Quantitative approximation by fractional smooth general singular operators

In this article we study the fractional smooth general singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. Furthermore we produce a fractional Voronovskaya type result giving the fractional asymptotic expansion of the basic error of our approximation.We finish with applications to fractional trigonometric singular integral operators. Our operators are not in general positive.     

Two-weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces

In this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two-weight norm inequalities for both operators on weighted Morrey spaces.     

Morrey spaces and fractional integral operators

The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities.     

WEAK TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATORS ON GENERALIZED NON-HOMOGENEOUS MORREY SPACES

We obtain weak type (1,q) inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces.The proofs use some properties of maximal operators.Our results are closely related to the strong type inequalities in [13,14,15].     

多线性分数次积分算子交换子的双权弱型不等式

多线性分数次积分算子定义为利用分数次Orlicz极大算子和sharp函数,得到了多线性分数次积分算子交换子的双权弱(p,p)型不等式成立的充分条件.     

A CHARACTERIZATION FOR FRACTIONAL INTEGRALS ON GENERALIZED MORREY SPACES

This paper concerns with the fractional integrals,which are also known as the Riesz potentials.A characterization for the boundedness of the fractional integral operators on generalized Morrey spaces will be presented.Our results can be viewed as a refinement of Nakai’s [7].     

Fractional integrals on radial functions with applications to weighted inequalities

We obtain operators pointwise equivalent to the fractional integral and the maximal fractional operator acting on radial functions. With the aid of these operators we obtain several weighted inequalities, assuming that either the functions or the weights are radial.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

Some estimates of certain integral operators in generalized fractional orlicz classes

We consider inequalities for composition of integral operators with homogeneous kernels in the Musielak-Orlicz classes. The main application are some inequalities with respect to the fractional modular.     

Positive solution of nonlinear fractional differential equations with Caputo-like counterpart hyper-Bessel operators

By studying a weakly singular integral whose kernel involves Mittag-Leffler functions, we obtain some new Gronwall-type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo-like counterpart hyper-Bessel operators are established.     

Fractional integrals on compact Riemannian symmetric spaces of rank one

In this paper we study mixed norm boundedness for fractional integrals related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.     

Orlicz–Morrey Spaces and Fractional Operators

By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz–Morrey spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz–Morrey spaces are established. The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also verified. The result obtained here sharpens the one in our earlier papers.     

Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s,?m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.     

A remark on fractional integrals on modulation spaces

In this paper, we give the necessary and sufficient conditions for the boundedness of fractional integral operators on the modulation spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transplantation and multiplier theorems for Fourier-Bessel expansions

Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.

     

多线性分数次积分迭代型交换子的加权不等式

研究了多线性分数次积分算子的迭代型交换子,给出了双权强型不等式的充分条件,即Fefferman -Phong型条件.对此迭代型交换子,还给出了Fefferman-Stein型加权不等式和Coifman型加权不等式.     

Fractional differential and integral operations via cumulative approach

In this study, a fractal operator model of cumulative processes is described. Accordingly, differential and integral operators of the fractional calculus are derived by the fractal operator model of a cumulative process. In order to exhibit the relation between our cumulative approach and fractional calculus, vertical motion of a body is handled within these frameworks. Thereby, regard to our assessments, the underlying physical mechanism of the success of the fractional differintegral operators in describing stochastic complex systems is uncovered to some extent.     

Sharp bounds for fractional one-sided operators

In this paper,we characterize the sharp boundedness of the one-sided fractional maximal function for one-weight and two-weight inequalities.Also a new two-weight testing condition for the one-sided fractional maximal function is introduced extending the work of Martín-Reyes and de la Torre.Improving some extrapolation result for the one-sided case,we get weak sharp bounded estimates for one-sided fractional maximal function and weak and strong sharp bounded estimates for one-sided fractional integral.     

分数次积分交换子的加权不等式

对分数次积分算子和BMO函数构成的高阶交换子, 该文给出了强型和弱型的加权不等式.     

Some inequalities involving Hadamard‐type k‐fractional integral operators

In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard‐type k‐fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2017 John Wiley & Sons, Ltd.