Fractional Maximal Operator Research Articles

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

Morrey Spaces were first introduced by C.B. Morrey in 1938. Morrey space can be considered as a generalization of the Lebesgue spaces. Morrey spaces were then generalized become the generalized Morrey spaces, the weighted Morrey spaces, and the generalized weighted Morrey spaces. One of the studies on Morrey spaces is the boundedness of certain operators on the spaces. One of the operators is the fractional integral. The boundedness of fractional integrals on the classical Morrey spaces, the weighted Morrey spaces, the generalized Morrey spaces, and the generalized weighted Morrey spaces had been known. One of the extensions of fractional integrals is generalized fractional integral. The operator was bounded on the generalized Morrey spaces. The purpose of this study is to investigate the boundedness of generalized fractional integrals on the generalized weighted Morrey spaces. The weight used is Muckenhoupt class. The results obtained show that the generalized fractional integral is bounded from generalized weighted Morrey space to another generalized weighted Morrey space under some assumptions. The main result obtained then implies the boundedness of the generalized fractional maximal operator on generalized weighted Morrey spaces under the same assumptions.

Regularity of local multilinear fractional maximal and strong maximal operators

Regularity of local multilinear fractional maximal and strong maximal operators

Bourgain–Morrey spaces and their applications to boundedness of operators

A class of function spaces related to Bourgain–Morrey spaces was recently introduced. Here, this class is investigated from the viewpoints of harmonic analysis and functional analysis. Specifically, this paper is oriented to certain inclusion, approximation, and interpolation properties as well as the boundedness of operators such as the Hardy–Littlewood maximal operator, fractional integral operators, fractional maximal operators, and singular integral operators. In particular, the embedding result by Bégout and Vargas is refined. Another important feature of this class of function spaces is that it can describe the convolution property. In addition, we give a description of the dual of Bourgain–Morrey spaces and we combine this with existing results to conclude the reflexivity of these spaces.

Fractional maximal and integral operators in variable exponent Morrey spaces

In this paper, we study the boundedness of the fractional maximal operator and fractional integral operator on the variable exponent Morrey spaces defined over spaces $(X,d,\mu)$ of homogeneous type.

Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div(A(x,∇u))=div(f)inΩ,andu=gon∂Ω,that related to Riesz potentials. Here, in our setting, Ω⊂Rn, n≥2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

$$L^{p}$$ Estimates and Weighted Estimates of Fractional Maximal Rough Singular Integrals on Homogeneous Groups

In this paper, we study the \(L^{p}\) boundedness and \(L^{p}(w)\) boundedness (\(1<p<\infty \) and w a Muckenhoupt \(A_{p}\) weight) of fractional maximal singular integral operators \(T_{\Omega ,\alpha }^{\#}\) with homogeneous convolution kernel \(\Omega (x)\) on an arbitrary homogeneous group \({\mathbb {H}}\) of dimension \({\mathbb {Q}}\). We show that if \(0<\alpha <{\mathbb {Q}}\), \(\Omega \in L^{1}(\Sigma )\) and satisfies the cancellation condition of order \([\alpha ]\), then for any \(1<p<\infty \), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}({\mathbb {H}})}\lesssim \Vert \Omega \Vert _{L^{1}(\Sigma )}\Vert f\Vert _{L_{\alpha }^{p}({\mathbb {H}})}. \end{aligned}$$where for the case \(\alpha =0\), the \(L^p\) boundedness of rough singular integral operator and its maximal operator were studied by Tao (Indiana Univ Math J 48:1547–1584, 1999) and Sato (J Math Anal Appl 400:311–330, 2013), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if \(0\le \alpha <{\mathbb {Q}}\) and \(\Omega \) satisfies the same cancellation condition but a stronger condition that \(\Omega \in L^{q}(\Sigma )\) for some \(q>{\mathbb {Q}}/\alpha \), then for any \(1<p<\infty \) and \(w\in A_{p}\), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}(w)}\lesssim \Vert \Omega \Vert _{L^{q}(\Sigma )}\{w\}_{A_p}(w)_{A_p}\Vert f\Vert _{L_{\alpha }^{p}(w)},\ \ 1<p<\infty . \end{aligned}$$

Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a} A_{p,q,\alpha}^{b'}$, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.

Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functions

In this paper, we explore the endpoint Sobolev regularity of the fractional maximal operators in the multilinear setting. Some new bounds and continuity for the derivatives of the multilinear fractional maximal functions are established, both in the centered and in the uncentered cases.

Singular and Fractional Integral Operators on Weighted Local Morrey Spaces

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón–Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results for these operators in the usual versions of weighted Morrey spaces, neither restricted to the Sobolev relation of the exponents nor to the one-weighted setting.

Commutators of fractional maximal operator in variable Lebesgue spaces over bounded quasi‐metric measure spaces

We study the fractional maximal commutators and the commutators of the fractional maximal operator with in the variable Lebesgue spaces over bounded quasi‐metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators and on the spaces when . Furthermore, we obtain some new characterizations for certain subspaces of .

Gradient potential estimates in elliptic obstacle problems with Orlicz growth

In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. Furthermore, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on \(C^{1,\alpha }\)-regularity of solutions.

Limiting Weak-Type Behaviors for Fractional Maximal Operators and Fractional Integrals with Rough Kernel

Limiting Weak-Type Behaviors for Fractional Maximal Operators and Fractional Integrals with Rough Kernel

BOUNDEDNESS CRITERION FOR SUBLINEAR OPERATORS AND COMMUTATORS ON GENERALIZED MIXED MORREY SPACES

<abstract> In this paper, the author studies the boundedness for a large class of sublinear operators $ T_\alpha, \alpha\in[0, n) $ generated by Calderón-Zygmund operators ($ \alpha=0 $) and generated by fractional integral operator ($ \alpha&gt;0 $) on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb R}^n) $. Moreover, the boundeness for commutators of $ T_\alpha, \alpha\in[0, n) $ on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb R}^n) $ is also studied. As applications, we obtain the boundedness for Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, fractional integral operator, fractional maximal operator and their commutators on generalzied mixed Morrey spaces. </abstract>

Fractional operators and their commutators on generalized Orlicz spaces

In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.

Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables). This work also extends some abstract results proved in Nguyen and Tran (2021) [43] by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimate of the gradient of weak solutions via fractional maximal operators and Riesz potential of data.

Sobolev Regularity of Multilinear Fractional Maximal Operators on Infinite Connected Graphs

Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and Hajłasz–Sobolev spaces on G.

Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data

This paper continues the development of regularity results for quasilinear elliptic equations −div(A(x,∇u))=μinΩ,andu=0on∂Ω,in Lorentz and Lorentz–Morrey spaces, where Ω⊂Rn (n≥2); A is a monotone Carathéodory vector valued operator acting between W01,p(Ω) and its dual W−1,p′(Ω); and μ is a datum in some Lebesgue space Lm(Ω), for m<p′. It emphasizes that in this paper, we restrict our study to the case of ‘very singular’ when 1<p≤3n−22n−1, and under mild assumption that the p-capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the ‘very singular’ case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces.

Boundedness of fractional integral operators in Herz spaces on the hyperplane

Our aim in this paper is to deal with the boundedness of the fractional maximal and fractional integral operators in Herz spaces with variable exponent on the hyperplane. We also discuss the integrability of the fractional maximal and fractional integrals in such spaces.

Extrapolation to product Herz spaces and some applications

Abstract This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Maximal and fractional maximal operators in the Lorentz-Morrey spaces and their applications to the Bochner-Riesz and Schrödinger-type operators

The aim of this paper is to obtain boundedness conditions for the maximal function Mf and to prove the necessary and sufficient conditions for the fractional maximal oparator Mα in the Lorentz-Morrey spaces which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner-Riesz operator and the Schrödinger-type operators Vγ (−Δ + V)−β and Vγ ∇(−Δ + V)−β in the Lorentz-Morrey spaces , where the nonnegative potential V belongs to the reverse Hölder class B∞ (ℝ n ).