Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic

2011 ◽  
Vol 284 (2-3) ◽  
pp. 355-369 ◽  
Author(s):  
Natasha Samko ◽  
Boris Vakulov
Keyword(s):  
Variable Order ◽  
Hölder Spaces ◽  
Hypersingular Integrals ◽  
Variable Characteristic ◽  
Holder Spaces

scholarly journals Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces

2010 ◽  
Vol 8 (3) ◽  
pp. 215-244 ◽  
Author(s):  
Natasha Samko ◽  
Stefan Samko ◽  
Boris Vakulov
Keyword(s):  
Homogeneous Spaces ◽  
Measure Zero ◽  
Fractional Integrals ◽  
Variable Order ◽  
Hölder Spaces ◽  
Continuity Modulus ◽  
Hypersingular Integrals ◽  
Potential Operators ◽  
Complex Valued ◽  
Holder Spaces

We consider non-standard Hölder spacesHλ(⋅)(X)of functionsfon a metric measure space (X, d, μ), whose Hölder exponentλ(x) is variable, depending onx∈X. We establish theorems on mapping properties of potential operators of variable orderα(x), from such a variable exponent Hölder space with the exponentλ(x) to another one with a “better” exponentλ(x) +α(x), and similar mapping properties of hypersingular integrals of variable orderα(x) from such a space into the space with the “worse” exponentλ(x) −α(x) in the caseα(x) <λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spacesHλ(⋅)(X), but also the generalized Hölder spacesHw(⋅,⋅)(X)of functions whose continuity modulus is dominated by a given functionw(x, h),x∈X, h> 0. We admit variable complex valued ordersα(x), whereℜα(x)may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Hölder spaces with the weightα(x).

More on Hypersingular Integrals and Embeddings into Hölder Spaces

2016 ◽  
pp. 439-454
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Humberto Rafeiro ◽  
Stefan Samko
Keyword(s):  
Hölder Spaces ◽  
Hypersingular Integrals ◽  
Holder Spaces

scholarly journals Kolmogorov type inequalities for hypersingular integrals with sign-alternating characteristic

2021 ◽  
Vol 18 ◽  
pp. 18
Author(s):  
V.F. Babenko ◽  
D.A. Levchenko
Keyword(s):  
Multivariate Functions ◽  
Hölder Spaces ◽  
Kolmogorov Type ◽  
Hypersingular Integrals ◽  
Holder Spaces

New sharp Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic of the form $\Omega(t) = \mathrm{sgn} \prod\limits_{k=1}^m t_k$ for multivariate functions from Hölder spaces are obtained.

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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