Multiresolution Approximations and Wavelet Bases of WeightedLpSpaces

We study boundedness and convergence on L ^p (R ⁿ ,d) of the projection operators P _j given by MRA structures with non-necessarily compactly supported scaling function. As a consequence, we prove that if w is a locally integrable function such that w ^-(1/p–1)(x) (1+|x|)^-N is integrable for some N > 0, then the Muckenhoupt A _p condition is necessary and sufficient for the associated wavelet system to be an unconditional basis for the weighted space L ^p (R ⁿ ,w(x) dx), 1 < p < .     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

     

The ρ-variation of the Hermitian Riesz transform

In this paper we prove the behaviour in weighted Lp spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps LP（R, w（x）dx） into itself when w is a weight in the Ap class for 1 〈 p 〈 ∞. For p = 1 we get weak type for the A1 class. Weighted estimated are also obtained in the extreme case p = ∞.     

Singular integrals in the Cesàro sense

The existence of the singular integral ∫K(x, y)f(y)dy associated to a Calderón-Zygmund kernel where the integral is understood in the principal value sense TF(x)=lim_ε→0+∫_|x−y|>εK(x, y)f(y)dy has been well studied. In this paper we study the existence of the above integral in the Cesàro-α sense. More precisely, we study the existence of

On the $$A_{\infty }$$ conditions for general bases

We discuss several characterizations of the \(A_\infty \) class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual \(A_\infty \) weights.     

Differential Transforms in Weighted Spaces

We extend the results by Jones and Rosenblatt about the series of the differences of differentiation operators along lacunary sequences to BMO and to the setting of weighted L^p spaces. We use a different approach which allows to establish that the one-sided Sawyer A_p weights are the natural ones to study the boundedness and convergence of that series in weighted spaces.     

Boolean Monomial Dynamical Systems

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For systems that can be described by monomials (including Boolean AND systems), one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system and can be verified in polynomial time.Received February 2, 2004     

A counting problem in ergodic theory and extrapolation for one-sided weights

The purpose of this paper is to prove that, given a dynamical system (X,M,μ, τ) and 0 < q < 1, the Lorentz spaces L^1,q(μ) satisfy the so-called Return Times Property for the Tail, contrary to what happens in the case q = 1. In fact, we consider a more general case than in previous papers since we work with a σ-finite measure μ and a transformation τ which is only Cesàro bounded. The proof uses the extrapolation theory of Rubio de Francia for one-sided weights. These results are of independent interest and can be applied to many other situations.