Monomial convergence on $\ell_r$

For $1<r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly the Marcinkiewicz sequence space $m_{\Psi_r}$ where the symbol $\Psi_r$ is given by $\Psi_r(n) := \log(n + 1)^{1 - \frac{1}{r}}$ for $n \in \mathbb N_0$. For the space of $m$-homogeneous polynomials on $\ell_r$, we prove that the set of monomial convergence $\mbox{mon} \mathcal P (^m \ell_r)$ contains the sequence space $\ell_{q}$ where $q=(mr')'$. Moreover, we show that for any $q\leq s<\infty$, the Lorentz sequence space $\ell_{q,s}$ lies in $\mbox{mon} \mathcal P (^m \ell_r)$, provided that $m$ is large enough. We apply our results to make an advance in the description of the set of monomial convergence of $H_{\infty}(B_{\ell_r})$ (the space of bounded holomorphic on the unit ball of $\ell_r$). As a byproduct we close the gap on certain estimates related with the \emph{mixed} unconditionality constant for spaces of polynomials over classical sequence spaces.


INTRODUCTION AND MAIN RESULTS
A basic fact taught on every course of one complex variable is that every function that is differentiable at all points of a disc centred at 0 can be represented as a power series, and vice-versa. In other words, the derivative f ′ (z) exists (i.e. f is differentiable at z) for every |z| < r if and only if there is a sequence of coefficients (c n ( f )) n ⊆ C so that c n ( f )z n for every |z| < r (i.e. it is analytic). In this case the coefficients can be computed either by differentiation or by the Cauchy integral formula, and the convergence is absolute and uniform on each compact subset of the disc. It also rather elementary to see that in fact this extends also to functions on several complex variables: a function defined on a Reinhardt domain R ⊂ C n (all needed definitions in this introduction can be found in Section 2), is differentiable at every z ∈ R if and only it is analytic (and has a power series representation as in (1)). So, differentiability and analiticity are two equivalent ways to define holomorphy in one and several complex variables. The idea of developing a sort of function theory in infinitely many variables (or, to put in nowadays terms, on infinite dimensional spaces) started at the beginning of the 20th century with the work, among others, of Hilbert, Fréchet and Gâteaux. Here the problem becomes much more subtle. To begin with, while a notion such as differentiability can be considered for functions on any Banach space the idea of analiticiy, where one needs power expansions with monomials of the form z α = z α 1 1 · · · z α n n , is much more restrictive. A Schauder basis, where an idea of 'coordinate' makes sense, is at least needed. This shows that the approach to holomorphy through differentiability is much more far reaching than the one through analiticity. We say, then, that a function f : U → C (where U is some open subset of a Banach space X ) is holomorphic if it is Fréchet differentiable at every point of U (or, equivalently, continuous and holomorphic when restricted to any one-dimensional affine subspace, see [Muj86,Din99]).
It is also worthy to explore the analytic approach whenever it makes sense (as, for example Banach sequences spaces, the definition is given below). Let us succinctly explain how this works (a detailed account on this can be found in [DGMSP19,Chapter 15]). Let f be a holomorphic function on some Reinhardt domain R in a Banach sequence space X . For each fixed n, the restriction of f to R n = R∩C n (which is a Reinhardt domain) is holomorphic and, therefore, has a monomial expansion with coefficients (c (n) α ( f )) α∈N n 0 . It is easy to check that c (n) α = c (n+1) α for α ∈ N n 0 ⊂ N n+1 0 . In other words, we have a a unique family (c α ( f )) α∈N (N) 0 , such that (2) f (z) = α∈N (N) 0 c α z α for all n ∈ N and all z ∈ R n . The coefficients can be computed, for each α = (α 1 , . . . , α n , 0, 0, . . .), by where r > 0 such that {|z| ≤ r } ⊂ R. As usual, the power series α c α z α is called the monomial expansion of f .
One could expect that in the settings where these two approaches coexist they are equivalent, just as in the finite dimensional setting. But this is not the case. When dealing with a totally different problem, related to the convergence of Dirichlet series, Toeplitz gave in [Toe13] an example that, to what we are concerned here, provided a holomorphic function on c 0 and a point in c 0 for which the monomial expansion does not converge absolutely. This shows that there are holomorphic functions that are not analytic (the converse, however, holds true: every analytic function is holomorphic).
