Carleson’s Theorem with Quadratic Phase Functions

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

L. Carleson’s celebrated theorem of 1965 [14] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. We give a proof of this fact, in particular the proof of Lacey and Thiele [47], as it can be presented in brief self contained manner, and a number of related results can be seen by variants of the same argument. We survy some of these variants, complements...

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...

On the quadratic support of strongly convex functions

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

A Variation Norm Carleson Theorem

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 equiv...

Carleson’s Theorem: Proof, Complements, Variations