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13 pages, 384 KiB

Open AccessArticle

Dynamical Analysis of Fractional Integro-Differential Equations

by Taher S. Hassan, Ismoil Odinaev, Rasool Shah and Wajaree Weera

Mathematics 2022, 10(12), 2071; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122071 - 15 Jun 2022

Cited by 2 | Viewed by 1282

In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo-operational matrix method confirm that Chebyshev Pseudospectral method is more accurate and straightforward as compared to other methods. Full article

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

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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://0-doi-org.brum.beds.ac.uk/10.3390/math10071112 - 30 Mar 2022

Viewed by 1283

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)

25 pages, 388 KiB

Open AccessArticle

Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain

by Jun Cao, Yongyang Jin, Yuanyuan Li and Qishun Zhang

Mathematics 2022, 10(4), 637; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040637 - 18 Feb 2022

Cited by 1 | Viewed by 861

Abstract

The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space

R^{n}

are well-established, whereas only partial results are known for the non-smooth domains. In this paper,

Ω

is a non-smooth domain of

R^{n}

that is bounded and uniform. Suppose p [...] Read more.

The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space

R^{n}

are well-established, whereas only partial results are known for the non-smooth domains. In this paper,

Ω

is a non-smooth domain of

R^{n}

that is bounded and uniform. Suppose p,

q \in [1, \infty)

and

s \in (n {(\frac{1}{p} - \frac{1}{q})}_{+}, 1)

with

n {(\frac{1}{p} - \frac{1}{q})}_{+} : = max {n (\frac{1}{p} - \frac{1}{q}), 0}

. The authors show that three typical types of fractional Triebel–Lizorkin spaces, on

Ω

F_{p, q}^{s} (Ω)

{\overset{˚}{F}}_{p, q}^{s} (Ω)

and

{\tilde{F}}_{p, q}^{s} (Ω)

, defined via the restriction, completion and supporting conditions, respectively, are identical if

Ω

is E-thick and supports some Hardy inequalities. Moreover, the authors show the condition that

Ω

is E-thick can be removed when considering only the density property

F_{p, q}^{s} (Ω) = {\overset{˚}{F}}_{p, q}^{s} (Ω)

, and the condition that

Ω

supports Hardy inequalities can be characterized by some Triebel–Lizorkin capacities in the special case of

1 \leq p \leq q < \infty

. Full article

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

103 pages, 868 KiB

Open AccessArticle

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

by Ali Behzadan and Michael Holst

Mathematics 2022, 10(3), 522; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030522 - 07 Feb 2022

Cited by 3 | Viewed by 1504

Abstract

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain

Ω

R^{n}

0 < t < 1

, and

1 < p < \infty

, it is not necessarily true that

W^{1, p} (Ω) ↪ W^{t, p} (Ω)

. This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here. Full article

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

13 pages, 269 KiB

Open AccessArticle

An Optimal Estimate for the Anisotropic Logarithmic Potential

by Shaoxiong Hou

Mathematics 2022, 10(2), 261; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020261 - 15 Jan 2022

Viewed by 929

Abstract

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate. Full article

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

19 pages, 353 KiB

Open AccessArticle

Maximal Function Characterizations of Hardy Spaces on

R^{n}

with Pointwise Variable Anisotropy

by Aiting Wang, Wenhua Wang and Baode Li

Mathematics 2021, 9(24), 3246; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243246 - 15 Dec 2021

Cited by 2 | Viewed by 1396

Abstract

In 2011, Dekel et al. developed highly geometric Hardy spaces

H^{p} (Θ)

, for the full range

0 < p \leq 1

, which were constructed by continuous multi-level ellipsoid covers

Θ

R^{n}

with high anisotropy in the [...] Read more.

In 2011, Dekel et al. developed highly geometric Hardy spaces

H^{p} (Θ)

, for the full range

0 < p \leq 1

, which were constructed by continuous multi-level ellipsoid covers

Θ

R^{n}

with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in

Θ

rapidly change shape from level to level, the authors further obtain some real-variable characterizations of

H^{p} (Θ)

in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik. Full article

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

12 pages, 288 KiB

Open AccessArticle

Calderón Operator on Local Morrey Spaces with Variable Exponents

by Kwok-Pun Ho

Mathematics 2021, 9(22), 2977; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222977 - 22 Nov 2021

Cited by 3 | Viewed by 1408

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)

18 pages, 356 KiB

Open AccessArticle

Spaces of Pointwise Multipliers on Morrey Spaces and Weak Morrey Spaces

by Eiichi Nakai and Yoshihiro Sawano

Mathematics 2021, 9(21), 2754; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212754 - 29 Oct 2021