Then the question arises in a natural way: for which z's does the monomial expansion of every holomorphic function converge absolutely? (note that when this is the case when the series converges to f (z)). From (2) we have that this happens for every z ∈ R n but, are there other ones? Ryan showed in [Rya80] that the monomial expansion of every holomorphic function on ℓ 1 converges at every z ∈ ℓ 1 . Later Lempert in [Lem99] proved that the monomial expansion of every holomorphic function on ρB ℓ 1 (for ρ > 0) converges at every z ∈ ρB ℓ 1 . This is a somewhat extremal case, where the analytic and differential approaches coincide. What happens in other spaces? or if we consider smaller families of holomorphic functions? To tackle this questions the set of monomial convergence of a family F (R) of holomorphic functions on R was defined in [DMP09] as mon F (R) = z ∈ C N : and a systematic study was started. We are mostly interested in studying the set of monomial convergence of the following three families: • H b (ℓ r ) (the space of holomorphic functions of bounded type on ℓ r ) • H ∞ (B ℓ r ) (the space of bounded holomorphic functions on the open unit ball of ℓ r ) • P ( m ℓ r ) (the space of m-homogeneous polynomials on ℓ r ).
The results of Ryan and Lempert mentioned before imply mon H b (ℓ 1 ) = mon P ( m ℓ 1 ) = ℓ 1 for every m and mon H ∞ (B ℓ 1 ) = B ℓ 1 . On the other endpoint of the scale (p = ∞) [BDF + 17] gives a precise description of mon P ( m ℓ ∞ ) as ℓ m−1 2m ,∞ and lower and upper inclusions for mon H ∞ (B ℓ ∞ ) that, although not optimal, are pretty tight. The study for 1 < r < ∞ was started in [DMP09] and continued in [BDS], where several interesting results in this direction for polynomials and bounded holomorphic functions were obtained. To our best knowledge, nothing has been done so far to describe the set of monomial convergence of the holomorphic functions of bounded type. In this note we make progress towards the description of these set of monomial convergence in the case 1 < r ≤ 2.
In Theorem 4.1 we provide a complete characterization of the set of monomial convergence of the space of holomorphic functions of bounded type for 1 < r ≤ 2 as The proof is given in Section 4 and the main tool developed is a decomposition of the multi-indices (in an even and a pure tetrahedral part), which allows us to split the monomial expansion in different pieces, for which we are able to find proper bounds.
Regarding set of monomial convergence of bounded holomorphic functions on B ℓ r is considered, there are a number of deep results (see [DMP09,Example 4.9 (1)(a)]) that in the case we are dealing with here (1 < r ≤ 2) imply We give here some upper and lower inclusions, in the spirit of the ones obtained for H ∞ (B ℓ ∞ ). We show in Theorem 5.1 that for every m ≥ 5 (we also give lower inclusions for m ≤ 4). The proof is technically involved and uses interpolation of linear operators defined on cones. All this is presented in Section 6.
Finally, as a byproduct, in Section 7 we provide correct estimates of the asymptotic growth of the mixed-(p, q) unconditional constant (a notion by Defant, Maestre and Prengel in [DMP09, Section 5]) as n tends to infinity for every 1 ≤ p, q ≤ ∞; closing the gap of the remaining cases of [GMMb].

PRELIMINARIES
For every x, y ∈ C N we denote by |x| the sequence (|x 1 |, |x 2 |, . . . , |x n |, . . .). If |x i | ≤ |y i | for every i ∈ N we write |x| ≤ |y|. A Banach sequence space is a Banach space (X , · X ) such that ℓ 1 ⊂ X ⊂ ℓ ∞ satisfying that, if x ∈ C N and y ∈ X with |x| ≤ |y|, then x ∈ X and x X ≤ y X . A non-empty open set R ⊂ X is called a Reinhardt domain if given x ∈ C N and y ∈ R such that |x| ≤ |y| then x ∈ R. Given a bounded sequence x its decreasing rearrangement x * is the sequence defined as For every x ∈ c 0 there is some injective mapping σ : N → N such that x * n = |x σ(n) | for all n ∈ N. We will say that a sequence x ∈ C N is decreasing whenever |x| is decreasing.
We are going to deal basically with three classes of Banach sequence spaces: the classical Minkowski ℓ r spaces, the Lorentz ℓ p,q spaces and the Marcinkiewicz sequence spaces. Let us recall some definitions. For 1 ≤ p, q ≤ ∞ the space ℓ p,q consists of those sequences z for which (we use the convention Observe that in general this is a quasi-norm and only defines a norm for 1 ≤ q ≤ p ≤ ∞. For z ∈ ℓ p,q we define It should be noted (see [BS88,Lemma 4.5]) that for 1 ≤ p, q ≤ ∞ and z ∈ ℓ p,q , it holds so we can always work with the quasi-norm · ℓ p,q and treat (ℓ p,q , · ℓ p,q ) as a Banach sequence space at the expense of p ′ (the conjugate exponent of p) as a price every time we do so . Let Ψ = (Ψ(n)) ∞ n=0 be an increasing sequence of nonnegative real numbers with Ψ(0) = 0 and Ψ(n) > 0 for every n ∈ N.
