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We characterize the commutators of fractional integral operator and Calderón–Zygmund operator on differential forms. Also, the Hölder continuity for commutator of fractional integral operator with double-phase functional is derived. Finally, some estimates for the solutions to A-harmonic equations on differential forms are obtained. [ABSTRACT FROM AUTHOR]

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R–L fractional operators are established. Additionally, some novel refinements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.

In this paper, an effcient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, effciency, and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Several classical integral inequalities have been proved in more general settings using fractional integral operator containing an extended Mittag-Leffler function in the kernel. A number of corollaries and consequences of the main results are also presented. [ABSTRACT FROM AUTHOR]

Integral operators are useful in real analysis, mathematical analysis, functional analysis and other subjects of mathematical approach. The goal of this paper is to study a unified integral operator via convexity. By using convexity and conditions of unified integral operators, bounds of these operators are obtained. Furthermore consequences of these results are discussed for fractional and conformable integral operators.

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$, and obtain a new estimate for their norm.

In the paper, the authors establish some new inequalities of the Grüss type for conformable fractional integrals. These inequalities generalize some known results.

The generalized fractional integral operators are shown to be bounded from an Orlicz–Hardy space HΦ(Rn) to another Orlicz–Hardy space HΨ(Rn), where Φ and Ψ are generalized Young functions. The result extends the boundedness of the usual fractional integral operator Iα from Hp(Rn) to Hq(Rn) for α,p,q∈(0,∞) and −n/p+α=−n/q, which was proved by Stein and Weiss in 1960. [ABSTRACT FROM AUTHOR]

The present investigation aims to extract a solution from the generalized fractional kinetic equations involving the generalized q-Bessel function by applying the Laplace transform. Methodology and results can be adopted and extended to a variety of related fractional problems in mathematical physics. [ABSTRACT FROM AUTHOR]

Summary: The aim of this article is to investigate an efficient computational method for solving distributed‐order fractional optimal control problems. In the proposed method, a new Riemann‐Liouville fractional integral operator for the Bernstein wavelet is given. This approach is based on a combination of the Bernstein wavelets basis, fractional integral operator, Gauss‐Legendre numerical integration, and Newton's method for solving obtained system. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed method. The error analysis of the proposed method is carried out. Examples reveal the applicability of the proposed technique. [ABSTRACT FROM AUTHOR]

In 1995, Pérez introduced B p -condition, which is necessary and sufficient condition for the boundedness of the Orlicz maximal operator on L p spaces. After, necessary and sufficient condition of the Hardy–Littlewood–Sobolev type inequality for Orlicz-fractional maximal operator is derived by Cruz-Uribe and Moen in 2013. In this paper, we investigate the boundedness of Orlicz maximal operator, Orlicz-fractional maximal operator and fractional integral operator in Morrey and Orlicz–Morrey spaces on the assumption that each Young function satisfies these conditions, respectively. In particular, one of the main results is based on the Adams inequality in the framework of Morrey spaces. [ABSTRACT FROM AUTHOR]

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Lately, many studies were offered to introduce the population dynamics of COVID-19. In this investigation, we extend different physical conditions of the growth by employing fractional calculus. We study a system of coupled differential equations, which describes the dynamics of the infection spreading between infected and asymptomatic styles. The healthy population properties are measured due to the social meeting. The result is associated with a macroscopic law for the population. This dynamic system is appropriate to describe the performance of growth rate of the infection and to verify if its control is appropriately employed. A unique solution, under self-mapping possessions, is investigated. Approximate solutions are presented by utilizing fractional integral of Chebyshev polynomials. Our methodology is based on the Atangana–Baleanu calculus, which provides various activity results in the simulation. We tested the suggested system by using live data. We found positive action in the graphs. [ABSTRACT FROM AUTHOR]

This paper presents an approximate method to solve a class of fractional partial differential equations (FPDEs). First, we introduce radial basis functions (RBFs) combined with wavelets. Next, we obtain fractional integral operator (FIO) of wavelets-radial basis functions (W-RBFs) directly. In the next step, the W-RBFs and their FIO are used to transform the problem under consideration into a system of algebraic equations, which can be simply solved to achieve the solution of the problem. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this work, we consider the norm inequalities for sublinear operators with rough kernel generated by fractional integrals and commutators on generalized vanishing local Morrey spaces including their weak versions under generic size conditions which are satisfied by most of the operators in harmonic analysis, respectively. [ABSTRACT FROM AUTHOR]

