Continuous embedding between P-de Branges spaces

: In this paper we study the continuity of the embedding operator ι : H p p E q ã → H q p E q when 0 ă p ă q ď 8 . The necessary and sufficient condition has already been described in [10] if p ą 1. In this work, we address the problem when p “ 1, using a new approach, but asking some additional hypothesis about the Hermite-Biehler function E . We give also a different proof for the case p ą 1.


Introduction
The continuity of the embedding operator between di erent p-Bernstein spaces is a well known result, [1] and [14]. In these notes, we study the same problem for the de Branges spaces: we ask which conditions the Hermite-Biehler function E must satisfy so that the embedding operator ι : H p pEq ã→ H q pEq when ă p ă q ď 8 (1.1) is continuous. This problem has already been investigated and solved in [8], [9], [10] by Professor K. Dyakonov, for p ą . Dyakonov proves that the boundedness of the derivative of the phase function of the meromorphic inner function Θ is a necessary and su cient condition for the continuity of the embedding operator in˚-invariant subspaces K p pΘq of Hardy space. This condition can be adapted also to de Branges spaces. Instead of focusing on the case p ą , we x our attention to the case p " , which was already been studied in [9] only for some kind of meromorphic inner functions. We prove that if and only if the embedding operator ι : H pEq ã→ H q pEq is continuous. However, to prove the theorem, we add further conditions to E, (2.5) and (2.6). For the case p ą , we present a proof longer than the theorem of Dyakonov. However, since it is similar and it uses the same approach to the case p " , we write it down. Furthermore, the whole reasoning can also be generalized to the case { ă p ă . These notes are divided in four short sections. The next section is introductory: we recall the main properties of the p-de Branges spaces. We skip almost all the details for which we refer to the classic texts on the subjects, such as [5], [6], [11], [14], [16] or [17]. At the end of this section, we state the two main results: Theorem 2.4 and Theorem 2.5 . In the third section we prove Theorem 2.4 and we will highlight where our proof is di erent from that of [10]. In the fourth section, we study the necessary condition for p " and we prove Theorem 2.5.
where f # pzq :" f pzq and H p pC`q is the p-Hardy space of C`.
If ă p ă , H p pEq is just a complete metric space and the distance is given by Every Hermite-Biehler function E can be associated to a meromorphic inner functions Θ, as shown in [15]; Indeed, if E is a Hermite-Biehler function, then is a meromorphic inner function of C`. Therefore, if p ą , the p-de Branges spaces are isometrically isomorphic to the˚-p invariant subspaces of H p pC`q, as proved for example in [2] when p " . The proof of Dyakonov in [10] is settled for˚-p invariant subspaces. When dealing with de Branges spaces, the phase function is an important instrument. D 2. 3 We denote by ϕpxq the phase function associated to Epzq, that is, the continuous function such that e iϕpxq Epxq " |Epxq| for all real x.
It can be shown that ϕpzq has an analytic continuation to an open neighborhood of R and it is possible to compute explicitly the expression of its derivative. Indeed, let tzn :" an´ibnu nPZ be the zeros of the Hermite-Biehler function, then where we know that the above series converges when t is xed due to the Blaschke condition [7]. If p ě , the spaces H p pEq are reproducing kernel Banach spaces. The kernel is de ned as If p ą , kzptq belongs to H p pEq for every z P C.
We are now ready for our main results which we prove in the next sections.

Proof of Theorem 2.4
In this section we focus our attention on the necessary and su cient condition for the continuity of the embedding operator when p ą . The proof that we present works directly at the level of the de Branges spaces. Let us start with su ciency. We need this preliminary lemma.

where Θ is the meromorphic inner function associated to the Hermite-Biehler function E. τ depends only on E and is speci ed in the proof.
Proof. Thanks to Lemma 1 of [10], if ϕ P L 8 pRq, then for some δ ą , Due to the subharmonicity of |f {E| p in a strip which contains the real line, we know that Thus, we obtain that Proof Su ciency 2.4. At this point, the proof is an easy consequence of (3.1). We recall that if f P H p pC`q X L q pRq, then f P H q pC`q. Consequently we have just to check the relation between the two norms. For every f P H p , we note that˜ż with the natural changes if q " 8.

