Weighted Bounds for Variational Walsh–Fourier Series

Do, Yen; Lacey, Michael T.

doi:10.1007/s00041-012-9231-8

Cited by 10 publications

(15 citation statements)

References 18 publications

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“…This fact has been recorded and well understood in several papers as for example in [11], [20], [15]. For the Banach space case that we are considering we will need the appropriate vector-valued extension proved by Pisier …”

Section: Vector-valued Lépingle Inequalitymentioning

confidence: 94%

“…In this section we linearize the variational norm of the partial Walsh-Fourier sums of a function f , using more or less standard arguments as in [28], [13], [11] and [20]. We reduce the statement of Theorem 1.2 to an analogous statement about some closely related linearized versions of the variational Carleson operator which are more amenable to the time-frequency analysis techniques and interpolation.…”

Section: Linearization Of the Variational Carleson Operatormentioning

confidence: 99%

“…We provide a step towards this direction by characterizing the UMD Banach spaces for which the finite tile-type assumption is true in terms of the boundedness of a variational Carleson operator. In particular we show that a variation norm version of Carleson's theorem for vector-valued Walsh-Fourier series, in the spirit of [11,20], is valid if and only if the UMD Banach space satisfies the finite tile-type hypothesis.…”

Section: Introductionmentioning

confidence: 96%

“…One of these notions is the variation norm Carleson theorem, proved in [20], which in particular shows the pointwise convergence of the Fourier series of f without having to resort to a dense subset where this convergence can be easily exhibited. Weighted versions of the corresponding results for Fourier and Walsh-Fourier series have recently appeared in [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…The question whether Carleson's theorem is valid on an arbitrary UMD Banach space remains open however. Our method and general strategy of proof is along the lines of [11,13,20], using the time-frequency analysis techniques and arguments introduced in [16].…”

Section: Introductionmentioning

confidence: 99%

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A variation norm Carleson theorem for vector-valued Walsh–Fourier series

Hytönen¹,

Lacey

Parissis³

2014

Rev. Mat. Iberoam.

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Abstract. We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in certain UMD Banach spaces, sharpening a recent result of Hytönen and Lacey. They proved the pointwise convergence of Walsh-Fourier series of X-valued functions under the qualitative hypothesis that X has some finite tile-type q < ∞, which holds in particular if X is intermediate between another UMD space and a Hilbert space. Here we relate the precise value of the tile-type index to the quantitative rate of convergence: tile-type q of X is "almost equivalent" to the L p -boundedness of the r-variation of the Walsh-Fourier sums of any function f ∈ L p ([0, 1); X) for all r > q and large enough p.

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Section: Vector-valued Lépingle Inequalitymentioning

confidence: 94%

Section: Linearization Of the Variational Carleson Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A variation norm Carleson theorem for vector-valued Walsh–Fourier series

Hytönen¹,

Lacey

Parissis³

2014

Rev. Mat. Iberoam.

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Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

Cruz-Uribe

Martell

2018

Math. Ann.

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We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights w 1 , w 2 for the bilinear Hilbert transform to satisfy weighted norm inequalities of the formThis improves the recent results of Culiuc et al. by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to 2 3 , the critical index from the unweighted theory of the bilinear Hilbert transform. Second, as an easy consequence of our method we obtain that the bilinear Hilbert transform satisfies some vector-valued inequalities with Muckenhoupt weights. This reproves and generalizes some of the vector-valued estimates obtained by Benea and Muscalu in the unweighted case. We also generalize recent results of Carando, et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators.

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Generalized Carleson embeddings into weighted outer measure spaces

Lewers

2022

Journal of Mathematical Analysis and Applications

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Weighted Bounds for Variational Walsh–Fourier Series

Cited by 10 publications

References 18 publications

A variation norm Carleson theorem for vector-valued Walsh–Fourier series

A variation norm Carleson theorem for vector-valued Walsh–Fourier series

Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

Generalized Carleson embeddings into weighted outer measure spaces

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