits_{\Omega}|f_{m}(Q)-f_{m}(Q+\mathbf{e}t)|^{2}dQ\right\\}^{\frac{1}{2}}=$
$=I_{1}+I_{2}+I_{3}<\varepsilon,$

where $f\in L_{2}(\Omega),\,\left\\{f_{n}\right\\}_{1}^{\infty}\subset
C_{0}^{\infty}(\Omega);$ $m$ is chosen so that $I_{1},I_{2}<\varepsilon/3$ and
$t$ is chosen so that $I_{3}<\varepsilon/3.$ Thus, there exists such a
positive number $t_{0}$ that

$\|T_{t}f-f\|_{L_{2}}<\varepsilon,\,t<t_{0},$

for arbitrary small $\varepsilon>0.$ Using the assumption that all functions
have the zero extension outside $\bar{\Omega},$ we have $\|T_{t}\|\leq 1.$
Hence we conclude that $T_{t}$ is a $C_{0}$ semigroup of contractions (see
[42]). ∎

The following theorem represented in [17] is formulated in terms of the
infinitesimal generator $-A$ of the semigroup $T_{t}.$ It is a central point
in the application of the spectral theory methods to the abstract integro-
differential constructions.

######  Theorem 5.

We claim that $L=Z^{\alpha}_{G,F}(A).$ Moreover if $\gamma_{a}$ is
sufficiently large in comparison with $\|\rho\|_{L_{\infty}},$ then $L$
satisfies conditions H1-H2, where we put
$\mathfrak{M}:=C_{0}^{\infty}(\Omega),$ if we additionally assume that
$\rho\in\mathrm{Lip}\lambda,\,\lambda>\alpha,$ then $\tilde{\mathcal{H}}=H.$

The meaning of the following lemma is rather significant since it establishes
a very useful property of the infinitesimal generator $-A$ of the semigroup
$T_{t},$ using which we can construct a Hilbert space corresponding to the
operator $A$ let alone the secondary fact establishing the core of the
operator $A.$

######  Lemma 4.

We claim that $A=\tilde{A_{0}},\,\mathrm{N}(A)=0,$ where $A_{0}$ is a
restriction of $A$ on the set $C^{\infty}_{0}(\Omega).$

In the following paragraph, we study generalized constructions originated from
the shift semigroup, they may be also interesting due to the applications
related to the multidimensional case as well as being themselves non-standard
constructions demonstrating one more class for which hypotheses H1,H2 hold.

3\. Further generalizations

Consider a linear space
$\mathbb{L}^{n}_{2}(\Omega):=\left\\{f=(f_{1},f_{2},...,f_{n}),\,f_{i}\in
L_{2}(\Omega)\right\\},$ endowed with the inner product

$(f,g)_{\mathbb{L}^{n}_{2}}=\int\limits_{\Omega}(f,g)_{\mathbb{E}^{n}}dQ,\,f,g\in\mathbb{L}^{n}_{2}(\Omega).$

It is clear that this pair forms a Hilbert space and let us use the same
notation $\mathbb{L}^{n}_{2}(\Omega)$ for it. Consider a sesquilinear form

$t(f,g):=\sum\limits_{i=1}^{n}\int\limits_{\Omega}(f,\mathbf{e_{i}})_{\mathbb{E}^{n}}\overline{(g,\mathbf{e_{i}})}_{\mathbb{E}^{n}}dQ,\,f,g\in\mathbb{L}^{n}_{2}(\Omega),$

where $\mathbf{e_{i}}$ corresponds to $P_{i}\in\partial\Omega,\,\,i=1,2,...,n$
(i.e. $Q=P_{i}+\mathbf{e_{i}}r$). The proofs of the propositions represented
in this paragraph are given in [24].

######  Lemma 5.

The points $P_{i}\in\partial\Omega,\,i=1,2,...,n$ can be chosen so that the
form $t$ generates an inner product.

