Related papers: On the growth of vector-valued Fourier series

We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and…

We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hyt\"onen and Lacey. Given q…

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge…

We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex…

We prove $L^p$-boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate of convergence of partial Fourier integrals of vector-valued functions. Our…

Carleson's Theorem asserts the pointwise convergence of Fourier series of square integrable functions. We give a complete proof, following joint work of the author and C. Thiele. Over 20 exercises are also detailed. We also discuss the…

We remark that a dyadic version of the Carleson embedding theorem for the Bergman space extends to vector-valued functions and operator-valued measures. This is in contrast to a result by Nazarov, Treil, Volberg in the context of the Hardy…

Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of…

The celebrated Carleson-Hunt theorem gives pointwise almost everywhere convergence for the Fourier series of a function in $L^p(\mathbb T)$. R. Oberlin, A. Seeger, T. Tao, C. Thiele and J. Wright (OSTTW) strengthened this theorem by proving…

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence…

Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach…

This paper is meant to be a gentle introduction to Carleson's Theorem on pointwise convergence of Fourier series.

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it…

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm…

In this paper we provide a general construction of a quaternionic Banach space of slice regular functions from a given Banach space of holomorphic functions, which we call its quaternionic lift. To the best of our knowledge, this…

A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $L^{p}-$multipliers, a rather general inequality…

Hyt\"onen, McIntosh and Portal (J. Funct. Anal., 2008) proved two vector-valued generalizations of the classical Carleson embedding theorem, both of them requiring the boundedness of a new vector-valued maximal operator, and the other one…

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an $m$-(sub)linear operator…

A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions (not radial in general) is considered…

In a recent paper by the authors, growth properties of the Fourier transform on Euclidean space and the Helgason Fourier transform on rank one symmetric spaces of non-compact type were proved and expressed in terms of of a modulus of…