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Open AccessArticle

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

by Asra Hadadfard, Mohammad Bagher Ghaemi and António M. Lopes

Axioms 2024, 13(10), 688; https://doi.org/10.3390/axioms13100688 - 3 Oct 2024

Viewed by 1105

Abstract

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. [...] Read more.

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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14 pages, 309 KB

Open AccessArticle

On the Commutators of Marcinkiewicz Integral with a Function in Generalized Campanato Spaces on Generalized Morrey Spaces

by Fuli Ku and Huoxiong Wu

Mathematics 2022, 10(11), 1817; https://doi.org/10.3390/math10111817 - 25 May 2022

Cited by 2 | Viewed by 2393

Abstract

This paper is devoted to exploring the mapping properties for the commutator

μ_{Ω, b}

generated by Marcinkiewicz integral

μ_{Ω}

with a locally integrable function b in the generalized Campanato spaces on the generalized Morrey spaces. Under the assumption that the [...] Read more.

This paper is devoted to exploring the mapping properties for the commutator

μ_{Ω, b}

generated by Marcinkiewicz integral

μ_{Ω}

with a locally integrable function b in the generalized Campanato spaces on the generalized Morrey spaces. Under the assumption that the integral kernel

Ω

satisfies certain log-type regularity, it is shown that

μ_{Ω, b}

is bounded on the generalized Morrey spaces with variable growth condition, provided that b is a function in generalized Campanato spaces, which contain the

B M O (R^{n})

and the Lipschitz spaces

{Lip}_{α} (R^{n})

(

0 < α \leq 1

) as special examples. Some previous results are essentially improved and generalized. Full article

(This article belongs to the Special Issue Recent Advances in Harmonic Analysis and Applications)

12 pages, 288 KB

Open AccessArticle

Calderón Operator on Local Morrey Spaces with Variable Exponents

by Kwok-Pun Ho

Mathematics 2021, 9(22), 2977; https://doi.org/10.3390/math9222977 - 22 Nov 2021

Cited by 6 | Viewed by 2177

Abstract

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

11 pages, 275 KB

Open AccessArticle

Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

by Yuexia Hou

Symmetry 2021, 13(11), 2061; https://doi.org/10.3390/sym13112061 - 1 Nov 2021

Viewed by 1436

Abstract

Let

X_{0}, X_{1}, \dots, X_{q} (q < N)

be real vector fields, which are left invariant on homogeneous group G, provided that

X_{0}

is homogeneous of degree two and [...] Read more.

Let

X_{0}, X_{1}, \dots, X_{q} (q < N)

be real vector fields, which are left invariant on homogeneous group G, provided that

X_{0}

is homogeneous of degree two and

X_{1}, \dots, X_{q}

are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift

L = \sum_{i, j = 1}^{q} a_{i j} (x) X_{i} X_{j} + a_{0} (x) X_{0}

, where the coefficients

a_{i j} (x)

a_{0} (x)

belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore,

a_{i j} (x)

satisfies the uniform ellipticity condition on

R^{q}

and

a_{0} (x) \neq 0

. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces. Full article

(This article belongs to the Section Mathematics)

57 pages, 641 KB

Open AccessSystematic Review

A Survey on Function Spaces of John–Nirenberg Type

by Jin Tao, Dachun Yang and Wen Yuan

Mathematics 2021, 9(18), 2264; https://doi.org/10.3390/math9182264 - 15 Sep 2021

Cited by 16 | Viewed by 2603

Abstract

In this systematic review, the authors give a survey on the recent developments of both the John–Nirenberg space

J N_{p}

and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO,

V J N_{p}

, and [...] Read more.

In this systematic review, the authors give a survey on the recent developments of both the John–Nirenberg space

J N_{p}

and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO,

V J N_{p}

, and

C J N_{p}

R^{n}

or a given cube

Q_{0} \subset R^{n}

with finite side length. In addition, some related open questions are also presented. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

13 pages, 801 KB

Open AccessArticle

Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

by Muhammad Zainul Abidin and Jiecheng Chen

Mathematics 2021, 9(5), 498; https://doi.org/10.3390/math9050498 - 28 Feb 2021

Cited by 11 | Viewed by 2448

Abstract

In this paper, we consider the generalized porous medium equation. For small initial data

u_{0}

In this paper, we consider the generalized porous medium equation. For small initial data

u_{0}

belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis, Semigroup Theory and Difference-Differential Equations)

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