Consider a pre Hilbert space
$\mathbf{L}^{n}_{2}(\Omega):=\\{f:\,f\in\mathbb{L}^{n}_{2}(\Omega)\\}$ endowed
with the inner product

$(f,g)_{\mathbf{L}^{n}_{2}}:=\sum\limits_{i=1}^{n}\int\limits_{\Omega}(f,\mathbf{e_{i}})_{\mathbb{E}^{n}}\overline{(g,\mathbf{e_{i}})}_{\mathbb{E}^{n}}dQ,\,f,g\in\mathbb{L}^{n}_{2}(\Omega),$

where $\mathbf{e_{i}}$ corresponds to
$P_{i}\in\partial\Omega,\,\,i=1,2,...,n\,,$ the following condition holds

$\Delta=\left|\begin{array}[]{cccc}P_{11}&P_{12}&...&P_{1n}\\\
P_{21}&P_{22}&...&P_{2n}\\\ ...&...&...&...\\\
P_{n1}&P_{n2}&...&P_{nn}\end{array}\right|\neq 0,$

where $P_{i}=(P_{i1},P_{i2},...,P_{in}).$ The following theorem establishes a
norm equivalence.

######  Theorem 6.

The norms $\|\cdot\|_{\mathbb{L}^{n}_{2}}$ and
$\|\cdot\|_{\mathbf{L}^{n}_{2}}$ are equivalent.

Consider a pre Hilbert space

$\mathfrak{\widetilde{H}}^{n}_{A}:=\left\\{f,g\in
C_{0}^{\infty}(\Omega),\,(f,g)_{\mathfrak{\widetilde{H}}^{n}_{A}}=\sum\limits_{i=1}^{n}(A_{i}f,A_{i}g)_{L_{2}}\right\\},$

where $-A_{i}$ is the infinitesimal generator corresponding to the point
$P_{i}.$ Here, we should point out that the form
$(\cdot,\cdot)_{\mathfrak{\widetilde{H}}^{n}_{A}}$ generates an inner product
due to the fact $\mathrm{N}(A_{i})=0,\,i=1,2,...,n$ proved in Lemma 4. Let us
denote a corresponding Hilbert space by $\mathfrak{H}^{n}_{A}.$

######  Corollary 1.

The norms $\|\cdot\|_{\mathfrak{H}^{n}_{A}}$ and $\|\cdot\|_{H_{0}^{1}}$ are
equivalent, we have a bounded compact embedding

$\mathfrak{H}^{n}_{A}\subset\subset L_{2}(\Omega).$

Bellow, we aim to represent an operator in terms of the infinitesimal
generator of the shift semigroup in a direction with the purpose to apply the
results [22], [26], [30] to the established representation. In this way we
come to natural conditions in terms of the infinitesimal generator of the
shift semigroup in a direction what gives us the desired result represented in
[23]. The following theorem allows us to express the construction of the
partial differential operator in terms of the semigroup theory (having chosen
the shift semigroup in the direction) what reveals a mathematical nature of
the operator $-\mathcal{T}.$

######  Theorem 7.

We claim that

$-\mathcal{T}=\frac{1}{n}\sum\limits^{n}_{i=1}A_{i}^{\ast}G_{i}A_{i},$ (30)

the following relations hold

$-\mathrm{Re}(\mathcal{T}f,f)_{L_{2}}\geq
C\|f\|_{\mathfrak{H}^{n}_{A}};\,|(\mathcal{T}f,g)_{L_{2}}|\leq
C\|f\|_{\mathfrak{H}^{n}_{A}}\|g\|_{\mathfrak{H}^{n}_{A}},\;f,g\in
C_{0}^{\infty}(\Omega),$

where $G_{i}$ are some operators corresponding to the operators $A_{i}.$

Thus, by virtue of Corollary 1 and Theorem 7, we are able to claim that
hypotheses H1,H2 [23] hold for the operator $-\mathcal{T}.$ It is rather
reasonable to represent analog of Theorem 5 which reflects connection between
the operator $-\mathcal{T}$ and its perturbation by the Kipriyanov operator.

