Operators In Harmonic Analysis Research Articles

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

AbstractIn this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space to and also from to the space of functions of bounded mean oscillation , with respect to the probability measure on when is a multi‐index in . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

Extrapolation in new weighted grand Morrey spaces beyond the Muckenhoupt classes

Rubio de Francia's extrapolation theorem for new weighted grand Morrey spaces Mwp),λ,θ(X) with weights w beyond the Muckenhoupt Ap classes is established. This result, in particular, implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. Necessary conditions for the boundedness of the Hardy–Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are also investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.

Quantitative weighted estimates for Harmonic analysis operators in the Bessel setting by using sparse domination

In this paper we obtain quantitative weighted L^p-inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain L^p(w)-operator norms in terms of the A_p-characteristic of the weight w. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type ((0,infty ),|cdot |,x^{2lambda }dx).

An Example of a Singular Integral and a Weight

Abstract Sharp weighted inequalities were recently proved for several classical operators in Harmonic analysis, however for the rough singular integral the sharp result remains open. The best bound so far was found by Hytönen, Roncal and Tapiola in [2]. For $A_2$ weight, it is quadratic, meaning $\|T_\Omega f\|_{L^2_{\omega }}\leq C [\omega ]_2^2 \|f\|_{L^2_{\omega }}.$ The authors also conjectured that the best bound is linear. We provide example of $A_2$ weights $\omega _n$, test functions $f_n$ and rough singular integrals $$ \begin{align*} & T_{\Omega_n}(f)(x)=\textrm{p.v.} \int_{\mathbb R^2} |y|^{-2} \Omega_n(y/|y|)f(x-y) dy, \end{align*}$$where $\Omega _n $ is a function in $L^\infty (\mathbb S^{1})$ with norm $1$ and vanishing integral such that $$ \begin{align*} &\|T_{\Omega_n} f_n\|_{L^2_{\omega_n}}\geq C [\omega_n]_2^{3/2} \|f_n\|_{L^2_{\omega_n}}\end{align*}$$and $[\omega _n]_2\approx n,$ disproving the conjecture.

Boundedness of certain bilinear operators on vanishing generalized Morrey spaces

In this article, we consider the bilinear operator T satisfying that there exists a positive constant C(T), depending on T, such that, for any measurable functions f and g with compact support, t∈ℝ with 0<|t|≤1, and x∈ℝn with 0∉supp(f(x−t⋅))∩supp(g(x−⋅)), T(f,g)(x)≤C(T)∫ℝnf(x−ty)g(x−y)|y|ndy. We investigate the boundedness of T on the vanishing generalized Morrey spaces V0Lp,φ(ℝn) and V∞Lp,φ(ℝn), and the boundedness of the subbilinear maximal operator ℳ on the vanishing generalized Morrey space V(∗)Lp,φ(ℝn), and their applications to some classical (sub)bilinear operators in harmonic analysis. As a byproduct, we also show that T is bounded on generalized Morrey spaces Lp,φ(ℝn). Some typical examples for the main results of this paper are also included.

On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces

In this paper, we establish necessary and sufficient conditions on monotone weight functions for the boundedness for Hausdorff operators of special kind in weighted Lebesgue spaces. In particular, we get similar results for important operators of harmonic analysis which are special cases of the Hausdorff operators. The weights are illustrated by examples at the end of the paper.

Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework

We prove sharp power-weighted Lp, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator Bν in the exotic range of the parameter −∞<ν<1. Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical g-function and fractional integrals (Riesz potential operators).

Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.

Sparse domination of weighted composition operators on weighted Bergman spaces

The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the A2 theorem for Calderón-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer–testing conditions. We also extend our results to the unit ball B in Cn.

