Operators In Harmonic Analysis Research Articles

Grand Lebesgue sequence spaces

Abstract We introduce grand Lebesgue sequence spaces and study various operators of harmonic analysis in these spaces, e.g., maximal, convolution, Hardy, Hilbert, and fractional operators, among others. Special attention is paid to fractional calculus, including the density of the discrete version of a Lizorkin sequence test space in vanishing grand spaces.

Adams-Spanne type estimates for parabolic sublinear operators and their commutators with rough kernels on parabolic generalized Morrey spaces

The aim of this paper is to give Adams-Spanne type estimates for parabolic sublinear operators and their commutators by with rough kernels generated by parabolic fractional integral operators under generic size conditions which are satisfied by most of the operators in harmonic analysis. Their endpoint estimates are also disposed.

Vector-valued boundedness of harmonic analysis operators

Vector-valued boundedness of harmonic analysis operators

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderon–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

Sublinear operators in generalized weighted Morrey spaces

Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmonic analysis, such as, e.g., maximal functions, singular and fractional integrals, Bochner–Riesz means, and so on.

Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

This is an ultimate completion of our earlier paper [3] where mapping properties of several fundamental harmonic analysis operators in the setting of symmetrized Jacobi trigonometric expansions were investigated under certain restrictions on the underlying parameters of type. In the present article we take advantage of very recent results due to Nowak, Sjogren and Szarek to fully release those restrictions, and also to provide shorter and more transparent proofs of the previous restricted results. Moreover, we also study mapping properties of analogous operators in the parallel context of symmetrized Jacobi function expansions. Furthermore, as a consequence of our main results we conclude some new results related to the classical non-symmetrized Jacobi polynomial and function expansions.

Weighted extrapolation in Iwaniec—Sbordone spaces. Applications to integral operators and approximation theory

We prove extrapolation theorems in weighted Iwaniec–Sbordone spaces and apply them to one-weight inequalities for several integral operators of harmonic analysis. In addition, in weighted grand Lebesgue spaces, we establish Bernstein and Nikol’skii type inequalities and prove direct and inverse theorems on the approximation of functions.

Mixed weak type estimates: Examples and counterexamples related to a problem of E. Sawyer

We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.

On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals

We reduce the boundedness of operators in Morrey spaces $$L_p^r\left( {\mathbb R}^n\right) $$ , its preduals, $$H^{\varrho }L_p ({\mathbb R}^n)$$ , and their preduals $$\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) $$ to the boundedness of the appropriate operators in Lebesgue spaces, $$L_p({\mathbb R}^n)$$ . Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.

Boundedness of integral operators on decreasing functions

We continue the study of the boundedness of the operatoron the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p &gt; 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

$$(\Phi ,\Psi )$$ ( Φ , Ψ ) -admissible potential operators and their commutators on vanishing Orlicz-Morrey spaces

We study the boundedness of $$(\Phi ,\Psi )$$ -admissible potential operators and their commutators on vanishing generalized Orlicz-Morrey spaces $$VM_{\Phi ,\varphi }(\mathbb {R}^n)$$ including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Riesz potential, fractional maximal operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities involving the Young functions $$\Phi $$ , $$\Psi $$ and the function $$\varphi (x,r)$$ defining the space, without assuming any monotonicity property of $$\varphi (x,r)$$ on $$r$$ .

Commutators of Singular Integrals and Fractional Integral Operators

The commutators of integral and differential operators are very important tools in harmonic analysis and PDE. Recently, the third author of this paper gave a survey (Front. Math. China, 2011, 6(5): 821-833) on some late results and put forward a few problems on commutators of many important operators in harmonic analysis. In this paper, we present some results about the commutators of singular integrals and fractional integrals with rough or variable kernels, which can be seen as a supplement of the aforementioned survey.

Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces

We study the boundedness of Φ-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderon-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights without assuming any monotonicity property of on r. MSC:42B20, 42B25, 42B35, 46E30.

Calderón-Zygmund operators in the Bessel setting for all possible type indices

We show that many harmonic analysis operators in the Bessel setting, including maximal operators, Littlewood-Paley-Stein type square functions, multipliers of Laplace or Laplace-Stieltjes transform type and Riesz transforms are, or can be viewed as, Calderon-Zygmund operators for all possible values of type parameter λ in this context. This extends results existing in the literature, but being justified only for a restricted range of λ.

Finitely strictly singular operators in harmonic analysis and function theory

We prove that the class of strictly singular operators and the class of finitely strictly singular operators are stable under perturbed domination. We prove a diagonal theorem, which is a vector valued version of the result due to V. Milman stating that the identity from ℓp to ℓq is finitely strictly singular when p<q. Some applications are given. The paper is illustrated with examples in harmonic analysis and function theory: we prove that the Fourier transform from Lp(T) into ℓp′ is finitely strictly singular if and only if p∈(1,2). In the case p=1, some examples of different kind of behavior involving thin sets are given. Another result deals with the identity from the Hardy space Hp to the Bergman space B2p: it is finitely strictly singular for every p⩾1. This result requires some new special inequalities between the norms on the Hardy and the Bergman spaces.

Φ-Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces

We study the boundedness ofΦ-admissible sublinear singular operators on Orlicz-Morrey spacesMΦ,φℝn. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operator.

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to Z2n (also negative values of the multiplicity function are admitted). Our investigations include maximal operators, g-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace–Stieltjes transform type. Using the general Calderón–Zygmund theory we prove that these objects are bounded in weighted Lp spaces, 1<p<∞, and from L1 into weak L1.

Harmonic analysis operators related to symmetrized Jacobi expansions

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood–Paley–Stein square functions and multipliers of Laplace and Laplace–Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.

Estimates of Some Operators on One-Sided Weighted Morrey Spaces

A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.