######  Theorem 8.

We claim that

$L=\frac{1}{n}\sum\limits^{n}_{i=1}A_{i}^{\ast}G_{i}A_{i}+FA_{1}^{\alpha},$
(31)

where $F$ is a bounded operator, $P_{1}:=P,$ and $G_{i}$ are the same as in
Theorem 7. Moreover if $\gamma_{a}$ is sufficiently large in comparison with
$\|\rho\|_{L_{\infty}},$ then the following relations hold

$\mathrm{Re}(Lf,f)_{L_{2}}\geq
C\|f\|_{\mathfrak{H}^{n}_{A}};\,|(Lf,g)_{L_{2}}|\leq
C\|f\|_{\mathfrak{H}^{n}_{A}}\|g\|_{\mathfrak{H}^{n}_{A}},\;f,g\in
C_{0}^{\infty}(\Omega).$

The theorem reveals a remarkable fact the perturbation preserves the property
being in the class satisfying hypotheses H1,H2 what makes the perturbed
operator interesting itself from the theoretical point of view let alone a
prospective applications determined by convenience, from the technical point
of view, in dealing with the invented operator construction in the
multidimensional space.

##  3 Integro-differential constructions

1\. Abel-Lidskii root vectors series expansion

In this section, we represent a theorem valuable from theoretical and applied
points. It is based upon the modification of the Lidskii method, this is why
following the the classical approach we divided it into three statements that
can be claimed separately. The first statement (Theorem 3 [30]) establishes a
character of the series convergence having a principal meaning within the
whole concept. The second statement (Theorem 3 [30]) reflects the name of
convergence - Abel-Lidskii since the latter can be connected with the
definition of the series convergence in the Abel sense, more detailed
information can be found in the monograph by Hardy G.H. [7]. The third
statement (Theorem 4 [30]) is a valuable application of the first one, it is
based upon suitable algebraic reasonings having been noticed by the author and
allowing to involve a fractional derivative in the first term. We should note
that previously, a concept of an operator function represented in the second
term was realized in the paper [27], where a case corresponding to a function
represented by a Laurent series with a polynomial regular part was considered.
Bellow, we consider a comparatively more difficult case obviously related to
the infinite regular part of the Laurent series and therefore requiring a
principally different method of study.

It is a well-known fact that each eigenvalue $\mu_{q},\,q\in\mathbb{N}$ of the
compact operator $B$ generates a set of Jordan chains containing eigenvectors
and root vectors. Denote by $m(q)$ a geometrical multiplicity of the
corresponding eigenvalue and consider a Jordan chain corresponding to an
eigenvector $e_{q_{\xi}},\;\xi=1,2,...,m(q),$ we have

$e_{q_{\xi}},e_{q_{\xi}+1},...,e_{q_{\xi}+k(q_{\xi})},$ (32)

where $k(q_{\xi})$ indicates a number of elements in the Jordan chain, the
symbols except for the first one denote root vectors of the operator $B.$ Note
that combining the Jordan chains corresponding to an eigenvalue, we obtain a
Jordan basis in the invariant subspace generated by the eigenvalue, moreover
we can arrange a so-called system of major vectors $\\{e_{i}\\}_{1}^{\infty}$
(see [33]) of the operator $B$ having combined Jordan chains. It is remarkable
that the eigenvalue $\bar{\mu}_{q}$ of the operator $B^{\ast}$ generates the
Jordan chains of the operator $B^{\ast}$ corresponding to (32). In accordance
with [26], we have

$g_{q_{\xi}+k(q_{\xi})},\;g_{q_{\xi}+k(q_{\xi})-1},...,g_{q_{\xi}},$

where the symbols except for the first one denote root vectors of the operator
$B^{\ast}.$ Combining Jordan chains of the operator $B^{\ast},$ we can
construct a biorthogonal system $\\{g_{n}\\}_{1}^{\infty}$ with respect to the
system of the major vectors of the operator $B.$ This fact is given in detail
in the paper [26]. The following construction plays a significant role in the
theory created in the papers [26],[27],[28] and therefore deserves to be
considered separately, denote

$\mathcal{A}_{\nu}(\varphi,t)f:=\sum\limits_{q=N_{\nu}+1}^{N_{\nu+1}}\sum\limits_{\xi=1}^{m(q)}\sum\limits_{i=0}^{k(q_{\xi})}e_{q_{\xi}+i}c_{q_{\xi}+i}(t),$
(33)

where $\\{N_{\nu}\\}_{1}^{\infty}$ is a sequence of natural numbers,

$c_{q_{\xi}+i}(t)=e^{-\varphi(\lambda_{q})t}\sum\limits_{j=0}^{k(q_{\xi})-i}H_{j}(\varphi,\lambda_{q},t)c_{q_{\xi}+i+j},\,i=0,1,2,...,k(q_{\xi}),$
(34)