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3.2 We specify that the proof of the su ciency condition is similar to that of [10]. The main di erence is the use of Lemma 3.1, instead of some estimates on the reproducing kernel. We think that this lemma is more similar to the techniques used in Bernstein spaces. Now, we move on to the proof of the necessary condition for the continuity of the embedding operator H p pEq ã→ H q pEq when ă p ă q ď 8. We need some preliminaries estimates. Proof. Let us consider since Epznq " for every n P Z. Since ι is continuous for every n P Z, which implies that the inf nPZ bn ą .
In the following estimates we will repeatedly use condition (2.6) and the constant M will be used for the lower bound of imaginary parts of the zeros of E.  Proof. We know that On the other hand, if |an´s| ą bn, and we know also that bn ě M ą M{ ą |s´t| due to (2.6), Therefore in any cases we have (3.4).
We can assume that On the other hand since we assumed that π ϕ ptnq ă M for tn large enough. Proof Necessity 2.4. First of all we prove the necessary condition for ι : H p pEq ã→ H 8 pEq to be continuous. Assume towards a contradiction that ϕ 8 " 8. Then, as proved in Proposition 3.6 there exists a sequence tt n u nPN such that (3.5) holds. Thanks to the boundedness of the embedding operator, we know that Consequently we would say that ϕ ptnq ď C , but it is impossible. For this reason ϕ 8 ă 8. Let us move on to the general statement for q ă 8. As done before, let us assume that ϕ 8 " 8. Then, as proved in Proposition 3.6 there exist ttnu nPN such that (3.5) holds. We know that for every tn, k tn H q ď C k tn H p and consequently, However this last inequality cannot hold if ϕ ptnq goes to in nity when n goes to in nity. For this reason ϕ ptnq ă 8.
The proof of the necessity condition is more complicated than the same proof of [10]. However, it can be generalized to the case { ă p ď as we show in the next section.

Proof of Theorem 2.5
In this section we prove that ϕ pxq P L 8 pRq is a necessary condition also for the continuity of the embedding ι : H ã→ H q when the function E satis es (2.5) and (2.6). The su ciency of Theorem 2.5 is equal to the proof of the su ciency of Theorem 2.4 and for this reason we omit the proof. It is clear that (2.5) implies that for every t, s P R such that ϕptq " ϕpsq`π, t´s " π{ϕ pζ q ď π{δ , where s ď ζ ď t .
In the proof of the following lemma, we use the relations (3.3) and (3.4).

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4.1. Let us assume that the function E satisfy (2.5) and (2.6). Furthermore, let us consider s, t P R so that ϕptq " ϕpsq`π .
The same calculations work also for pDq, for which we obtain the same estimate. Finally pCq ď Considering all the estimates, we proved relation ( The only integral we have not already considered, is pCq: Considering all the estimates, we proved relation (4.1) also in this nal case. For sake of clarity, from now on, we assume ϕ psnq ě ϕ ptnq. We start by the proof of the continuity of H pEq ã→ H 8 pEq. Thanks to the boundedness of the embedding operator, we know that Therefore, @ sn, which is impossible. Consequently, ϕ psnq ď C , and ϕ 8 ă 8 .
Let us move on to the general statement for q ă 8. We know that for every n, Λn H q ď C Λn H . We know also that, when n is large enough, However this last inequality cannot hold if ϕ psnq goes to in nity when n goes to in nity. Consequently ϕ 8 ă 8.

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4.2 We note that the previous proof with some natural changes shows also that Theorem 2.5 holds even when { ă p ă .

Conclusion
In our opinion, the most innovative part of these notes is the fourth section where we proved that the boundedness of the derivative of the phase function is a necessary condition for the boundedness of the embedding operator also when p " , if conditions (2.5) and (2.6) hold. We strongly believe that this result can be further improved, considering even p ď { . In order to do this, we would need an atomic description of the p-de Branges spaces, probably using Clark's measures.

Con ict of interest:
The author states no con ict of interest. No data-set was used.