Elijah Liflyand

MULTIDIMENSIONAL HAUSDORFF OPERATORS ON THE REAL HARDY SPACE

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family is proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Moricz.

Full length article: Conditions for the absolute convergence of Fourier integrals

New sufficient conditions for the representation of a function via an absolutely convergent Fourier integral are obtained in the paper. In the main result, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension d>=2.

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgues sense on either R or R2, and acting on the real Hardy space H1(R), or the product Hardy space H11(R×R), or one of the hybrid Hardy spaces H10(R2) and H01(R2), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform.

Interaction between the Fourier transform and the Hilbert transform

Well-known and recently observed situations where the two main transforms in harmonic analysis, the Fourier transform and the Hilbert transform, show their adjacency and interplay in a specic interesting manner are overviewed. Some relations of that kind are new, while in other cases well-known formulas are considered in a different setting.

Two spaces conditions for integrability of the Fourier transform

Abstract Tests for the integrability of the Fourier transform of a function are given in terms of belonging of the function simultaneously to two spaces of smooth functions. These are, in a sense, generalizations of Zygmund´s test for the absolute convergence of Fourier series.

On Quasi-Monotone Functions and Sequences

Two types of spaces of sequences as well as their analogs for functions are compared. One of them was inspired by results of A. Beurling in spectral synthesis. The other has appeared in the work of R. P. Boas in trigonometric series. It turns out that natural additional assumptions provide the equivalence of these two types of spaces. Applications are given to the study of behavior of the Fourier transform and integrability of trigonometric series.

Asymptotic Behavior of the Fourier Transform of a Function of Bounded Variation

A recent result on the asymptotic behavior of the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation is extended to the case of several variables. For this, the initial one-dimensional result is reconsidered and refined. To even one-dimensional asymptotics and their multidimensional generalizations, a new balance operator is introduced.

On weighted conditions for the absolute convergence of Fourier integrals

Abstract In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivatives.

The Fourier Transform of Convex and Oscillating Functions

What may be referred to as an initial point for the subject of this chapter is Trigub’s result of the 1970’s on the asymptotics of the Fourier transform of a convex function (see, e.g., [198]).

The Fourier Transform of a Radial Function

Spherical symmetry is a very interesting and important property of a function. Theorem 1.5 gives that if f(x) is radial (depending only on ‌x‌), then \( \hat{f} \) is radial too.