Carleson Operator Research Articles

Weak-[formula omitted] bounds for the Carleson and Walsh–Carleson operators

We prove a weak-Lp bound for the Walsh–Carleson operator for p near 1, improving on a theorem of Sjölin. We relate our result to the conjectures that the Walsh–Fourier and Fourier series of a function f∈LlogL(T) converge for almost every x∈T.

On the boundedness of the Carleson operator near $L^1$

Based on the tile discretization elaborated in [14], we develop a Calderón–Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of extrapolation techniques, we recover previously known results regarding the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series.

On a trilinear form related to the Carleson theorem

Demeter and Thiele introduced the two-dimensional bilinear Hilbert transform in Demeter and Thiele (2010) [3] and showed that the Carleson operator can be identified in particular instances of the corresponding trilinear form. Demeter considered the Walsh model of one such form in Demeter (2012) [2], in relation to the discussion of the Walsh–Carleson theorem. We prove boundedness of this trilinear form for a single triple of exponents at the boundary of the previously established range. For this purpose, we adapt the multilinear Bellman function technique from Kovač (2011) [5] to the one-and-a-half-dimensional time–frequency space.

On Generalized Carleson Operators of Periodic Wavelet Packet Expansions

Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied.

On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if Snf stands for the n-th partial Fourier sum of f and {nj}j⊂N is a lacunary sequence, then Snjf is a.e. pointwise convergent for any f∈LloglogL. In this paper we will show that ‖supj|Snj(f)|‖1,∞⩽C‖f‖1loglog(10+‖f‖∞‖f‖1). As a direct consequence we obtain that Snjf→f a.e. for f∈LloglogLlogloglogL. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do not (re)cover the (continuous) Fourier setting. The key ingredient for our proof is a tile decomposition of the operator supj|Snj(f)| which depends on both the function f and on the lacunary structure of the frequencies. This tile decomposition, called (f,λ)-lacunary, is directly adapted to the context of our problem, and, combined with a canonical mass decomposition of the tiles, provides the natural environment to which the methods developed by the author in “On the boundedness of the Carleson operator near L1” apply.

PP* estimates for the truncated Carleson operator

We consider two Hilbert transforms, with real phases a and b, smoothly truncated at \({{2^{-k_0}}}\) where k0 is a non-negative integer. We decode the operator resulting from composing one of them with the adjoint of the other one. Then the case of a similarly truncated Carleson operator is dealt with. The case of Hilbert transforms, as well as the Carleson operator, truncated away from the origin is also considered. An application is mentioned.

The (Weak-L 2) Boundedness of the Quadratic Carleson Operator

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase.

Pseudodifferential Operators with Rough Symbols

In this work, we develop L p boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(L p) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the ξ variable. As a corollary, we obtain L p bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator.

The Walsh model for $M_2^*$ Carleson

We study the Walsh model of a certain maximal truncation of Carleson's operator related to the Return Times Theorem.

Weighted norm inequalities for maximally modulated singular integral operators

We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderon-Zygmund singular (and maximal singular) integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators. We discuss two approaches, one based on the good-λ method of Coifman and Fefferman [CF] and an alternative method employing the sharp maximal operator. As an application we obtain new weighted and vector-valued inequalities for the Carleson operator proving that it is controlled by a natural maximal function associated with the Orlicz space L(log L)(log log log L). This control is in the sense of a good-λ inequality and yields strong and weak type estimates as well as vector-valued and weighted estimates for the operator in question.

