Necessary and sufficient conditions for Calderon-Zygmund singular integral operators to be bounded operators on //' (R2) are investigated. Let m be a bounded measurable function on the circle, extended to R2 by homogeneity (m(rx) = m(x)). If the Calderon-Zygmund singular integral operator Tm , defined by Tmf = y_1C"^"(/)), is bounded on //'(R2), then it is proved that S'm has bounded variation ...

The degree of approximation of a function f belonging to Lipschitz class by the Cesàro mean and f ∈ Hα by the Fejér means has been studied by Alexits [4] and Prössdorf [7] respectively. But till now no work seems to have been done to obtain best approximation of functions belonging to generalized Zygmund class, Z (w) r , (r ≥ 1) by product summability means of the form (∆.E1). Z (w) r class is ...

Understood in a wide sense, square functions play a central role in classical Littlewood-Paley theory. This entails for instance dyadic type decompositions of Fourier series, Stein’s theory for symmetric diffusion semigroups or Burkholder’s martingale square function. All these topics provide a deep technique when dealing with quasi-orthogonalitymethods, sums of independent variables, Fourier m...

1. Introduction. The Calderón-Zygmund theory of singular integrals has been traditionally considered with respect to a measure satisfying a doubling condition. Recently, Tolsa [T] and, independently, Nazarov, Treil, and Volberg [NTV] have shown that this standard doubling condition was not really necessary. Likewise, in the homogeneous spaces setting, functions of bounded mean oscillation, BMO,...

We obtain converse Marcinkiewicz-Zygmund inequalities such as k P kLp[ 1;1] C 0@ n X j=1 j jP (tj)j 1A1=p for polynomials P of degree n 1, under general conditions on the points ftjgj=1 and on the function . The weights f jgj=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [ 1; 1].

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of sub...