Differential Transforms in Weighted Spaces

Bernardis, A. L.; Lorente, María; Martín-Reyes, F. J.; Martínez, Minerva; Torre, A. de la; Torrea, José L.

doi:10.1007/s00041-005-5064-z

Cited by 27 publications

(32 citation statements)

References 8 publications

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“…Some singular integrals with kernel supported in (−∞, 0) appear in the study of the convergence of the ergodic averages associated to a measurable flow or to a measure-preserving transformation [27], [5]. We are going to state the problem in the real line which is enough to see how these kernels appear.…”

Section: Further Examples Of One-sided Singular Kernelsmentioning

confidence: 98%

Weights for One–Sided Operators

Martín-Reyes¹,

Ortega²,

Torre³

2009

Recent Developments in Real and Harmonic Analysis

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Summary. We present a survey about weights for one-sided operators, one of the areas in which Carlos Segovia made significant contributions. The classical DunfordSchwartz ergodic theorem can be considered as the first result about weights for the one-sided Hardy-Littlewood maximal operator. From this starting point, we study weighted inequalities for one-sided operators: positive operators like the Hardy averaging operator, the one-sided Hardy-Littlewood maximal operator, singular approximations of the identity, one-sided singular integrals. We end with applications to ergodic theory and with some recent results in dimensions greater than 1.

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Section: Further Examples Of One-sided Singular Kernelsmentioning

confidence: 98%

Weights for One–Sided Operators

Martín-Reyes¹,

Ortega²,

Torre³

2009

Recent Developments in Real and Harmonic Analysis

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“…Once we have Corollary 6.3 and Theorem 1.6 the proof of Theorem 1.7 is straightforward (see for instance [1]). We include it for the sake of completeness.…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

“…In [6], Jones and Rosenblatt work in Lebesgue spaces L p (dx) while in [1] the results were extended to weighted Lebesgue spaces.…”

Section: Introductionmentioning

confidence: 99%

Differential transforms of Cesàro averages in weighted spaces

Bernardis

Martín-Reyes²

2008

Publicacions Matemàtiques

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In this paper we obtain convergence results for the series of differences of Cesàro averages along lacunary sequences in the setting of weighted L p -spaces. These results give some information about how the Cesàro averages converge. The paper extends results of an earlier work by R. L. Jones and J. Rosenblatt. The operators considered are essentially convolution operators given by kernels more singular than the ones in the article by Jones and Rosenblatt.

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“…In [2], using a different approach, the result is extended to the context of L p (w dx), where w is a one-sided A p weight, and the index k takes values in Z. Recently C. Demeter [4] has studied a similar operator but in his case f is not periodic and D k f (x) = 2 k+1 2 k f (x − y)ϕ(y) dy, where ϕ is a function defined on (0, ∞) such that:…”

Section: Introductionmentioning

confidence: 99%

Weighted and Ergodic Theorems for Series of Differences of Convolutions

Lorente

Torre

2008

J Fourier Anal Appl

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We extend some results of C. Demeter about the boundedness of series of difference of convolutions to the setting of one-sided A p weights. We need to use a different approach. To be precise, we prove that the kernel of the associated operator satisfies a generalized Hörmander condition. The weighted inequalities allow us to transfer the result to the ergodic case, when the operator is induced by a mean bounded, invertible, positive groups.

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Differential Transforms in Weighted Spaces

Cited by 27 publications

References 8 publications

Weights for One–Sided Operators

Weights for One–Sided Operators

Differential transforms of Cesàro averages in weighted spaces

Weighted and Ergodic Theorems for Series of Differences of Convolutions

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