Riesz Potential With Logarithmic Kernel in Generalized Hölder Spaces

2021 ◽  
pp. 275-296
Author(s):  
Boris Grigorievich Vakulov ◽  
Yuri Evgenievich Drobotov
Keyword(s):  
Financial Analysis ◽  
Riesz Potential ◽  
Inverse Operator ◽  
Mathematical Physics ◽  
Hölder Spaces ◽  
Fractal Sets ◽  
Logarithmic Kernel ◽  
Potential Type ◽  
Holder Spaces ◽  
Potential Type Operators

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.

Riesz Potential in Generalized Hölder Spaces

2018 ◽  
pp. 249-273 ◽  
Author(s):  
Boris Grigorievich Vakulov ◽  
Galina Sergeevna Kostetskaya ◽  
Yuri Evgenievich Drobotov
Keyword(s):  
Riesz Potential ◽  
Theoretical Physics ◽  
Potential Operator ◽  
Variable Order ◽  
Hölder Spaces ◽  
Continuity Modulus ◽  
Potential Type ◽  
Riesz Potential Operator ◽  
Point To Point ◽  
Holder Spaces

The chapter provides an overview of the advanced researches on the multidimensional Riesz potential operator in the generalized Hölder spaces. While being of interest within mathematical modeling in economics, theoretical physics, and other areas of knowledge, the Riesz potential plays a significant role for analysis on fractal sets, and this aspect is briefly outlined. The generalized Hölder spaces provide convenient terminology for formalizing the smoothness concept, which is described here. There are constant and variable order potential type operators considered, including a two-pole spherical one. As a sphere is, in some sense, a convenient set for analysis, there are two results, proved in detail: the conditions for the spherical fractional integral of variable order to be bounded in the generalized Hölder spaces, whose local continuity modulus has a dominant, which may vary from point to point, and the ones for the constant-order two-pole spherical potential type operator.

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

Solvability in Hölder spaces of an integral equation which models dynamics of the capillary rise

2020 ◽  
Vol 490 (1) ◽  
pp. 124237
Author(s):  
Hanna Okrasińska-Płociniczak ◽  
Łukasz Płociniczak ◽  
Juan Rocha ◽  
Kishin Sadarangani
Keyword(s):  
Integral Equation ◽  
Capillary Rise ◽  
Hölder Spaces ◽  
Holder Spaces
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