Reverse Carleson measures in Hardy spaces

We give a necessary and sufficient condition for a measure $\mu$ in the closed unit disk to be a reverse Carleson measure for Hardy spaces. This extends a previous result of Lef\'evre, Li, Queff\'elec and Rodr\'{\i}guez-Piazza \cite{LLQR}. We also provide a simple example showing that the analogue for the Paley-Wiener space does not hold. This example can be generalised to model spaces associated to one-component inner functions.


INTRODUCTION
For 1 ≤ p < ∞ let H p be the Hardy space on the unit disk D equipped with its usual norm where S I = {z ∈ D : 1 − |I| ≤ |z| ≤ 1, z/|z| ∈ I} is the usual Carleson window. This theorem has been extended to several other spaces, like Bergman, Fock, model spaces etc., and we refer the reader to the huge bibliography on this topic for further information.
Note that H p contains a dense set of continuous functions for which the embedding (1.1) still makes sense when the measure has a part supported on the boundary. Then (1.2) implies that the restriction of the measure µ to the boundary has to be absolutely continuous with respect to Lebesgue measure and with bounded Radon-Nikodym derivative. It is thus possible to consider more generally positive, finite Borel measures supported on the closed unit disk: M + (D).
Here, we are interested in reverse Carleson inequalities f p f L p (D,µ) , f ∈ C(D) ∩ H p (D), 1 < p < ∞. In [LLQR] Lefèvre et al. proved that when µ is already a Carleson measure these hold if and only it there exists C > 0 such that for all arcs I ⊂ ∂D µ(S I ) ≥ C|I|.
Our elementary proof actually shows that the reverse inequalities hold without the Carleson condition. It turns out that the interesting part of the measure has to be supported on the boundary, while the part supported in the disk can be dropped.
The embedding problem is closely related with the reproducing kernel thesis: if the embedding holds on the reproducing kernels, then it actually holds for every function. We also show that the reproducing kernel thesis holds for the reverse Carleson embedding.
Finally, we provide a simple example showing that the analogous reproducing kernel thesis for the reverse embedding in the Paley-Wiener space does not hold. The construction can be generalised to model spaces associated to one-component inner functions.
We shall use the following standard notation: f g means that there is a constant C independent of the relevant variables such that f ≤ Cg, and f ≃ g means that f g and g f .
Our main result reads as follows.
Theorem 2.1. Let 1 < p < ∞ and let µ ∈ M + (D). Then the following assertions are equivalent: (1) There exists C 1 > 0 such that for every function f ∈ H p ∩ C(D), (3) There exists C 3 > 0 such that for every arc I ⊂ ∂D, (4) There exists C 4 > 0 such that the Radon-Nikodym derivative of µ| ∂D with respect to the length measure is bounded below by C 4 .
Observe that in this theorem we do not require absolute continuity of the restriction µ| ∂D . Still, if we want to extend (1) to the entire H p -space, then, in order that D |f | p dµ makes sense for every function in H p , we need to impose absolute continuity on µ| ∂D . Note that the integral D |f | p dµ can be infinite for certain f ∈ H p when the Radon-Nikodym derivative of µ| ∂D is not bounded.
(3) ⇒ (4). Take h > 0 so that |I|/h is a large integer N and consider the modified Carleson window [Ru,Theorem 2.18]) we thus have We deduce that the Lebesgue measure on ∂D denoted by m is absolutely continuous with respect to the restriction of µ to ∂D and that the corresponding Radon-Nikodym derivative of µ is bounded below by C 3 . In particular one can choose C 4 = C 3 .
(2) ⇒ (3). By hypothesis, integrating over S I,h with respect to area measure dA on D we get so that the previous estimate becomes We claim that Indeed, if z / ∈ I, then there are δ, h 0 > 0 such that for every 0 < h < h 0 and for every λ ∈ S I,h , we have |1 − λz| ≥ δ > 0, and the result follows from the estimate Suppose now that z = e iθ 0 ∈ I. Let h ≤ |I|, then setting λ = (1 − t)e iθ for λ ∈ S I,h we have Since 0 ≤ t ≤ h ≤ |I| and z = e it ∈ I, the set {e iθ : |θ − θ 0 | ≤ t, e iθ ∈ I} contains an interval of length at least t/2, we get On the other hand, integrating in polar coordinates, we get Hence ϕ h converges pointwise to a function comparable to χ I , and ϕ h is uniformly bounded in h. Now, from (2.1) and by dominated convergence we finally deduce that Remark. The following example shows that the reproducing kernel thesis fails for the reverse Carleson inequalities in the Paley-Wiener space P W π , the space of Fourier transforms of square integrable functions on [−π, π]. In Section 2 we will show how it can be adapted to any model space associated to a one-component inner function.
Consider the sequence S = {x n } n∈Z\{0} , where x n = n + 1/8 if n is even n − 1/8 if n is odd.
By the Kadets-Ingham theorem (see e.g. [Nik,Theorem D4.1.2]) S would be a minimal sampling sequence if we added the point 0. Since S is not sampling the discrete measure µ := n =0 δ xn does not satisfy the reverse inequality f L 2 (R) f L 2 (µ) , f ∈ P W π .

