uniformly in x as h—»0, we shall say briefly that/ belongs to Lip a. Following 4](1)) we shall say that/ belongs to a) uniformly as *—>0. a notation already used (see Zygmund lipa, 0<a<l, if \f(x+h)-f(x)\=o(\h It is a classical result of Lebesgue (see Zygmund [5, p. 61 ] ; hereafter this book will be denoted by T.S.) that if / belongs to Lip a and if sn denotes the nth partial sum of the Fourie...

Some recent results for bilinear or multilinear singular integrals operators are presented. The focus is on some of the results that can be viewed as natural counterparts of classical theorems in Calderón-Zygmund theory, adding to the already existing extensive literature in the subject. In particular, two different classes of operators that can be seen as bilinear counterparts of linear Calder...

Abstract. As a step in developing a non-commutative Calderón–Zygmund theory, J. Parcet (J. Funct. Anal., 2009) established a new pseudo-localisation principle for classical singular integrals, showing that Tf has small L norm outside a set which only depends on f ∈ L but not on the arbitrary normalised Calderón–Zygmund operator T . Parcet also asked if a similar result holds true in L for p ∈ (...

The main purpose of this paper is to briefly review the earlier works of multiparameter Hardy space theory and boundedness of singular integral operators on such spaces defined on product of Euclidean spaces, and to describe some recent developments in this direction. These recent works include discrete multiparameter Calderón reproducing formulas and Littlewood-Paley theory in the framework of...

We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space R, using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi–projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to ...

For n + 1 = 2, these results had been established earlier by Plessner (1923) and by Marcinkiewicz and Zygmund (1938), who exploited the connection with holomorphic functions and mappings, including the conformal invariance of ∫ |f ′|2 for holomorphic f , and properties of conformal mappings of domains bounded by rectifiable curves. Concerning his motivation, Calderón discloses only that he aims...

In this work, we give new sufficient conditions for Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces H p with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and the authors show that they are sufficient for the associated square function to ...

The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main...

Abstract In this paper, we characterize the discrete Hölder spaces by means of heat and Poisson semigroups associated with Laplacian. These characterizations allow us to get regularity properties fractional powers Laplacian Bessel potentials along these also in Zygmund a more direct way than using pointwise definition spaces. To obtain our results, it has been crucial boundedness kernels their ...