43 research outputs found

    Maximal polynomial modulations of singular integrals

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    Let KK be a standard H\"older continuous Calder\'on--Zygmund kernel on Rd\mathbb{R}^{\mathbf{d}} whose truncations define L2L^2 bounded operators. We show that the maximal operator obtained by modulating KK by polynomial phases of a fixed degree is bounded on Lp(Rd)L^p(\mathbb{R}^{\mathbf{d}}) for 1<p<∞1 < p < \infty. This extends Sj\"olin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.Comment: v5: small corrections, more reference

    Cancellation for the simplex Hilbert transform

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    We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the multilinear Hilbert transform. Our main tool is the Hilbert space regularity lemma due to Gowers, which enables a very short proof.Comment: 8 page

    A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages

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    We show that a kk-linear pointwise ergodic theorem on an ergodic measure-preserving system implies a uniform kk-linear nilsequence Wiener-Wintner theorem on that system. The assumption is known to hold for arbitrary systems and k=2k=2 (due to Bourgain) and for distal systems and arbitrary kk (due to Huang, Shao, and Ye).Comment: v2: 4 pages, characterization of good weights for L^2 convergence added, uniformity seminorm in the main result correcte

    Typical operators admit common cyclic vectors

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    Given a countable dense subset D of an infinite-dimensional separable Hilbert space H the set of operators for which every vector in D except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology in the unit ball of L(H) multiplied by R>1 (resp. R>0)Comment: 7 pages, incorporating the referee's suggestion

    Kakeya-Brascamp-Lieb inequalities

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    We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp-Lieb inequalities.Comment: v4: revised following referee reports, 18 page

    Variation estimates for averages along primes and polynomials

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    We prove qq-variation estimates, q>2q>2, on β„“p\ell^{p} spaces for averages along primes (with 1<p<∞1<p<\infty) and polynomials (with ∣1pβˆ’12∣<12(d+1)\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}, where dd is the degree of the polynomial). This improves the pointwise ergodic theorems for these averages in the corresponding ranges of LpL^{p} spaces.Comment: v4: final, revised following referee's suggestion

    Intrinsic square functions with arbitrary aperture

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    We consider intrinsic square functions defined using (log-)Dini continuous test functions on spaces of homogeneous type. We prove weighted estimates with optimal (at least in the Euclidean case) dependence on the aperture of the cone used to define the square function and linear dependence on the (log-)Dini modulus of continuity.Comment: v3: now on spaces of homogeneous type, 12 page

    Cube spaces and the multiple term return times theorem

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    We give a new proof of Rudolph's multiple term return times theorem based on Host-Kra structure theory. Our approach provides characteristic factors for all terms, works for arbitrary tempered F{\o}lner sequences and also yields a multiple term Wiener-Wintner-type return times theorem for nilsequences.Comment: v2: 13 p., main result has been extended to tempered F{\o}lner sequence

    Return times theorem for amenable groups

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    We extend Bourgain's return times theorem to arbitrary locally compact second countable amenable groups. The proof is based on a version of the Bourgain--Furstenberg--Katznelson--Ornstein orthogonality criterion.Comment: v2: 8 p., small correction

    Decoupling for certain quadratic surfaces of low co-dimensions

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    We prove sharp β„“pLp\ell^{p}L^{p} decoupling inequalities for 22 quadratic forms in 44 variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.Comment: v2: revised following referee reports, 22 page
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