Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

Sharp general local estimates for dyadic-like maximal operators and related Bellman functions

For each 1?q<p we precisely evaluate the main Bellman functions associated with the local Lp→Lq estimates of the dyadic maximal operator on Rn. Actually we do that in the more general setting of tree-like maximal operators and with respect to general convex and increasing growth functions. We prove that these Bellman functions equal to analogous extremal problems for the Hardy operator which can be viewed as a symmetrization principle for such operators. Under certain mild conditions on the growth functions we show that for the latter extremals exist (although for the original Bellman functions do not) and analyzing them we give a determination of the corresponding Bellman function.     

Criteria of strong type two-weighted inequalities for fractional maximal functions

A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.     

On the uniqueness of maximal functions

The uniqueness theorem for the one-sided maximal operator has been proved.     

Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces

Given and a Young function η, we consider the generalized fractional maximal operator defined by where the supremum is taken over every ball B contained in . In this article, we give necessary and sufficient Dini type conditions on the functions , and η such that is bounded from the Orlicz space into the Orlicz space . We also present a version of this result for open subsets of with finite measure. Both results generalize those contained in 6 and 14 when , respectively. As a consequence, we obtain a characterization of the functions involved in the boundedness of the higher order commutators of the fractional integral operator with BMO symbols. Moreover, we give sufficient conditions that guarantee the continuity in Orlicz spaces of a large class of fractional integral operators of convolution type with less regular kernels and their commutators, which are controlled by .     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

Weighted norm inequalities for geometric fractional maximal operators

For 0 < let Tf denote one of the operators

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given $ we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from into We approach the poblem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1,1) for the “global” part of the centered maximal function. Received May 18, 1999 / Revised December 9, 1999 Published online July 20, 2000     

Extremal problems related to maximal dyadic-like operators

We obtain sharp estimates for the localized distribution function of the dyadic maximal function Md?, when ? belongs to Lp,∞. Using this we obtain sharp estimates for the quasi-norm of Md? in Lp,∞ given the localized L¹-norm and certain weak Lp-conditions.     

Weak type inequalities of maximal Hankel convolution operators

In this paper we characterize weak type (1,1) inequalities for Hankel convolution operators by means of discrete methods. Partially supported by DGICYT Grant PB 94-0591 (Spain).     

Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

We prove L^p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, , when the parameter α is greater than -1. Namely, the maximal operator is of strong type (p,p) if p>1 and , when -1<α<0. If α?0 there is strong type for 1<p?∞. The behavior at the end points is studied in detail.     

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Applications of generalized perron trees to maximal functions and density bases

In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate L^p-functions or give Hardy-Littlewood type maximal functions which are bounded on L^p, p>1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et al. [6] regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above. Acknowledgements and Notes. This research was partially supported by NSERC.     

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator _Mα$M_{\alpha }$, 0<α<n$0<\alpha <n$, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted _Lp$L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.     

Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces

The necessary and sufficient conditions are derived in order that a strong type weighted inequality be fulfilled in Orlicz classes for scalar and vector-valued maximal functions defined on homogeneous type spaces. A weak type problem with weights is solved for vector-valued maximal functions.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I

Criteria of various weak and strong type weighted inequalities are established for singular integrals and maximal functions defined on homogeneous type spaces in the Orlicz classes.     

Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. II

This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [1].