$c_{q_{\xi}+i}=(f,g_{q_{\xi}+k-i})/(e_{q_{\xi}+i},g_{q_{\xi}+k-i}),$
$\lambda_{q}=1/\mu_{q}$ is a characteristic number corresponding to
$e_{q_{\xi}},$

$H_{j}(\varphi,z,t):=\frac{e^{\varphi(z)t}}{j!}\cdot\lim\limits_{\zeta\rightarrow
1/z}\frac{d^{j}}{d\zeta^{\,j}}\left\\{e^{-\varphi(\zeta^{-1})t}\right\\},\;j=0,1,2,...\,,\,.$

More detailed information on the considered above Jordan chains can be found
in [26].

2\. Decomposition theorem

Denote by $\mathfrak{H}$ the abstract separable Hilbert space and assume that
the hypotheses H1,H2 hold for the operator $W$ acting in $\mathfrak{H}.$ We
should point out that such chose of the operator class justified by both
abstract theoretical relevance related to the spectral properties of non-
selfadjoint operators and the concrete applications including ones involving
the Kipriyanov operator. Denote by

$\varphi(W):=\sum\limits_{n=l}^{k}c_{n}W^{n},\;\;-\infty\leq l,k\leq\infty$
(35)

a formal construction called by an operator function, where $c_{n}$ are the
coefficients corresponding to the function of the complex variable $\varphi.$
Here, we ought to make a bibliographic digression and remind that the case
$l=-\infty,\,k<\infty$ was considered in [28]. In this case, the complex
function $\varphi$ was supposed to have a decomposition into the Laurent
series about the point zero with the coefficients $c_{n}$ satisfying the
additional assumption

$\max\limits_{n=0,1,...,k}(|\mathrm{arg}c_{n}|+n\theta)<\pi/2,$ (36)

where $\theta$ is the semi-angle of the sector containing the numerical range
of values of the operator $W.$ We should note that the problem connected with
the representation (35) can be divided on two parts $l\geq-\infty,\,k=0$ and
$l=0,\,k\leq\infty,$ thus the first one was properly studied in [28], the
second one was studied in [30], with the following assumptions (we represent a
technical variant, the expended variant can be found in [30]): the complex
function $\varphi$ of the order less than a half maps the ray
$\mathrm{arg}\,z=\theta_{0}$ within a sector
$\mathfrak{L}_{0}(\zeta),\,0<\zeta<\pi/2,$ the condition holds

$\mathrm{Re}\varphi(z)>Ce^{H(\theta_{0})r^{\varrho}},\mathrm{arg}\,z=\theta_{0},$
(37)

where $H(\theta_{0})$ is a positive number in accordance with the Lemma 1
[30]. Taking into account the above, we can consider a function represented by
a Laurent series with the arbitrary principal part and the regular part
satisfying (36),(37) respectively to the finite, infinite cases. This
statement can be proved by repetition of the reasonings represented in Lemma 5
[28], thus we leave the proof to the reader.

Bellow, we consider a Hilbert space consists of element-functions
$u:\mathbb{R}_{+}\rightarrow\mathfrak{H},\,u:=u(t),\,t\geq 0,$ we understand
the differentiation and integration operations in the generalized sense, i.e.
the derivative is defined as a limit in the sense of the norm e.t.c. (see
[26], [31]). Combining the operations, we can define a generalized fractional
derivative in the Riemann-Liouville sense (see [27],[53]), in the formal form,
we have

$\mathfrak{D}^{1/\alpha}_{-}f(t):=-\frac{1}{\Gamma(1-1/\alpha)}\frac{d}{dt}\int\limits_{0}^{\infty}f(t+x)x^{-1/\alpha}dx,\;\alpha\geq
1,$

here we should note that facts
$\mathfrak{D}^{1}_{-}f(t)=-du/dt,\,\mathfrak{D}^{0}_{-}f(t)=f(t)$ can be
obtained due to the definition of the operator (see [53]). In terms of the
expression (35), consider a Cauchy problem

$\mathfrak{D}^{1/\alpha}_{-}u=\varphi(W)u,\;u(0)=f\in\mathrm{D}(W^{n}),\,n=1,2,...k.$
(38)

Taking into account the above, combining results [28], [30], we can formulate
the following theorem.

