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.
