María Cristina Pereyra

Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r 1 p 1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

Data-driven and optimal denoising of a signal and recovery of its derivative using multiwavelets

Multiwavelets are relative newcomers into the world of wavelets. Thus, it has not been a surprise that the used methods of denoising are modified universal thresholding procedures developed for uniwavelets. On the other hand, the specific of a multiwavelet discrete transform is that typical errors are not identically distributed and correlated, whereas the theory of the universal thresholding is based on the assumption of identically distributed and independent normal errors. Thus, we suggest an alternative denoising procedure based on the Efromovich-Pinsker algorithm. We show that this procedure is optimal over a wide class of noise distributions. Moreover, together with a new cristina class of biorthogonal multiwavelets, which is introduced in this paper, the procedure implies an optimal method for recovering the derivative of a noisy signal. A Monte Carlo study supports these conclusions.

SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES

We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator (b;T ) with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = (b;T k 1 b ) will obey a bound that is a power (k + 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding L p (w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the Ap constant of the weight for all 1 < p <1, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.

Haar multipliers, paraproducts, and weighted inequalities

In this paper we present a brief survey on Haar multipliers, dyadic paraproducts, and recent results on their applications to deduce scalar and vector valued weighted inequalities. We present a new proof of the boundedness of a Haar multiplier in Lp(ℝ). The proof is based on a stopping time argument suggested by P. W. Jones for the case p = 2, that it is adapted to the case 1 < p < ∞ using an new version of Cotlar’s Lemma for Lp. We then prove some weighted inequalities for simple dyadic operators.

Wavelets, Their Friends, and What They Can Do for You

These notes were created by Maŕia Cristina Pereyra for the short course Wavelets: Theory and Applications, at the I Panamerican Advanced Studies Institute in Computational Science and Engineering (PASI), Universidad Nacional de Cordoba, Cordoba, Argentina, June 24–July 5, 2002. They were modified and extended by Martin J. Mohlenkamp for the short course Wavelets and Partial Differential Equations, at the II Panamerican Advanced Studies Institute in Computational Science and Engineering, Universidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras, June 14–18, 2004. They are being slightly modified and actualized once more by by Maŕia Cristina Pereyra for the short course From Fourier to Wavelets at the III Panamerican Advanced Studies Institute in Computational Science and Engineering (PASI), Universidad Tecnologica de la Mixteca, Huajuapan de Leon, Oaxaca, Mexico, July 16–21, 2006. We would like to thank the organizers for the invitations to participate in these institutes. We hope that these notes will help you to begin your adventures with wavelets.

On the two weights problem for the Hilbert transform

In this paper, we prove sufficient conditions on pairs of weights (u,v) (scalar, matrix or operator valued) so that the Hilbert transform H f(x) = p.v. ? [f(y) / x - y] dy, is bounded from L2(u) to L2(v).

Harmonic analysis : from Fourier to wavelets

In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fouriers study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.

Haar multipliers meet Bellman functions

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in L2(R) of the Haar multipliers T t w on the corresponding RHd p or Ap characteristic of the weight w, for t = 1,±1/2. These results can be viewed as particular cases of estimates on homogeneous spaces L2(vdσ), for σ a doubling positive measure and v ∈ A2(dσ), of the weighted dyadic square function Sd σ. We show that the operator norms of such square functions in L2(vdσ) are bounded by a linear function of the A2(dσ) characteristic of the weight v, where the constant depends only on the doubling constant of the measure σ. We also show an inverse estimate for Sd σ. Both results are known when dσ = dx. We deduce both estimates from an estimate for the Haar multiplier (T σ v ) 1/2 on L2(dσ) when v ∈ A2(dσ), which mirrors the estimate for T 1/2 w in L2(R) when w ∈ A2. The estimate for the Haar multiplier adapted to the σ measure, (T σ v ) 1/2, is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all σ, since the particular case dσ = dx, v = w, correspond to the estimates for the Haar multipliers T 1/2 w proven to be sharp.

On the resolvents of dyadic paraproducts

We consider the boundedness of certain singular integral operators that arose in the study of Sobolev spaces on Lipschitz curves, [P1]. The standard theory available (David and Journes T1 Theorem, for instance; see [D]) does not apply to this case becuase the operators are not necessarily Calderon-Zygmund operators, [Ch]. One of these operators gives an explicit formula for the resolvent at ? = 1 of the dyadic paraproduct, [Ch].

Sharp bounds for

We show that if a weight