On the convergence of lacunary Walsh-Fourier series

Abstract

Abstract. We show that for function f on [0, 1], with |f|(log log + |f|)(log log log + |f|) dx < ∞, and lacunary subsequence of integers {n j }, it holds that S n j f −→ f a.e., where S m f is the m-th Walsh-Fourier partial sum of f. According to a result of Konyagin, the sharp integrability condition would not have the triple-log term in it. The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderón-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-Martín, which generalizes the work of Antonov and Arias-de-Reyna.

Endpoint bounds for the bilinear Hilbert transform

“…This motivates the study of endpoint estimates for the bilinear Hilbert transform at the boundary of the hexagon. Similar endpoint questions for Carleson's operator, the other stalwart of time-frequency analysis, have enjoyed some popularity as well in recent years [8,9,10,22,23].…”

“…It is our opinion that Proposition 3.2, or variants thereof, could be employed as well in the translation to the continuous case of the arguments of [8,9,10] Outline of the article. Sections 2 and 3 are concerned with the multi-frequency Calderón Zygmund decomposition in general.…”

“…The bad part b is such that the Carleson measure norms of the coefficients |〈b, ϕ J 〉| 2 , associated to collections ϕ (I ,ξ) of u-adapted wave packets with mean zero are simultaneously exponentially small in k. The constant C appearing in the statement can be taken equal to 10 3 a −1 , where a is the uniform adaptation rate of the ϕ (I ,ξ) 's. 1 Here, and in the remainder of the article, we adopt an L 2 normalization for our adapted functions.…”

Lacunary Walsh series in rearrangement invariant spaces

“…Let us see that the operators P n : X n → X n have uniformly bounded norm (in n). From (7), since T j = 1, we have for any f ∈ X n that…”

The pointwise convergence of Fourier Series (II). Strong L1 case for the lacunary Carleson operator

“…We say that Y is a C−space iff ∃ C 0 = C 0 (Y ) > 0 such that the Carleson operator defined by 9 Moreover, there exists a continuous function for which its Fourier Series is divergent at a given point (and in fact at any rational point). 10 Recall that the weak-L 1 quasinorm is given by f 1,∞ := sup λ>0 λ |{x | |f (x)| > λ}|.…”