Cited by 2 | Viewed by 1138

Abstract

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the earlier works of the first author’s papers written in 1997 and 2000, as well as Lemarié-Rieusset’s 2013 paper. As a corollary, the main result in the present paper shows that different quasi-Banach lattices can create the same vector-valued Morrey spaces. The goal of the present paper is to provide a complete picture of the pointwise multiplier spaces. Full article

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

46 pages, 576 KiB

Open AccessArticle

Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

by Ziwei Li, Dachun Yang and Wen Yuan

Mathematics 2021, 9(21), 2724; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212724 - 27 Oct 2021

Cited by 4 | Viewed by 1364

Abstract

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to [...] Read more.

γ

-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets. Full article

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

26 pages, 427 KiB

Open AccessArticle

Boundedness of Some Paraproducts on Spaces of Homogeneous Type

by Xing Fu

Mathematics 2021, 9(20), 2591; https://0-doi-org.brum.beds.ac.uk/10.3390/math9202591 - 15 Oct 2021

Viewed by 1076

Abstract

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations [...] Read more.

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of

Π_{3}

on Hardy spaces and then, via the methods of interpolation and the well-known

T (1)

theorem, establishes the endpoint estimates for

{Π_{j}}_{j = 1}^{3}

. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of

{Π_{j}}_{j = 1}^{3}

, which has independent interests. It is also remarked that, throughout this article,

μ

is not assumed to satisfy the reverse doubling condition. Full article

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

16 pages, 301 KiB

Open AccessArticle

Weighted Estimates for Iterated Commutators of Riesz Potential on Homogeneous Groups

by Daimei Chen, Yanping Chen and Teng Wang

Mathematics 2021, 9(19), 2421; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192421 - 29 Sep 2021

Viewed by 915

Abstract

In this paper, we study the two weight commutators theorem of Riesz potential on an arbitrary homogeneous group

H

of dimension N. Moreover, in accordance with the results in the Euclidean space, we acquire the quantitative weighted bound on homogeneous group. [...] Read more.

In this paper, we study the two weight commutators theorem of Riesz potential on an arbitrary homogeneous group

H

of dimension N. Moreover, in accordance with the results in the Euclidean space, we acquire the quantitative weighted bound on homogeneous group. Full article

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

11 pages, 274 KiB

Open AccessArticle

Wavelets and Real Interpolation of Besov Spaces

by Zhenzhen Lou, Qixiang Yang, Jianxun He and Kaili He

Mathematics 2021, 9(18), 2235; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182235 - 12 Sep 2021

Cited by 1 | Viewed by 1454

Abstract

In view of the importance of Besov space in harmonic analysis, differential equations, and other fields, Jaak Peetre proposed to find a precise description of [...] Read more.

In view of the importance of Besov space in harmonic analysis, differential equations, and other fields, Jaak Peetre proposed to find a precise description of

{(B_{p_{0}}^{s_{0}, q_{0}}, B_{p_{1}}^{s_{1}, q_{1}})}_{θ, r}

. In this paper, we come to consider this problem by wavelets. We apply Meyer wavelets to characterize the real interpolation of homogeneous Besov spaces for the crucial index p and obtain a precise description of

{({\dot{B}}_{p_{0}}^{s, q}, {\dot{B}}_{p_{1}}^{s, q})}_{θ, r}

. Full article

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

24 pages, 433 KiB

Open AccessArticle

Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

by Jun Liu, Long Huang and Chenlong Yue

Mathematics 2021, 9(18), 2216; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182216 - 09 Sep 2021

Cited by 4 | Viewed by 1389

Abstract

Let

\vec{p} \in {(0, \infty)}^{n}

be an exponent vector and A be a general expansive matrix on

R^{n}

. Let

H_{A}^{\vec{p}} (R^{n})

be the anisotropic mixed-norm Hardy spaces associated with A [...] Read more.

Let

\vec{p} \in {(0, \infty)}^{n}

be an exponent vector and A be a general expansive matrix on

R^{n}

. Let

H_{A}^{\vec{p}} (R^{n})

be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of

H_{A}^{\vec{p}} (R^{n})

, the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from

H_{A}^{\vec{p}} (R^{n})

to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on

H_{A}^{\vec{p}} (R^{n})

. In addition, the boundedness of anisotropic Calderón–Zygmund operators from

H_{A}^{\vec{p}} (R^{n})

to the mixed-norm Lebesgue space

L^{\vec{p}} (R^{n})

is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on