######  Theorem 9.

Assume that conditions (36),(37) hold respectively to the cases corresponding
to the finite, infinite regular part of the series (35), then there exists a
solution of the Cauchy problem (38) in the form

$u(t)=\sum\limits_{\nu=0}^{\infty}\mathcal{A}_{\nu}(\varphi^{\alpha},t)f,\;\sum\limits_{\nu=0}^{\infty}\|\mathcal{A}_{\nu}(\varphi^{\alpha},t)f\|<\infty.$
(39)

Moreover, the existing solution is unique if the operator
$\mathfrak{D}^{1-1/\alpha}_{-}\\!\varphi(W)$ is accretive.

###### Proof.

To avoid any kind of repetition, we does not represent the complete proof
having restricted reasonings by the scheme appealing to Lemma 5, Theorem 1
[28], Lemma 3, Theorem 4 [30]. Thus, the detailed calculation is left to the
reader. ∎

Further, considering an operator function, we will assume that conditions
(36),(37) hold respectively to the case. Now consider transform (29)

$Z^{\alpha}_{G,F}(J)=J^{\ast}GJ+FJ^{\alpha},\,\alpha\in[0,1),$

assuming that the conditions of Theorem 6 hold, we can consider a Cauchy
problem involving the transform which represents an integro-differential
construction in the generalized sense. The latter problem appeals to a plenty
of concrete evolution equations which form a base for the modern engineering
sciences. Let us contemplate then a magnificent representative of the
generators creating the transform - the directional derivative which
fractional power is the Kipriyanov operator. Consider a Cauchy problem

$\mathfrak{D}^{1/\alpha}_{-}u=\sum\limits_{n=l}^{k}c_{n}L^{n}u,\;u(0)=f\in
C_{0}^{\infty}(\Omega),\,n=1,2,...k,$ (40)

where we are dealing with the following integro-differential construction with
the sufficiently smooth coefficients for which the conditions (25) hold

$L:=-\mathcal{T}\,+\mathfrak{I}^{\sigma}_{0+}\rho\,\mathfrak{D}^{\gamma}_{d-},\;\sigma,\gamma\in[0,1).$

Thus, we can easily see that the case
$\sigma=0,\,\gamma=0,\,-\infty<l,k<\infty$ leads to the class of integro-
differential equations of the integer order and the obtained results give us a
method to solve the corresponding Cauchy problems (40). Certainly, we can
consider a closure of the defined operator function on the set
$C_{0}^{\infty}(\Omega),$ the fact that it admits closure is proved in [29].
The case corresponding to arbitrary values of $\sigma,\gamma$ within the
range, requires more peculiar technique that was considered in paragraph 3.1
[27]. However, Theorem 9 becomes relevant in solving evolution equations with
an integro-differential operator in the second term to say nothing on the far
reaching generalizations corresponding to an operator function with the
infinite principal or regular part of its Laurent series.

##  4 Conclusions

In the paper, there was represented a historical survey devoted to the
achievements of Kipriyanov I.A. where the exclusively constructed fractional
calculus theory was discussed convexly. We produced a comparison analysis in
the framework of ways and means related to further prospective generalizations
of fractional derivative as a notion. The qualitative properties of the
Kipriyanov fractional differential operator were studied by the methods of the
classical fractional calculus theory in contrast to the exclusive approach
invented by Kipriyanov I.A. Having taken as a basis the concept of
multidimensional generalization of the fractional differential operator in the
sense of Marchaud, we adapted the previously known technique of the proofs
related to the theory of fractional calculus of one variable. Along with the
previously known definition of a fractional derivative introduced by
Kipriyanov I.A. we used a new definition of a multidimensional fractional
integral in the direction what allows to describe the range of the Kipriyanov
adjoint operator. A number of statements having analogues in the theory of
fractional calculus of one variable and previously proved by the author were
discussed. In particular the classical result - a sufficient condition for
representability by a fractional directional integral in the direction was
observed. The strictly accretive property of the Kipriyanov operator, being an
outstanding author’s result, was observed. On the base of the given technique,
there were developed methods of the semigroup theory and the spectral theory
of non-selfadjoint operators what leads us to significant applications to the
abstract evolution equations in the Hilbert space. The latter gives us an
opportunity to solve a whole class of problems related to integro-differential
equations of the real order wherein the one related to the semigroup connected
with the Kipriyanov operator was studied properly.