These functions are usually known as symbols. The Marcinkiewicz sequence space associated to the symbol Ψ, denoted by m Ψ , is the vector space of all bounded sequences (z n ) n such that An m-homogeneous polynomial in n variables is a function P of the form Given α ∈ N n 0 we write |α| = α 1 + · · · + α n and Λ(m, n) = {α ∈ N n 0 : |α| = m}. We also consider the set J (m, 2, . . . , n, α n . . . n) ∈ J (m, n). Conversely, each j ∈ J (m, n) defines α ∈ Λ(m, n) by α k = card{i : j i = k}. In this way these two indexing sets are injective and, denoting z α 1 1 · · · z α n n = z α and z j 1 · · · z j m = z j we can write each homogeneous polynomial in two alternative ways c j z j .
We will freely change from the α to the j notation whenever it is more convenient (always assuming that α and j are related to each other). We write |j| = card{i ∈ N m : there exists a permutation σ of 1, . . . , m so that i σ(k) = j k for all k} .
Note that if j and α are associated to each other, then We will sometimes denote this by |[α]|. We write P ( m C n ) for the space of all m-homogeneous polynomials in n variables. Each ℓ r -norm on C n induces a different (though all equivalent) norm We follow the theory of holomorphic functions on arbitrary Banach spaces as presented in [Muj86,Din99]. If X is a (finite or infinite dimensional) Banach space, a function P : X → C is a (continuous) . The space of m-homogeneous polynomials on X is denoted by P ( m X ), and is endowed with the norm P = sup x ≤1 |P (x)|. Every homogeneous polynomial is entire (holomorphic on X ) and, then, its coefficients can be computed through (3). Let us note that c α (P ) = 0 only if |α| = m and that, if j ∈ J (m, n) is associated to α, then c α (P ) = m! α!P (e j 1 , . . . , e j m ) .
An entire function is said to be of bounded type if it is bounded on every bounded set of X . The space of entire functions of bounded type is denoted by H b (X ). It is a Fréchet space with the family of seminorms defined by p n ( f ) = sup x ≤n | f (x)|.
We denote by N (N) 0 the set of eventually zero multi-indices. In other words, . From now on we will identify N n 0 × {0} with N n 0 without further notice.

REARRANGEMENT FAMILIES OF HOLOMORPHIC FUNCTIONS.
A very useful tool in the study of sets monomial convergence (see [BDF + 17]) is that usually, a sequence belongs to the set of monomial convergence if and only if its decreasing rearrangement does (see also [DGMPG08]). We isolate this property, and say in this case that F ⊂ H(R) is a rearrangement family (where R is a Reinhardt domain in a Banach sequence space X ). In [BDF + 17] it was proved that H ∞ (B c 0 ) and P ( m c 0 ) are rearrangement families. The fact that this is also the case for ℓ r for 1 ≤ r < ∞ is implicitly used in [BDS]. Our aim now is to find other rearrangement families of holomorphic functions (compare this with [Sch15,Chapter 7] where similar results appear).
To this purpose we introduce another concept. We say a family F ⊂ H(R) is linearly balanced if f • T | R ∈ F for every f ∈ F and T : X → X linear with T = 1 and T (R) ⊂ R. We give a series of preliminary results needed for the proof of Theorem 3.2. Given an injective mapping σ : N → N we define two mappings in the following way. First Second, S σ : C N → C N is defined for x ∈ C N by Both are clearly linear and T σ (S σ x) = x for every x.
Remark 3.3. Let us see now how these two mappings behave with the decreasing rearrangement of a bounded sequence x. Fixed n ∈ N and J ⊂ N such that card(J ) < n we have That is, T σ (x) * ≤ x * . A similar argument shows that (S σ x) * = x * .
The following lemma shows that the restrictions of S σ and T σ to symmetric Banach sequence spaces are endomorphisms of norm 1.
Lemma 3.4. Let X be a symmetric Banach sequence space and σ : N → N an injective mapping. Then T σ , S σ : X → X defined by (7) and (8) respectively are well defined, T σ = 1 and S σ is an isometry.
Proof. Remark 3.3 together with the symmetry of the space imply that both operators are well defined, that S σ is an isometry and T σ ≤ 1. The fact that T σ = 1 follows from the equality T σ (S σ x 0 ) = x 0 . Now we are able to give the proof of Theorem 3.2.