Real Interpolation of Hardy-Type Spaces an Announcement with some Remarks

We consider couples (XA, YA) of Hardy-type spaces defined for quasi-Banach lattices of measurable functions on 𝕋×Ω. Under certain fairly general assumptions, the following conditions are shown to be equivalent: (XA, YA) is K-closed in (X, Y), this couple is stable with respect to the real interpolation in the sense that (XA, YA)θ,p = (XA + YA) ∩ (X, Y)θ,p, the inclusion (X1−θYθ)A ⊂ (XA, YA)θ,∞ holds true, and lattices (L1, (Xr)′Yr)δ,q are BMO-regular for some values of the parameters. The last property is weaker than the BMO-regularity of (X, Y), and it requires further study. Some new (compared to the main article) results are given concerning the characterization of this property in terms of the boundedness of the standard harmonic analysis operators such as the Hilbert transform and the Hardy-Littlewood maximal operator.

Improvements on Sawyer type estimates for generalized maximal functions

AbstractIn this paper we prove mixed inequalities for the maximal operator , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy–Littlewood maximal operator proved by E. Sawyer. We show that given , if are weights belonging to the A1‐Muckenhoupt class and Φ is a Young function as above, then the inequality holds for every positive t.A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of . Moreover, it is well‐known that for the particular case with these maximal functions control, in some sense, certain operators in harmonic analysis.

Boundedness of iterated spherical average on modulation spaces

The spherical average A1(f) and its iteration (A1)N are important operators in harmonic analysis and probability theory. Also Δ(A1)N is used to study the K functional in approximation theory, where Δ is the Laplace operator. In this paper, we obtain the sufficient and necessary conditions to ensure the boundedness of Δ(A1)N from the modulation space Mp1,q1s1 to the modulation space Mp2,q2s2 for 1≤p1,p2,q1,q2≤∞ and s1,s2∈R.

Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaces

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

Limiting Weak-Type Behaviors for Certain Classical Operators in Harmonic Analysis

This paper explores the limiting weak-type behaviors of certain classical operators in harmonic analysis including maximal operators, singular and fractional integral operators and maximal truncated singular integrals as well as the general convolution operators with weak-type Young’s inequalities. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous results. As applications, several characterizations on the weak-type endpoint boundedness for fractional integral operators and maximal operators with homogeneous kernels, and the weak-type Young’s inequality for the general convolution operator are also obtained.

Extrapolation in weighted classical and grand Lorentz spaces. Application to the boundedness of integral operators

We establish weighted extrapolation theorems in classical and grand Lorentz spaces. As a consequence we have the weighted boundedness of operators of Harmonic Analysis in grand Lorentz spaces. We treat both cases: diagonal and off-diagonal ones.

On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $$C_0^\infty ({\mathbb {R}^n})$$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.

Discrete Hardy Spaces and Heat Semigroup Associated with the Discrete Laplacian

In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian \(\Delta _d\) in discrete Hardy spaces \({\mathcal {H}}^p({\mathbb {Z}})\). We prove that the maximal operator and the Littlewood–Paley g function defined by the semigroup generated by \(\Delta _d\) are bounded from \({\mathcal {H}}^p({\mathbb {Z}})\) into \(\ell ^p({\mathbb {Z}})\), \(0<p\le 1\). Also, we establish that every \(\Delta _d\)-spectral multiplier of Laplace transform type is a bounded operator from \({\mathcal {H}}^p({\mathbb {Z}})\) into itself, for every \(0<p\le 1\).

Weighted extrapolation in grand Morrey spaces and applications to partial differential equations

In this paper we derive weighted extrapolation results in grand Morrey spaces. In particular, the main statements say that if for a class of pairs of measurable functions the one-weight inequality holds in the classical weighted L^{p_0}_w space for some p_0 and for all Muckenhoupt A_{p_0} weights, then the one-weight inequality is also true in weighted grand Morrey space L^{p), r, \theta}_w with A_p weights for all p . The spaces under consideration are defined on quasi-metric measure spaces. The obtained results are applied to derive one-weight estimates for some operators of Harmonic Analysis and to study regularity properties of solutions of second order partial differential equations with discontinuous coefficients.

On the Behaviors of Rough Fractional Type Sublinear Operators on Vanishing Generalized Weighted Morrey Spaces

The aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples.

Extrapolation in Grand Lebesgue Spaces with A∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ1 ×· · ·×μn onX1 ×· · ·×Xn, where (Xi, di, μi), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A∞ weights for operators of harmonic analysis are derived.