Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.     

Wavelet expansions forBMOρ (w)‐functions

We give necessary and sufficient conditions on the wavelet coefficients of a function for being a member of some BMO_φ (w) space. We achieve this characterization for a wide variety of wavelet systems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Look at BMOϕ(ω) through Carleson Measures

As Fefferman and Stein showed, there is a tight connection between Carleson measures and BMO functions. In this work we extend this type of results to the more general scope of the BMO_ϕ(ω) spaces. As a byproduct a weighted version of the Triebel-Lizorkin space is introduced, which turns out to be isomorphic to BMO(ω) as in the unweighted case.     

Global W 2, p estimates for nondivergence elliptic operators with potentials satisfying a reverse H?lder condition

In this article, we give some a priori ${L^{p}(\mathbb{R}^{n})}$ estimates for elliptic operators in nondivergence form with VMO coefficients and a potential V satisfying an appropriate reverse H?lder condition, generalizing previous results due to Chiarenza?CFrasca?CLongo to the scope of Schr?dinger-type operators. In particular, our class of potentials includes unbounded functions such as nonnegative polynomials. We apply such a priori estimates to derive some global existence and uniqueness results under some additional assumptions on V.     

BMO spaces related to Laguerre semigroups

For the system of Laguerre functions we define a suitable BMO space from the atomic version of the Hardy space considered by Dziubański in 7 , where is the maximal operator of the heat semigroup associated to that Laguerre system. We prove boundedness of over a weighted version of that BMO, and we extend such result to other systems of Laguerre functions, namely and . To do that, we work with a more general family of weighted BMO‐like spaces that includes those associated to all of the above mentioned Laguerre systems. In this setting, we prove that the local versions of the Hardy‐Littlewood and the heat‐diffusion maximal operators turn to be bounded over such family of spaces for weights. This result plays a decisive role in proving the boundedness of Laguerre semigroup maximal operators.     

Weighted inequalities and a.e. convergence for Poisson integrals in light-cones

We show that the Poisson maximal operator for the tube over the light-cone, P ^*, is bounded in the weighted space L ^p (w) if and only if the weight w(x) belongs to the Muckenhoupt class A _p . We also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v(x) for which P ^* is bounded from L ^p (v) into L ^p (u), for some other weight u(x) > 0. Some applications to a.e. restricted convergence of Poisson integrals are given.     

Some estimates for maximal functions on Köthe function spaces

IfA is an invertiblen×n matrix with entries in the finite field F_q, letT _n (A) be its minimum period or exponent, i.e. its order as an element of the general linear group GL(n,q). The main result is, roughly, that $T_n (A) = q^{n - } (log n)^{2 + 0(1)} $ for almost everyA.     

Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure. In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case. The first and third authors were partially supported by CONICET (Argentina) and Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral. The second author was partially supported by the European Commission via the TMR network “Harmonic Analysis”.