Variable order fractional integrals in variable generalized Hölder spaces of holomorphic functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Alexey Karapetyants ◽  
Stefan Samko
Keyword(s):  
Holomorphic Functions ◽  
Fractional Integrals ◽  
Variable Order ◽  
Hölder Spaces ◽  
Spaces Of Holomorphic Functions ◽  
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).

scholarly journals Hardy-Littlewood-Type Theorem for Mixed Fractional Integrals in Hölder Spaces

2021 ◽  
pp. 15-19
Author(s):  
TulkinMamatov ◽  
◽  
Nemat Mustafoev ◽  
Dilshod Barakaev ◽  
Rano Sabirova ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Fractional Integration ◽  
Type Theorem ◽  
Fractional Integrals ◽  
Power Weight ◽  
Hölder Spaces ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight.

scholarly journals Hardy-Littlewood-Type Theorem for Mixed Fractional Integrals in Hölder Spaces

2021 ◽  
Vol 1 (2) ◽  
pp. 15-19
Author(s):  
Tulkin Mamatov ◽  
Nemat Mustafoev ◽  
Dilshod Barakaev ◽  
Rano Sabirova
Keyword(s):  
Fractional Derivative ◽  
Fractional Integration ◽  
Type Theorem ◽  
Fractional Integrals ◽  
Power Weight ◽  
Hölder Spaces ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight.

scholarly journals Fractional Type Operators in Weighted Generalized Hölder Spaces

1994 ◽  
Vol 1 (5) ◽  
pp. 537-559
Author(s):  
S. G. Samko ◽  
Z. U. Mussalaeva
Keyword(s):  
Convolution Type ◽  
Fractional Integrals ◽  
Hölder Spaces ◽  
Continuity Modulus ◽  
Type Spaces ◽  
Holder Spaces ◽  
Fractional Type

Abstract Weighted Zygmund type estimates are obtained for the continuity modulus of some convolution type integrals. In the case of fractional integrals this is strengthened to a result on isomorphism between certain weighted generalized Hölder type spaces.

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.

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