141 research outputs found

    On weighted norm inequalities for the Carleson and Walsh-Carleson operators

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    We prove Lp(w)L^p(w) bounds for the Carleson operator C{\mathcal C}, its lacunary version Clac\mathcal C_{lac}, and its analogue for the Walsh series \W in terms of the AqA_q constants [w]Aq[w]_{A_q} for 1≤q≤p1\le q\le p. In particular, we show that, exactly as for the Hilbert transform, ∥C∥Lp(w)\|{\mathcal C}\|_{L^p(w)} is bounded linearly by [w]Aq[w]_{A_q} for 1≤q<p1\le q<p. We also obtain Lp(w)L^p(w) bounds in terms of [w]Ap[w]_{A_p}, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near L1L^1. Our approach works in the general context of maximally modulated Calder\'on-Zygmund operators.Comment: A major revision of arXiv: 1310.3352. In particular, the main result is proved under a different assumption, and applications to the lacunary Carleson operator and to the Walsh-Carleson operator are give

    On the Convergence of Lacunary Walsh-Fourier Series

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    We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality for lacunary subsequences of integers, and (4) the extrapolation method of Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.Comment: 18 pages. v2: Several typos corrected. Final version of the paper, accepted to LM

    A Fefferman-Stein inequality for the Carleson operator

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    We provide a Fefferman-Stein type weighted inequality for maximally modulated Calder\'on-Zygmund operators that satisfy \textit{a priori} weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of P\'erez. Applying it to the Hilbert transform we obtain the corresponding Fefferman-Stein inequality for the Carleson operator C\mathcal{C}, that is C:Lp(M⌊p⌋+1w)→Lp(w)\mathcal{C}: L^p(M^{\lfloor p \rfloor +1}w) \to L^p(w) for any 1<p<∞1<p<\infty and any weight function ww, with bound independent of ww. We also provide a maximal-multiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by P\'erez.Comment: Revised version. To appear in Rev. Mat. Ibe

    Weighted multiple interpolation and the control of perturbed semigroup systems

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    In this paper the controllabillity and admissibility of perturbed semigroup systems are studied, using tools from the theory of interpolation and Carleson measures. In addition, there are new results on the perturbation of Carleson measures and on the weighted interpolation of functions and their derivatives in Hardy spaces, which are of interest in their own right

    Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

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    Let HωfH_\omega f be the Fourier restriction of f∈L2(R)f\in L^2(\mathbb{R}) to an interval ω⊂R\omega\subset \mathbb{R}. If Ω\Omega is an arbitrary collection of pairwise disjoint intervals, the square function of {Hωf:ω∈Ω}\{H_\omega f: \omega \in \Omega\} is termed the Rubio de Francia square function TΩT^\Omega. This article proves a pointwise bound for TΩfT^\Omega f by a sparse operator involving local L2L^2-averages. A pointwise bound for the smooth version of TΩT^\Omega by a sparse square function is also proved. These pointwise localization principles lead to quantified Lp(w)L^p(w), p>2p>2 and weak Lp(w)L^p(w), p≥2p\geq 2 norm inequalities for TΩT^\Omega. In particular, the obtained weak Lp(w)L^p(w) norm bounds are new for p≥2p\geq 2 and sharp for p>2p>2. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, local orthogonality and very elementary time-frequency analysis techniques. The paper also contains two results related to the outstanding conjecture that TΩT^\Omega is bounded on L2(w)L^2(w) if and only if w∈A1w\in A_1. The conjecture is verified for radially decreasing even A1A_1 weights, and in full generality for the Walsh group analogue of TΩT^\Omega.Comment: 27 pages; submitted. v2: typos fixed and more details provide

    Sparse domination via the helicoidal method

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    Using exclusively the localized estimates upon which the helicoidal method was built, we show how sparse estimates can also be obtained. This approach yields a sparse domination for multiple vector-valued extensions of operators as well. We illustrate these ideas for an nn-linear Fourier multiplier whose symbol is singular along a kk-dimensional subspace of Γ={ξ1+…+ξn+1=0}\Gamma=\lbrace \xi_1+\ldots+\xi_{n+1}=0 \rbrace, where k<n+12k<\dfrac{n+1}{2}, and for the variational Carleson operator.Comment: 60 page
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