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is drawn from the posterior law given $({\bf A},{\bf B})$. Note that in the
last expression the ground-truth expression does not appear explicitly.

We turn now to the estimator
$\boldsymbol{\widehat{\pi}}(i)=\arg\max_{i^{\prime}}\widehat{P}_{i,i^{\prime}}$
based on the approximation of the posterior probabilities through the message-
passing algorithm, which is the only one we can use in practice, and define
what we shall call the true average overlap,

${\rm ov}=\mathbb{E}[{\rm
ov}(\boldsymbol{\widehat{\pi}},\boldsymbol{\pi_{\star}})]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[\mathbb{I}(\boldsymbol{\pi_{\star}}(i)=\arg\max_{i^{\prime}}\widehat{P}_{i,i^{\prime}})]\
,$ (47)

and the estimated average overlap,

$\widehat{{\rm
ov}}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[\max_{i^{\prime}}\widehat{P}_{i,i^{\prime}}]\
,$ (48)

where in the last equation we do not use explicitly the knowledge of the
ground-truth permutation, $\max_{i^{\prime}}\widehat{P}_{i,i^{\prime}}$ being
the estimation by the algorithm itself of the probability that its prediction
$\boldsymbol{\widehat{\pi}}(i)$ is correct. According to the discussion above
we would have ${\rm ov}=\widehat{{\rm ov}}$ if the approximated probabilities
$\widehat{P}_{i,i^{\prime}}$ coincided with the exact ones $P_{i,i^{\prime}}$.

We present in Fig. 14 the numerical results of the comparison of the two
quantities ${\rm ov}$ and $\widehat{{\rm ov}}$; to account for the dependency
of $\widehat{P}$ on the depth parameter $d$ of the message-passing algorithm
we plot both quantities as a function of $d$ for fixed values of
$n,\lambda,s$. One can see on the figure that they coincide for small $d$ but
differ from each other for larger values of $d$; as a matter of fact when $d$
increases the estimated probability laws $\widehat{P}_{i,\cdot}$ become more
and more concentrated on their mode $\boldsymbol{\widehat{\pi}}(i)$, i.e.,
close to
$\widehat{P}_{i,i^{\prime}}=\delta_{i^{\prime},\boldsymbol{\widehat{\pi}}(i)}$.
This means that the algorithm becomes overconfident about its predictions, its
estimation of the probability that the predicted match is correct is larger
than the true one. The plots on the bottom of Fig. 14 study the dependency of
the separation point between the two curves on $\lambda$ and $n$. Larger $n$
leads to agreement up to higher values of $d$, while higher $\lambda$ makes
the two curves depart at smaller $d$. Let us finally underline that if
$P=\hat{P}$ implies that ${\rm ov}=\widehat{{\rm ov}}$, the converse
implication is far from being true: suppose for instance that
$\widehat{P}_{i,i^{\prime}}=1/n$ for all $i,i^{\prime}$, i.e., that the
estimation does not extract any information from the observations and that
$\boldsymbol{\widehat{\pi}}(i)$ is a uniformly random vertex (the ties in the
$\arg\max$ being broken at random). Then from (47) and (48) it follows that
${\rm ov}=\widehat{{\rm ov}}=1/n$. However in a hard phase where partial
recovery is information theoretically possible this situation would occur with
a non-trivial $P\neq\widehat{P}$.

Figure 14: A comparison of the true average overlap (dashed line) of Eq. (47)
and of the estimated overlap (solid line) of Eq. (48). (Top) $n=2048$,
$\lambda=2.4$: for small $d$ the estimated and true overlap agree, however
increasing $d$ the estimated overlap grows towards $1$ showing that the
algorithm is overconfident of its predictions. (Bottom left) when $n$ is
increased the value of $d$ at which the true and estimated overlap ceases to
agree also increases. (Bottom right) when $\lambda$ is increased the value of
$d$ at which the two curves separates decreases, possibly because of stronger
finite size effects for larger $\lambda$.

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