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11 pages, 312 KiB

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 1016

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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16 pages, 317 KiB

Open AccessArticle

Compactness of Commutators for Riesz Potential on Generalized Morrey Spaces

by Nurzhan Bokayev, Dauren Matin, Talgat Akhazhanov and Aidos Adilkhanov

Mathematics 2024, 12(2), 304; https://doi.org/10.3390/math12020304 - 17 Jan 2024

Cited by 3 | Viewed by 1262

Abstract

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces

M_{p}^{w (\cdot)}

. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces [...] Read more.

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces

M_{p}^{w (\cdot)}

. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces

L_{p}, p > 0 .

As an application, we prove the compactness of the commutator of the Riesz potential

[b, I_{α}]

in generalized Morrey spaces, where

b \in V M O

(

V M O (R^{n})

denote the

B M O

-closure of

C_{0}^{\infty} (R^{n})

). We prove auxiliary statements regarding the connection between the norm of average functions and the norm of the difference of functions in the generalized Morrey spaces. Such results are also of independent interest. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

14 pages, 309 KiB

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 2314

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)

22 pages, 377 KiB

Open AccessArticle

Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

by Tianjun Shen and Bo Li

Mathematics 2022, 10(7), 1112; https://doi.org/10.3390/math10071112 - 30 Mar 2022

Cited by 1 | Viewed by 1852

Abstract

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

[...] Read more.

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

and V be a Muckenhoupt weight belonging to a reverse Hölder class

R H_{q} (X)

for some

q \geq (Q + 1) / 2

. In this paper, we consider the Dirichlet problem for the Schrödinger equation

- \partial_{t}^{2} u + 𝓛 u + V u = 0

on the upper half-space

X \times R_{+}

, which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space

R^{Q}

to the metric measure space X and improves the reverse Hölder index from

q \geq Q

q \geq (Q + 1) / 2

. Full article

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

13 pages, 801 KiB

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 2354

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)

16 pages, 294 KiB

Open AccessArticle

Boundedness of a Class of Oscillatory Singular Integral Operators and Their Commutators with Rough Kernel on Weighted Central Morrey Spaces

by Yongliang Zhou, Dunyan Yan and Mingquan Wei

Mathematics 2020, 8(9), 1455; https://doi.org/10.3390/math8091455 - 30 Aug 2020

Cited by 1 | Viewed by 2230

Abstract

In this paper, we establish the boundedness of a class of oscillatory singular integral operators with rough kernel on central Morrey spaces. Moreover, the boundedness for each of their commutators on weighted central Morrey spaces was also obtained. We generalized some existing results. [...] Read more.

(This article belongs to the Special Issue New Trends in Function Spaces and Partial Differential Equations: Theory and Applications to Physical Processes)

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