Related papers: Carleson's Theorem: Proof, Complements, Variations

This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to,…

In this article, we prove a quantitative version of Carleson's $\varepsilon^2$ conjecture in higher dimension: we characterise those Ahlfors-David regular domains in $\mathbb{R}^{n+1}$ for which the Carleson's coefficients satisfy the…

It is proved that the complex double Fourier series of an integrable function $f(x,y)$ with coefficients $\{c_{jk}\}$ satisfying certain conditions, will converge in $L^{1}$-norm. The conditions used here are the combinations of Tauberian…

This work proves pointwise convergence of the truncated Fourier double integral of non-Lebesgue integrable bounded variation functions. This leads to the Dirichlet-Jordan theorem proof for non-Lebesgue integrable functions, which has not…

Carleson's corona theorem is used to obtain two results on cyclicity of singular inner functions in weighted Bergman-type spaces on the unit disk. Our method proof requires no regularity conditions on the weights.

Following Semmes and Zinsmeister, we continue the study of Carleson measures and their invariance under pull-back and push-forward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show…

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of…

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds…

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$…

We consider the pointwise convergence problem for the solution of Schr\"odinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was…

We prove the $L^p$-boundedness, $1<p<\infty$, of the Polynomial Carleson operator in general dimension. This follows the author's resolution of the one dimensional case as well as the work of Zorin-Kranich on the higher dimensional case in…

We prove $L^p$ bounds in the range $1<p<\infty$ for a maximal dyadic sum operator on $\rn$. This maximal operator provides a discrete multidimensional model of Carleson's operator. Its boundedness is obtained by a simple twist of the proof…

We introduce a scale of weighted Carleson norms, which depend on an integrability parameter p, where p=2 corresponds to the classical Carleson measure condition. Relations between the weighed BMO norm of a vector-valued function f:R->X, and…

Lacey and Thiele have recently obtained a new proof of Carleson's theorem on almost everywhere convergence of Fourier series. This paper is a generalization of their techniques (known broadly as time-frequency analysis) to higher…

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. We also consider little Hankel operators on these Bergman…

Using Carleson measure theorem of weighted Bergman spaces, we provide a complete characterization of embedding theorem for Dirichlet type spaces. As an application, we study the Volterra integral operator and multipliers for Dirichlet type…

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…

The goal of this paper is to formalize the notion of The Compositional Integral in The Complex Plane. We prove a convergence theorem guaranteeing its existence. We prove an analogue of Cauchy's Integral Theorem--and suggest an approach at…

In this paper we prove a new version of Krein-Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [D. Alpay, F. Colombo, I. Sabadini, Krein-Langer factorization and related topics…