Multilinear singular integrals

We survey the thoery of multilinear singular integral operators with modulation symmetry. The basic example for this theory is the bilinear Hilbert transform and its multilinear variants. We outline a proof of boundedness of Carleson's operator which shows the close connection of this operator to multilinear singular inte- grals. We discuss particular multilinear singular integrals which historically arose in the study of eigenfunctions of Schrodinger operators. This survey article arose from a series of three expository lectures given at the 6th International Conference on Harmonic Analysis and Partial Di! erential Equations at El Escorial 2000. I would like to thank the organizers for organizing this stimulating and successful conference. The article is divided into three chapters following the three lectures at El Escorial. The first section gives an introduction into the subject of multilinear singular integrals with modulation symmetries as it has evolved over the past five years. The second section presents a proof of boundedness of the Carleson operator, the essential ingredient in the proof of Carleson's theorem on almost everywhere convergence of Fourier series. We follow closely the work (29), with additional comments as pre- sented during the lecture. While Carleson's operator is not a multilinear operator itself, it serves as a good model for the type of arguments used in the theory: thus the theory extends to other objects than multilinear operators. The third section is again purely expository and gives an overview of how the discussed theory of multilinear singular integrals plays a role in eigenfunction expansions of one dimensional Schrodinger operators. In fact, some of the most prominent objects in the theory such as the bilinear Hilbert transform and Carleson's operator enter directly into these multilinear expansions.

A littlewood-paley inequality for the Carleson operator

The Carleson operator is closely related to the maximal partial sum operator for Fourier series. We study generalizations of this operator in one and several variables.

Weights of Exponential Type

We consider the operators where T is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator. It has been relevant, in questions of harmonic analysis on noncompact rank one symmetric spaces, the boundedness of from to , where the weights and . In this paper we extend the above result to a much larger class of weights that include for instance , with . The case is also studied. Operators are bounded for to itself, if and only if .

Singular integrals with exponential weights

We study the operators Vf (t) = V(f (r)w (r)) (t) w(t) where V is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator. Under suitable conditions on the weight w(t) of exponential type, we prove boundedness of V from LP spaces, defined on [1, +oo) with respect to the measure w2(t)dt, to LP + L2, 1 < p < 2, with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from LP to LP, 1 < p < oo, if and only if p = 2. The study of convergence properties of inverse spherical transforms of radial functions on noncompact symmetric spaces [1], [6], [7] requires estimates on singular integrals with exponential weights, as defined below, due to the exponential growth at infinity of the radial part D(t)dt of the measure, where D(t) = (Sht)p (Sh 2t)y, with p and q suitable nonnegative integers that depend upon the geometry of the symmetric space. In what follows we define a class of functions w(t), that include exponential, and prove boundedness of the operators wt) V (f(r)w(r)) (t) from LP (w2 (t)dt) to LP(w2 (t)dt) + L2 (w2 (t)dt), 1 < p < 2, where V is the classical Hardy-Littlewood maximal function, the maximal Hilbert transform or the maximal Carleson operator. Boundedness from LP to LOP 1 < p < oc, w.r. to the measure w2 (t)dt, holds for p = 2 only. Theorem 1. We denote by LP = LP ([1, +oo), w2 (t)dt) the space of functions defined on [1, +oo) that are LP with respect to the density measure w2 (t)dt. Received by the editors October 7, 1994. 1991 Mathematics Subject Classification. Primary 42A50; Secondary 43A80. (@)1996 American Mathematical Society 1171 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:42:59 UTC All use subject to http://about.jstor.org/terms

Higher order commutators for vector-valued Calderón-Zygmund operators

Weighted norm estimates for higher order commutators are obtained. The proof, that remain valid in the vector-valued case, are obtained as an application of some extrapolation results. The vector-valued version of the commutator theorem is applied to the Carleson operator, U.M.D. Banach spaces, approximate identities and maximal operators.

ESTIMATES OF THE SINGULAR NUMBERS OF THE CARLESON IMBEDDING OPERATOR

Let be the Hardy class in the unit disc and a finite Borel measure in . Carleson's theorem describes conditions on under which the corresponding imbedding operator (the Carleson operator) is bounded. From this theorem follows a criterion for compactness of in terms of .This paper is devoted to further study of the Carleson operator. Almost sharp upper bounds on the singular numbers of are presented in terms of the intensity of . For measures whose support is a set of nonzero linear measure adjacent to the unit circle (and when certain other conditions), an asymptotic formula is obtained. A study is begun of measures whose support has just one point on the unit circle. A solution of a problem from the theory of rational approximation, posed by A. A. Gonchar, is also presented.Bibliography: 17 titles.

A proof of boundedness of the Carleson operator

A proof of boundedness of the Carleson operator