Two-weight Inequalities Research Articles

Weighted bilinear multiplier theorems in Dunkl setting via singular integrals

Abstract The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.

Two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type

In this paper we give a characterization of two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type \(\mathcal{M}_{\omega }^{p,\varphi }(X)\). We prove the boundedness of maximal commutators \([M,b]\) from the spaces \(\mathcal{M}_{\omega _{1}^{\delta }}^{p,\varphi _{1}}(X)\) to the spaces \(\mathcal{M}_{\omega _{2}^{\delta }}^{p,\varphi _{2}}(X)\), where \(1<p<\infty \), \(0<\delta <1\) and \((\omega _{1},\omega _{2})\in \widetilde{A}_{p}(X)\).

Weak and strong type estimates for square functions associated with operators

Let L be a linear operator on L2(Rn) which generates a semigroup e−tL whose kernels pt(x,y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function Sα,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding to Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p,p) for Sα,L and the endpoint estimate for commutators of Sα,L. Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.

Two-weight inequalities for multilinear maximal functions in Orlicz spaces

In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.

Two-weighted inequalities of maximal commutators in the modified Morrey spaces, associated with the Laplace-Bessel differential operator

We consider the generalized shift operator, associated with the LaplaceBessel differential operator ∆B =∑ni=1∂2∂x2i+∑ki=1γixi∂∂xi. The maximal commutators Mb,γ, associated with the generalized shift operator are investigated. At first, we prove that the maximal commutators is bounded from the modified Morrey space Lep,λ,φ1,γ(Rnk,+) to Lep,λ,φ1,γ(R n k,+) for all 1 < p < ∞, b ∈ BMOγ(Rnk,+), (φ1, φ2) ∈ Aep,γ(Rnk,+).

Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces Gℳωp,θ,φ(ℝn)

Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces Gℳωp,θ,φ(ℝn)

Two-weighted inequalities for maximal operator in generalized weighted Morrey spaces on spaces of homogeneous type

In this paper we give a characterization of two-weighted inequalities for maximal operators in generalized weighted Morrey spaces on spaces of homogeneous type Mp,φω (X). We prove the boundedness of maximal operator M from the spaces Mp,φ1ωδ 1(X)to the spaces Mp,φ2ωδ2(X), where 1 < p < ∞, 0 < δ < 1 and (ω1, ω2) ∈ Aep(X).

Bump conditions and two-weight inequalities for commutators of fractional integrals

This paper gives new two-weight bump conditions for sparse operators related to iterated commutators of fractional integrals. As applications, two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtain

Two-Weight Inequalities for Multilinear Commutators in Product Spaces

This article is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.

A characterization of two-weighted inequalities for maximal, singular operators andtheir commutators in generalized weighted Morrey spaces

In this paper we give a characterization of two-weighted inequalities for maximal, singular operators and their commutators in generalized weighted Morrey spaces $\mathcal{M}^{p,\varphi}_{\omega}(\mathbb{R}^n)$. We prove the boundedness of maximal operator $M$ and maximal commutators $[M,b]$ from the spaces $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to the spaces $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$, where $1< p<\infty$, $0<\delta<1$ and $(\omega_1,\omega_2)\in \widetilde{A}_{p}(\mathbb{R}^n)$. We also prove the boundedness of the Calderón--Zygmund singular operators $T$ and their commutators $[b,T]$ from $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$. Finally we give generalized weighted Morrey a priori estimates as applications of our results.

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.

Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators

Let \(S_{\alpha }\) be the multilinear square function defined on the cone with aperture \(\alpha \ge 1\). In this paper, we investigate several kinds of weighted norm inequalities for \(S_{\alpha }\). We first obtain a sharp weighted estimate in terms of aperture \(\alpha \) and \(\vec {w} \in A_{\vec {p}}\). By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman–Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer’s conjecture, for which a Coifman–Fefferman inequality with the precise \(A_{\infty }\) norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood–Paley \(g^*_{\lambda }\) function. Some results are new even in the linear case.

A NOTE ON TWO WEIGHT INEQUALITIES FOR THE DYADIC PARAPRODUCT

A NOTE ON TWO WEIGHT INEQUALITIES FOR THE DYADIC PARAPRODUCT

Necessary conditions for two-weight inequalities for singular integral operators

We prove necessary conditions on pairs $(\mu ,\nu )$ of measures for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p \lt \infty $, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Ste

Weighted Alpert Wavelets

In this paper we construct a wavelet basis in $$L^{2}({\mathbb {R}}^{n};\mu )$$ possessing vanishing moments of a fixed order for a general locally finite positive Borel measure $$\mu $$ . The approach is based on a clever construction of Alpert in the case of the Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón–Zygmund operator on $${\mathbb {R}}$$ and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from $$L^{2}({\mathbb {R}};\sigma )$$ to $$L^{2}({\mathbb {R}};\omega )$$ if certain stronger testing conditions hold on polynomials. An example is provided showing that this conjecture is logically different from existing results in the literature.

English

Let − (n + 1) < m ⩽ − (n + 1) (1 − ϱ) and let $${T_a} \in {\cal L}_{ϱ}^m,\delta $$ be pseudo-differential operators with symbols a(x, ξ) ∈ ℝn × ℝn, where 0 < ϱ ⩽ 1,0 ⩽ δ < 1 and δ ⩽ ϱ. Let μ, λ be weights in Muckenhoupt classes Ap, ν = (μλ−1)1/p for some 1 < p < ∞. We establish a two-weight inequality for commutators generated by pseudo-differential operators Ta with weighted BMO functions b ∈ BMOν, namely, the commutator [b, Ta] is bounded from Lp(μ) into Lp(λ). Furthermore, the range of m can be extended to the whole m ⩽ − (n + 1)(1 − ϱ).

Two-weighted inequalities for maximal operators related to Schrödinger differential operator

Abstract We introduce classes of pairs of weights closely related to Schrödinger operators, which allow us to get two-weight boundedness results for the Schrödinger fractional integral and its commutators. The techniques applied in the proofs strongly rely on one hand, boundedness results in the setting of finite measure spaces of homogeneous type and, on the other hand, Fefferman–Stein-type inequalities that connect maximal operators naturally associated to Schrödinger operators.

The two-weight inequality for the Poisson operator in the Bessel setting

Fix λ>0. Consider the Bessel operator Δλ:=−d2dx2−2λxddx on R+:=(0,∞) and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator Pt[λ]=e−tΔλ in this Bessel setting. In particular, we prove that for a measure μ on R+,+2:=(0,∞)×(0,∞) and σ on R+:‖Pσ[λ](f)‖L2(R+,+2;μ)≲‖f‖L2(R+;σ), if and only if testing conditions hold for the Poisson operator and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in the testing conditions.

Commutators of certain fractional type operators with Hörmander conditions, one-weighted and two-weighted inequalities

Fil: Ibanez Firnkorn, Gonzalo Hugo. Universidad Nacional de Cordoba; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina

Two-weighted inequalities for the fractional integral associated to the Schrödinger operator

Two-weighted inequalities for the fractional integral associated to the Schrödinger operator