The Hausdorff mean of a Fourier-Stieltjes transform

Abstract

It is shown that the integral Hausdorff mean Tp of the Fourier-Stieltjes transform of a measure on the real line is the Fourier transform of an L1 function if and only if Tp. vanishes at infinity or the kernel of T has mean value zero. Also a sufficient condition on the kernel of T and a necessary and sufficient condition on the measure is established in order for-i sï%r{x)T p(x) to be the Fourier transform of an L1-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators. Let Af (/?) be the space of complex bounded regular Borel measures on the real line R and ßix) be the Fourier-Stieltjes transform of a measure p in MiR), pix)= [ e~'xt dpit), xeR. Jr We consider the integral Hausdorff mean Tp generated by a Borel measurable kernel ip in LliR), which is defined for x e R by Tpix)= / fiixs)y/is)ds =-/ tp (?-) piy)dy, Jr \x\ Jr xx/

Hausdorff operators on Hardy type spaces

“…The paper [15] plays a special role in this topic not because such operators were introduced there or studied for the first time. In fact, in the one-dimensional case, Hausdorff operators on the real line were introduced in [9] (in a sense, they can be found in a dual form in [10]). The main feature of [15] is that such operators in a more or less full generality were studied on the real Hardy space.…”

Boundedness of Multidimensional Dunkl-Hausdorff Operators

“…We are now in a position to introduce the Dunkl transform that is taken with respect to the measure μ κ defined by (1)…”

“…By definition, the atomic Hardy space H 1 atom consists of all functions f ∈ L 1 k (R n ) that can be written as f � ℓ λ ℓ a ℓ , where the a ℓ 's are atoms and ℓ |λ ℓ | < + ∞, and the norm is given by…”

“…e Hausdorff operator is one of the most important operators in harmonic analysis, and it is used to solve certain classical problems in analysis. In the one-dimensional case, Hausdorff operators on the real line were introduced in [1] and studied on the Hardy space in [2]. e natural generalization in several dimensions was introduced and studied in [3][4][5].…”

Hausdorff operators on homogeneous spaces of locally compact groups

“…Hausdorff operators were introduced by G. Hardy [1, chapter XI] on the segment, and by C. Georgakis [2] and independently by E. Liflyand and F. Moricz [3] on the whole real line. Their multidimensional generalizations were considered later by G. Brown and F. Moricz [4], and E. Liflyand and A. Lerner [5].…”