Abstract
The multidimensional Riesz potential type operators are of interest within mathematical modelling in economics, mathematical physics, and other, both theoretical and applied, disciplines as they play a significant role for analysis on fractal sets. Approaches of operator theory are relevant to researching various equations, which are widespread in financial analysis. In this chapter, integral equations with potential type operators are considered for functions from generalized Hölder spaces, which provide content terminology for formalizing the concept of smoothness, briefly described in the presented chapter. Results on potentials defined on the unit sphere are described for convenience of the analysis. An inverse operator for the Riesz potential with a logarithmic kernel is carried out, and the isomorphisms between generalized Hölder spaces are proven.
TopIntroduction
Let a natural n≥2 stand for the dimension of Rn – the set of all vectors 
with real coordinates, thus xi∈R, 
. Being supplied with the metric
,
Rn is a Euclidean space, and |x|=|x–0|, while an inner product in Rn could be defined as
,or, due to the parallelogram identity,
. This allows to express the metric through the inner product:
A unit sphere 
in Rn is considered as a set of points, equidistant from the origin by a distance of one, i.e.
.
It is clear, that, for an arbitrary x∈Rn, 
. For 
, the relation (1) becomes
The purpose of the paper is to investigate solutions of an integral equation with a spherical convolution operator
,which is to be achieved through applying the Fourier—Laplace multiplier theory. In a nutshell, the latter assume, that if the preimage
f can be decomposed into a Fourier series, then the coefficients of such a decomposition for the image
Kf are defined in the specific way, briefly described below.
Eventually, the approach allows to achieve theorems on reflections between various function spaces and outline the properties of integral equations. In the present paper, isomorphisms between generalized Hölder spaces are proved for a kind of Kf – the Riesz potential type operator with a logarithmic kernel
,where
r∈R is a fixed constant. Due to the relation (2),
Kα is indeed a particular instance of
K with
.
Let the common definition of a potential operator be reminded according to a monograph by N.S. Landkof (1973). Suppose G is a region in Rn, which is to be considered further coincident with Rn, K(x,σ) is a real function, defined on G×G, and μ is a signed measure with support in G. Then, for any μ-measurable domain Ω(μ), the formal expression
is called “the
K-potential of the signed measure μ” (p. 58). In this paper, the measure μ is fixed implicitly, and
dσ=
dμ(σ) for short. In fact,
dσ is the area element of the sphere.
An M. Riesz kernel is defined as a function of the point x∈Rn, n≥2,
,where the positive constant
A(
n,α) is a normalization factor (Landkof, 1973, p. 43). Assuming
and, which follows from applying the Fourier—Laplace multiplier theory,
,provides the definition of the Riesz potential of order α:
Key Terms in this Chapter
Jacobi Polynomials: The Jacobi polynomials are defined as , Re ?, Re s> -1, where , k =1,2,3,….
Euler Beta-Function: The Euler integral of the first kind , Re z >0, Re w >0, is known as the beta function. It relates to the gamma-function as .
Pochhammer Symbol: The Pochhammer symbol is ( z ) n = z ( z +1) … ( z + n –1), n - 1,2,3,… .
Bounded Operator: Let U and V be two vector spaces over the same ordered field, and they are equipped with norms. A linear operator K from U to V is called bounded, it there exists a constant C >0 such that , x ? U .
Isomorphism: An isomorphism is a homomorphism that is a bijection, i.e. it is a mapping, that preserves sets and relations among elements and can be reversed by an inverse mapping.
Psi-Function: The psi-function is defined as . It is an analytical in the whole z plane function, except at the points z =0, -1, -2,…, at which it has simple poles.
Euler Gamma-Function: The Euler integral of the second kind , Re z >0, with is known as the gamma-function. It is well defined for all complex z with Re z >0 and can be extended to the half-plane Re z =0, z ?0, -1, -2,…, by analytic continuation via the reduction formula G( z +1)= z G( x ), Re z >0.