Proof of Theorem 3.2. To begin with we take z ∈ mon F and see that z * ∈ mon F . As mon F ⊂ c 0 there is some injective mapping σ : for every α. Take, then, some α ∈ N (N) 0 and set N = max{k : α k = 0}. On one hand we have The uniqueness of the Taylor coefficients gives (9). Once we have this we obtain (recall that f •T σ ∈ F and z ∈ mon F ) which proves our claim.
For the converse, suppose z * ∈ mon F . Again, as mon F ⊂ c 0 , there is some injective mapping Besides, . By the uniqueness of the coefficients of the Taylor expansion for f • S σ : as we wanted.
Remark 3.5. Let R be a symmetric Reinhardt domain in a Banach sequence space X and consider a family of homolorphic functions F ⊂ H(R) such that for some m ≥ 2 the space P ( m X ) lies inside F .
Corollary 3.6. For every symmetric Banach sequence space X the families of holomorphic functions Proof. Each of these families satisfies the condition in Remark 3.5. Then Remark 3.1 and Theorem 3.2 give the conclusion.
Remark 3.7. As we have already pointed out, we are mainly interested in H ∞ (B ℓ r ), H b (ℓ r ) and P ( m ℓ r ). The set of monomial convergence of each one of these spaces is, by Remark 3.5 contained in c 0 . But, as matter of fact, we can say more. By [DGMSP19, Proposition 20.3] we have Noting that every functional f ∈ ℓ * r belongs to H b (ℓ r ) and using the definition of the set of monomial convergence we have mon H b (ℓ r ) ⊆ ℓ r . Finally, exactly the same argument as in [DGMSP19, Remark 10.7] shows that mon P ( m ℓ r ) ⊆ P ( 1 ℓ r ) = mon ℓ * r = ℓ r .

MONOMIAL CONVERGENCE FOR HOLOMORPHIC FUNCTIONS OF BOUNDED TYPE ON ℓ r
We can now describe the set of monomial convergence of H b (ℓ r ) for 1 < r ≤ 2. It happens to be a Marcinkiewicz space m Ψ r where the symbol is given by We handle the upper and the lower inclusions separately in the following two sections. is a constant C r > 0 such that for all n and m ≥ 2 we can find a choice of signs (ε α ) α so that

The upper inclusion mon
These polynomials are the main tool for the proof of the upper inclusion. We also need the following result, an extension of [DMP09, Lemma 4.1] whose proof follows the same lines.
Lemma 4.2. Let R be a Reinhardt domain in a Banach sequence space X and let (F , (q n ) n ) be a Fréchet space of holomorphic functions continuously included in H b (R). Then, for each z ∈ mon(F ), there exist C > 0 and n such that for every P ∈ P ( m X ).
We have now everything at hand to proceed with the proof of the upper inclusion.
Proof of the upper inclusion in Theorem 4.1. Fix 1 < r ≤ 2 and choose z ∈ mon H b (ℓ r ). Now fix n, m, choose signs as in (12) and define the polynomial P (w) := α∈Λ(m,n) ε α m! α! w α . By Corollary 3.6 we know that z * ∈ mon H b (ℓ r ). Using first the multinomial formula, then Lemma 4.2 and finally (12) we Taking the power 1/m and using Stirling's formula (m! ≤ 2πme 1 12m m m e −m ) yield Finally, choosing m = ⌊log(n +1)⌋ gives that the term We face now the proof of the lower inclusion in Theorem 4.1. The main tool is the following result, the proof of which requires some work, that we perform all along this section. Theorem 4.3. Fix 1 < r ≤ 2. For every ε > 0 there is C r = C r (ε) > 0 such that for every m, n ∈ N, every m-homogeneous polynomial in n complex variables P and every z ∈ C n , we have Before we start with the proof of this result, let us see how, having it at hand, we can prove the lower inclusion we are aiming at.
Proof of the lower inclusion in Theorem 4.1. Choose z ∈ m Ψ r and let us see that z ∈ mon H b (ℓ r ). By Corollary 3.6 we may assume without loss of generality z = z * . Given f ∈ H b (ℓ r ) (recall that we denote P m ( f ) for the m-homogeneous part of the Taylor expansion) and Theorem 4.3 (with ε = 1) gives Let us see that this sum is finite.
where the last step is due to Cauchy's inequality. This completes the proof.
We start now the way to the proof of Theorem 4.3. We begin with a simple remark.
That is 1/r (note that this series is convergent for 1 < r ).
