Hardy Spaces Generated by an Integrability Condition

Abstract

an integrability condition for cosine series. No condition superior to that has been given so far. In this paper we identify the atomic structure of the Hardy type space that can be associated with this condition. As a consequence, we conclude that Telyakovskiȋ's condition is equivalent to certain Sidon type inequalities. Then on the basis of this equivalence we show how the atomic technique can be used to extend Telyakovskiȋ's condition to several systems, including Walsh series and integrals, in a uniform way.

The Wiener algebra of absolutely convergent Fourier integrals: an overview

“…The T -transform first appeared in [82], and in [44] it is called the Telyakovskii transform to designate that for obtaining results for the Fourier transform in [82] and [44] it is used to generalize one of the most general result for the integrability of trigonometric series (see, e.g., [136]). It is clear that the T -transform, being a truncated Hilbert transform, should have much in common with the latter; this is revealed and discussed in [82] and later on in, e.g., [44,85], etc. For certain related notions, see also Sect.…”

On Hardy spaces associated with Bessel operators

“…Fridli [7] studied Hardy spaces generated by the Telyakovskii transform and introduced a class of atoms and the corresponding Hardy space H 1 F (0, ∞); cf. 1.9.…”

“…In [7], Fridli introduced an atomic Hardy type space H 1 F (0, ∞) as follows. A measurable function a defined on (0, ∞) is said to be an F -atom if (a) a = 1 δ χ (0,δ) , for some δ > 0, where χ (0,δ) denotes the characteristic function on the interval (0, δ), or (b) there exists a bounded interval I ⊂ (0, ∞) such that supp a ⊂ I, I a(x)dx = 0, and a L ∞ ((0,∞), dx) ≤ |I| −1 , where |I| denotes the length of I.…”

Odd BMO $${(\mathbb{R})}$$ Functions and Carleson Measures in the Bessel Setting

“…The space H 1 o (R) was considered by Fridli[6], who described it by using what he called Telyakovskii transform, a local Hilbert transform studied by Andersen and Muckenhoupt[1]. Fridli also obtained a description Paley g-functions associated to the Poisson semigroup for the Bessel operators ∆ λ .…”