The paper presents generalizations of Minkowski type integral inequalities for a generalized fractional integral operator containing an extended Mittag-Leffler function in the kernel. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the Linear fractional differential and integral operator given by Srivastava and Owa [8]. In addition, we study some geometric properties of this new operator for the unified class U [Φ, Ψ;α, β, λ, n] , which was defined by Darus [2] like growth, distortion and the radii of starlikeness. [ABSTRACT FROM AUTHOR]

This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators. [ABSTRACT FROM AUTHOR]

: In this paper, author introduces some new integral inequalities by using the Caputo-Fabrizio (CF) fractional integral and functions with the same sense of variation. Also an open problem concerning an integral inequality is discussed. [ABSTRACT FROM AUTHOR]

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

Convexity is crucial in obtaining many forms of inequalities. As a result, there is a significant link between convexity and integral inequality. Due to the significance of these concepts, the purpose of this study is to introduce a new class of generalized convex interval-valued functions called LR-preinvex interval-valued functions (LR-preinvex I-V-Fs) and to establish Hermite–Hadamard type inequalities for LR-preinvex I-V-Fs using partial order relation (≤p). Furthermore, we demonstrate that our results include a large class of new and known inequalities for LR-preinvex interval-valued functions and their variant forms as special instances. Further, we give useful examples that demonstrate usefulness of the theory produced in this study. These findings and diverse approaches may pave the way for future research in fuzzy optimization, modeling, and interval-valued functions.

Abstract In this article, a new general integral identity involving generalized fractional integral operators is established. With the help of this identity new Hermite-Hadamard type inequalities are obtained for functions whose absolute values of derivatives are convex. As a consequence, the main results of this paper generalize the existing Hermite-Hadamard type inequalities involving the Riemann-Liouville fractional integral.

We establish the boundedness of some Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. [ABSTRACT FROM AUTHOR]

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, µ) of homogeneous type. [ABSTRACT FROM AUTHOR]

In this article, we first establish a new general integral identity for differentiable functions with the help of generalized fractional integral operators introduced by Raina [8] and Agarwal et al. [1]. As a second, by using this identity we obtain some new fractional Hermite-Hadamard type inequalities for functions whose absolute values of first derivatives are convex. Relevant connections of the results presented here with those involving Riemann-Liouville fractional integrals are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of fractional functions based on Chelyshkov wavelets for solving one- and two-variable distributed-order fractional differential equations. The concept of fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on fractional integral operator of fractional-order Chelyshkov wavelets with composite collocation method. This operator and collocation method are utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. The convergence of the fractional-order Chelyshkov wavelets bases is discussed. The efficiency and the applicability of the new methodology are illustrated by eight examples, in addition our findings in comparison with the existing results show the advantage of our method. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for the boundedness of commutators of fractional integral operators on mixed Morrey spaces. We construct the predual spaces of mixed Morrey spaces. Furthermore, we show the sharp maximal inequality of the mixed Morrey norm. This inequality is one of the key inequalities of the main theorem. [ABSTRACT FROM AUTHOR]

Suppose L = - Δ + V is a Schrödinger operator on R n with a potential V belonging to certain reverse Hölder class R H σ with σ ≥ n / 2 . The aim of this paper is to study the A p weights associated to L, denoted by A p L , which is a larger class than the classical Muckenhoupt A p weights. We first prove the quantitative A p L bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative A p , q L bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical A p , q constant. However, since A p , q ⊂ A p , q L , the A p , q L constants are smaller than A p , q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the "exp–log" link between A p L and B M O L (the BMO space associated with L), and show that for w ∈ A p L , log w is in B M O L , and that the reverse is not true in general. [ABSTRACT FROM AUTHOR]

We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman–Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Abstract We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.

We investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

In this present work, the authors establish a new integral identity involving generalized fractional integral operators and by using this fractional-type integral identity, obtain some new Hermite-Hadamard type inequalities for functions whose first derivatives in absolute value are convex. Relevant connections of the results presented here with those earlier ones are also pointed out.

Abstract The purpose of this paper is to investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main result generalizes the Sawano, Sugano, and Tanaka result to a weighted setting.

In this article, a general integral identity for twice differentiable mapping involving fractional integral operators is derived. Secondly by using this identity we obtain some new generalized Hermite-Hadamards type inequalities for functions whose absolute values of second derivatives are s-convex and concave. The main results generalize the existing Hermite-Hadamard type inequalities involving the Riemann-Liouville fractional integral. Also we point out, some results in this study in some special cases such as setting s = 1, λ = λ, ς(0) = 1 and w = 0, more reasonable than those obtained in [8]. [ABSTRACT FROM AUTHOR]

In the present work we find inequalities that generalize some results found by Iqbal in [4, 5] regarding Riemann-Liouville fractional integral, by means of a Hardy inequality, using fractional integral operators defined by R.K. Raina in [9]. [ABSTRACT FROM AUTHOR]

In this paper we prove the boundedness of certain sublinear operators , , generated by fractional integral operators with rough kernels , , from one generalized local Morrey space to another , , , and from the space to the weak space , , . In the case b belongs to the local Campanato space and is a linear operator, we find the sufficient conditions on the pair which ensures the boundedness of the commutator operators from to , , , , . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in r. [ABSTRACT FROM AUTHOR]

Abstract Unsteady nonlinear boundary layer convection flows and heat transfer of a fractional Maxwell fluid near a vertical plate with constant thermal flux are studied. The fractional constitutive equations for the shear stress and thermal flux are formulated for the first time with the integral time-fractional operator of type Caputo–Fabrizio. These types of fractional relationships provide a weighted average exponential to the velocity gradient and temperature gradient. The studied model is constituted from a system of nonlinear partial differential equations coupled with fractional differential equations with initial and boundary conditions. The solutions for the shear stress, velocity, thermal flux and temperature are obtained by using the Laplace transform along with a suitable transformation of variables. In order to finding solutions of the studied model, two inverse Laplace transforms of exponential type are obtained. The particular cases when only the stress or thermal flux are damped, as well as the convection flows of the ordinary fluid are obtained as particular cases when the stress/thermal fractional parameter tends to 1. [ABSTRACT FROM AUTHOR]

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent 1, and obtain a new estimate for their norm. [ABSTRACT FROM AUTHOR]

This paper studies fractional integral operator for vector fields in weighted L1. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg inequalities and establish new div-curl inequalities for vector fields. [ABSTRACT FROM AUTHOR]

We analyze local (central) Morrey spaces, generalized local (central) Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalized Bessel-Riesz operators and generalized fractional integral operators in generalized local (central) Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalized fractional integral operator and Olsen type inequalities in Campanato spaces and generalized local (central) Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidean settings, however, some of them are new already in the standard Euclidean cases. [ABSTRACT FROM AUTHOR]

Abstract: We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces ( X , d , μ ) without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen. [ABSTRACT FROM AUTHOR]

In the present paper, the notion of generalized (r;g,s,m,ϕ)-preinvex function is applied for establish some new generalizations of Ostrowski type inequalities via fractional integral operators. These results not only extend the results appeared in the literature ([38]) but also provide new estimates on these type. At the end, some applications to special means are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce higher order Riesz-Bessel transforms which we can express partial derivatives of order α of Im,νf for f ∈ Lp,ν. In addition, we establish relationship between Riesz potential with higher order Riesz-Bessel transform related to generalized shift operator. By using this relationship, we make some improvements of integral estimates for Im,νf and higher order Riesz-Bessel transform Rmν in the Beppo Levi space BLmp,v We prove an estimate for the singular integral operator with convolution type generated by generalized shift operator in the Beppo Levi spaces. [ABSTRACT FROM AUTHOR]

The object of this paper is to obtain certain Hermite-Hadamard type integral inequalities involving general class of fractional integral operators and the fractional integral operators with exponential kernel by using harmonically convex functions. [ABSTRACT FROM AUTHOR]