A variation norm Carleson theorem

By Richard Oberlin, Andreas Seeger, Terence Tao, C. Thiele, James Wright. 2009

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By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type:

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7ba2b77467c98607168892f90ca0debd55a16dda

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Citations in the corpus

Weighted variation inequalities for differential operators and singular integrals in higher dimensions (2013, background): “m. Bourgain’s work has been considerably improved by subsequent works and largely extended to many other operators in ergodic theory (see, for instance, [12, 14, 16]) and harmonic analysis (cf. e.g., [3, 5, 8, 13, 19, 20]). To state our results we need to recall some deﬁnitions. Let 1 ≤ q < ∞ and”
Strong q-variation inequalities for analytic semigroups (2011, background): “Many classical sequences of operators and semigroups have been proved to satisfy strong variational bounds, see in particular [9, 16, 17, 18, 25] and references therein.”
Anisotropic Hardy-Lorentz spaces and their applications (2015, background): “It is well known that, due to their fine structures, Lorentz spaces play an irreplaceable role in the study on various critical or endpoint analysis problems from many different research fields and there exists a lot of literatures on this subject, here we only mention several recent papers from har”
Multiple Vector Valued Inequalities via the Helicoidal Method (2015, background): “y/, estimates for the variational Carleson from [Oberlin et al. 2012] yield the same result whenever p > 4 3 .”
Some Problems in Harmonic Analysis (2017, methodology): “E-mail address: J.R.Wright@ed.ac.uk”
Lp theory for outer measures and two themes of Lennart Carleson united (2013, methodology): “For variational estimates of the Carleson operator, as in [16], one has variational wave packets”
Discrete maximal functions in higher dimensions and applications to ergodic theory (2014, background): “Variational inequalities in harmonic analysis and ergodic theory have been the subject of many recent articles, see especially [14, 15, 16, 30, 41] and the references given therein (see also [27, 29]).”