Fefferman Stein Type Inequalities Research Articles

Fefferman-Stein type inequalities for maximal function of Choquet integrals associated with weighted capacities

Fefferman-Stein type inequalities for maximal function of Choquet integrals associated with weighted capacities

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.

Block Decomposition and Weighted Hausdorff Content

AbstractBlock decomposition of$L^{p}$spaces with weighted Hausdorff content is established for$0<p\leqslant 1$and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

The Fefferman–Stein Type Inequalities for Strong Fractional Maximal Operators

We prove the Fefferman–Stein type inequalities for strong fractional maximal operators by additional compositions of certain maximal operators instead of using the strong Muckenhoupt weight. With an arbitrary weight, in $${{\mathbb {R}}}^2$$ , we establish an endpoint estimate and in $${{\mathbb {R}}}^n$$ , $$n\ge 2$$ , we give a weak (p, p) type estimate for $$p>1$$ . We also investigate the case $$p=1$$ in higher dimensions.

Weighted Norm Inequalities for Rough Singular Integral Operators

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_\Omega$ with $\Omega\in L^\infty(\mathbb{S}^{n-1})$ and the Bochner-Riesz multiplier at the critical index $B_{(n-1)/2}$. More precisely, we prove qualitative and quantitative versions of Coifman-Fefferman type inequalities and their vector-valued extensions, weighted $A_p-A_\infty$ strong and weak type inequalities for $1<p<\infty$, and $A_1-A_\infty$ type weak $(1,1)$ estimates. Moreover, Fefferman-Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 90's. As a corollary, we obtain the weighted $A_1-A_\infty$ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function $\Omega\in L^q(\mathbb{S}^{n-1})$, $1<q<\infty$, and provide Fefferman-Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde-Alonso et.al. [CACDPO], results by the first author in [L], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $A_{\infty}$ weights [CMP,CGMP] and ideas contained in previous works by A. Seeger in [S] and D. Fan and S. Sato [FS].

Fractional maximal operators with weighted Hausdorff content

Let $n\ge 2$ be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator ${\mathfrak M}{\alpha}$ of order $\alpha$, $0\le\alpha<n$, on the weighted Choquet-Lorentz space $L^{p,q}(H_{w}^{d})$, where the weight $w$ is arbitrary and the underlying measure is the weighted $d$-dimensional Hausdorff content $H^{d}_{w}$, $0<d\le n$. Concerning a dependence of two parameters $\alpha$ and $d$, we establish a general form of the Fefferman-Stein type inequalities for ${\mathfrak M}_{\alpha}$. Our results contain the works of Adams, \cite{Ad} and of Orobitg and Verdera \cite{OV} as the special cases. Our results also imply the Tang result \cite{Ta}, if we assume the weight $w$ is in the Muckenhoupt $A_{1}$-class.

The Fefferman–Stein Type Inequalities for Strong and Directional Maximal Operators in the Plane

AbstractThe Fefferman–Stein type inequalities for strongmaximal operators and directional maximal operators are verified with an additional composition of the Hardy–Littlewood maximal operator in the plane.

Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

Abstract We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

Maximal estimates for the Cesàro means of weighted orthogonal polynomial expansions on the unit sphere

Estimates of the maximal Cesàro means at the “critical index” are established for the orthogonal polynomial expansions (OPEs) with respect to the weight function ∏j=1d|xj|2κj on the unit sphere of Rd. These estimates allow us to improve several known results in this area, including the almost everywhere (a.e.) convergence of the Cesàro means at the “critical index”, the sufficient conditions for the Marcinkiewitcz multiplier theorem, and a Fefferman–Stein type inequality for the Cesàro operators. In addition, several similar results for the weighted OPEs on the unit ball and on the simplex are deduced from the corresponding weighted results on the sphere.

The spherical maximal operator on radial functions

We use simple one-dimensional operators to bound pointwise the spherical maximal operator acting on radial functions. With this bounds we obtain weighted inequalities, which are sharp for power weights. We also discuss boundedness results on Morrey spaces and give a Fefferman–Stein type inequality. The bounds of the two-dimensional case can be use for the universal maximal operator as well.

Weighted Inequalities for Multilinear Potential Operators and their Commutators

We prove weighted strong inequalities for the multilinear potential operator \({\cal T}_{\phi}\) and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator \(\mathcal{M}_{\varphi,L^{B}}\) associated to an essentially nondecreasing function φ and to the Orlicz space LB for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator \({\cal T}_{\phi}\).

Dyadic sets, maximal functions and applications on ax + b-groups

Let S be the Lie group \({{\mathbb R}^n\ltimes {\mathbb R}}\), where \({{\mathbb R}}\) acts on \({{\mathbb R}^n}\) by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H 1 and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].

Commutators of singular integrals on spaces of homogeneous type

In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on L p (w) when w belongs to the Muckenhoupt’s class A p , p > 1. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff’s range

Let K δ , 0 < δ << 1, be the Kakeya (Nikodym) maximal operator defined as the supremum of averages over tubes of eccentricity δ. The (so-called) Fefferman-Stein type inequality: ∥K δ f∥ Lp(Rd(Rd,w) ≤ C(1/δ) d/p-1 (log(1/δ)) α ∥f∥ Lp(Rd,Kδw) is shown in the range 1 < p < (d + 2)/2, where C and a are some constants depending only on p and the dimension d and w is a weight. The result is a sharp bound up to log(1/δ)-factors.