FAILURE IN OTHER MODEL SPACES
The previous construction can be generalised to certain model spaces in the disk. The model space associated to an inner function Θ is K Θ = H 2 ⊖ ΘH 2 , and the reproducing kernel corresponding to λ ∈ D is given by A particular class of model spaces is given by the so-called one-component inner functions, those for which the sub-level set {z ∈ D : |Θ(z)| < ε} is connected for some 0 < ε < 1 .
The Paley-Wiener space corresponds, after a conformal mapping of D into the upper halfplane, to the inner function Θ 2π (z) = e i2πz . More precisely K Θ 2π = e iπz P W π .
Here we show the following result.
Theorem 3.1. If Θ is a one-component inner function, then the reverse reproducing kernel thesis does not hold in K Θ .
We refer the reader to [BFGHR] for sufficient conditions for reverse Carleson measures in model spaces.
For one-component inner functions the set ∂D \ σ(Θ) is a countable union of arcs where Θ is analytic (and on which the argument of Θ increases by 2π). Moreover, for any |α| = 1, is countable and the system (K Θ ζn ) ζn∈Eα is an orthonormal basis of K Θ , a so-called Clark basis (see [Cl], and [BaDy,Section 4] for the material needed here). For such ζ ∈ ∂D \ σ(Θ) the reproducing kernel is defined as Its norm is |Θ ′ (ζ)|, so that the corresponding normalised reproducing kernel is With these elements we follow the scheme of the Paley-Wiener case to prove Theorem 3.1.
Proof. Pick the Clark basis (K Θ ζn ) n≥0 for α = 1 and set where we choose ξ 1 sufficiently close to ζ 1 (and in particular different from ζ n , n = 1) but different from ζ 1 , implying in particular K Θ ξ 1 , K Θ ζn = 0 for every n, so that (K Θ ξn ) n≥0 is an unconditional basis (see [BaDy]; it is actually not far from being orthogonal). It will be clear from the proof below how close to ζ 1 we have to choose ξ 1 .
We now consider the measure where we have taken away the very first point ξ 0 , so that (K Θ ξn ) n>0 is an incomplete family. Notice that this is a perturbation of the Clark measure σ = n≥0 k Θ ζn −2 δ ζn with one mass point deleted. Thus µ is not a reverse Carleson measure since there are functions vanishing in all the points ξ n , n > 0, but not in ξ 0 . which is not possible unless ζ = ζ 0 .
Then either the vectors (g(ζ), g(ζ 0 )) and (h(ζ), h(ζ 0 )) are linearly independent (and we are done) or they are not, in which case the solution of the linear dependence gives ζ = ζ 0 .