2 00 3 Carleson ’ s Theorem : Proof , Complements , Variations

ar X iv : m at h / 04 05 40 1 v 1 [ m at h . G N ] 2 1 M ay 2 00 4 VARIATIONS ON KURATOWSKI ’ S 14 - SET THEOREM

Kuratowski’s 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. In this article we consider the analogous questions for any possible subcollection of the operations {closure, complement, interior, intersection, union}, and any number of initially given sets. We use the al...

L Bounds for a Maximal Dyadic Sum Operator

The authors prove L bounds in the range 1 < p < ∞ for a maximal dyadic sum operator on R. This maximal operator provides a discrete multidimensional model of Carleson’s operator. Its boundedness is obtained by a simple twist of the proof of Carleson’s theorem given by Lacey and Thiele [6] adapted in higher dimensions [8]. In dimension one, the L boundedness of this maximal dyadic sum implies in...

The characterization of the Carleson measures for analytic Besov spaces: a simple proof

In this note we return on the characterization of the Carleson measures for the class of weighted analytic Besov spaces Bp(ρ) considered in [ARS1], Theorem 1. Since our first paper on this topic, we have found shorter or better proofs for some of the intermediate results, which were used in proving various extensions and generalizations of the characterization theorem, [A][ARS2][ARS3]. Here we ...

Carleson’s Theorem: Proof, Complements, Variations

Wiener-wintner for Hilbert Transform

We prove the following extension of the Wiener–Wintner Theorem and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows (X,μ, Tt) and f ∈ L(X,μ), there is a set Xf ⊂ X of probability one, so that for all x ∈ Xf we have lim s↓0 ∫ s<|t|<1/s e f(Tt x) dt t exists for all θ. The proof is by way of establishing an appropriate oscillation inequality which ...