Inequalities on tent spaces and closed range integration operators on spaces of average radial integrability

Abstract

We deal with a reverse Carleson measure inequality for the tent spaces of analytic functions in the unit disc D of the complex plane. The tent spaces of measurable functions were introduced by Coifman, Meyer and Stein. Let 1 ≤ p, q < ∞ and consider the measurable set G ⊆ D. We prove a necessary and sufficient condition on G in order to exist a constant K > 0 such that

T

β (ξ )∩G | f (z)| p dm(z) 1 − |z| q/p |dξ | ≥ K

T

1/2(ξ ) | f (z)| p dm(z) 1 − |z| q/p |dξ |, for any analytic function f in D with the property, the right term of the inequality above is finite. Here T stands for the unit circle, dm(z) is the area Lebesgue measure in D and β(ξ ) is the cone-like region β(ξ ) = {z ∈ D |z| < β} ∪ |z|<β [z, ξ ), β ∈ (0, 1), with vertex at ξ ∈ T. This work extends the study of D. Luecking on Bergman spaces to the analytic tent spaces. We apply this result in order to characterize the closed range property of the integration operator Tg( f )(z) = z 0 f (w)g (w) dw, z ∈ D, when acting on the average radial integrability spaces. The Hardy and the Bergman spaces form part of this family. The function g is a fixed analytic function in the unit disc. The operator Tg is known as Pommerenke operator. Moreover, for the first time, we provide examples of symbols g that introduce or not a closed range operator Tg in these spaces.