Our first ingredient is the following lemma, that follows with a careful analysis of the proof of [BDS,Lemma 3.5], that relates the summability of certain coefficients of a polynomial and its uniform norm in ℓ n r . It has been very useful to provide a proof 'at an elementary level' (in the sense that it does not require tools from the local theory of Banach space) of the asymptotic growth of the unconditional constant of the space of m-homogeneous polynomials on ℓ n r as n goes to infinite with suitable care on the dependence of m (in fact this has been proved for general index sets, see [BDS, Theorem 3.2]). As a consequence the behaviour of the Bohr radii of holomorphic functions on ℓ r for 1 ≤ r ≤ 2 has been described in [BDS,Theorem 3.9]. It has recently been used also to study the asymptotic growth of the mixed Bohr radii in [GMMa]. In some sense, for 1 ≤ r ≤ 2, it plays the role of the Bohnenblust-Hille inequality for the case r = ∞.
Lemma 4.5. Let 1 ≤ r ≤ ∞ and P be an m-homogeneous polynomial in n variables. Then for each This is in fact the statement of [BDS, Lemma 3.5.]. With it we can give the first step towards the proof of Theorem 4.3.
Lemma 4.6. Let 1 < r ≤ 2, there is A r > 0 such that for every m, n ∈ N, every P ∈ P ( m C n ) and every Proof. Consider P ∈ P ( m C n ) as in (5) and z ∈ C n decreasing. Using first Hölder's inequality and then (16) we have where the last inequality is due to the fact that We now bound the factor |z j m−1 | n j m =j m−1 |z j m | r 1 r . For each 1 ≤ j ≤ n we use Remark 4.4 to obtain (note that r r ′ − 1 = r − 2 ≤ 0).
We deal with the last sum n k=j log(k + 1)

This and (17) give the conclusion
In view of Lemma 4.6, now we need to bound i∈J (m−2,k) |z i ||i| 1 r in a suitable way (depending on k). To this purpose we switch to the α-notation of multi-indices (recall (5)), that is going to be more convenient. Then the sum reads and the strategy is to decompose this sum into two sums: a tetrahedral and an even part and, then, bound each one of these. This lies in the general philosophy of decomposing index sets into some smaller subset in which a certain problem results easier and, at the same time, are the bricks in which any general index can be recovered. This philosophy has alredy been used in [GMMa].
Let us be more precise and introduce some notation. A multi-index α is tetrahedral if all its entries are either 0 or 1. We consider the set of tetrahedral multi-indices A multi-index is called even if all its non-zero entries are even (note that this forces the multi-index to have even order). We consider then the set Λ E (m, n) = α ∈ Λ(m, n) : α i is even for every i = 1, . . . , n .
Remark 4.7. Given α ∈ Λ(M, N ) define α T (the tetrahedral part) and α E (the even part) as If 0 ≤ k ≤ M is the number of odd entries in α, then clearly α T ∈ Λ T (k, N ) and α E ∈ Λ E (M − k, N ) and Our next step is to bound a sum as in (18) when we just consider even or tetrahedral indices. We start with the latter.
for every ε > 0 and Proof. We begin with the first inequality, observing that it is obvious if N = 1. We may, then, assume A simple calculus argument shows that the function f : For the proof of the second inequality let us recall first that for each α ∈ Λ E (M, N ) there is a unique β ∈ Λ(M/2, N ) such that α = 2β and, moreover, where last inequality holds because 2 k ≤ (2k)! k! 2 ≤ 2 2k and then Then (note that, since 2/r ≥ 1, the ℓ 1 norm bounds the ℓ 2/r norm) Lemma 4.9. Given 1 < r ≤ 2 there is a constant K r ≥ 1 such that for every M, N ∈ N, and every decreas- for every ε > 0.
Proof. Choose some decreasing z and use by Remark 4.7 and Lemma 4.8 to get For r = 2 the last sum is exactly M + 1. If 1 < r < 2 the series converges to 2 2/r 2 2/r −2 . This completes the proof We are finally in the position to give the proof of Theorem 4.3 from which (as we already saw) the lower inclusion in Theorem 4.1 follows.
Proof of Theorem 4.3. Fix 1 < r ≤ 2 and n, m. Pick then P ∈ P ∈ P ( m C n ) and z ∈ C n . Since z m Ψ r = z * m Ψ r , we may assume z = z * . Applying Lemma 4.9 with M = m − 2 and N = k after Lemma 4.6 yields j∈J (m,n) is convergent. This completes the proof.

MONOMIAL CONVERGENCE FOR BOUNDED HOLOMORPHIC FUNCTIONS ON B ℓ r
We change now our focus to the space H ∞ (B ℓ r ) of bounded holomorphic functions on B ℓ r . Our main contribution in this side is the following theorem, that provides with lower and upper inclusions for the set of monomial convergence of these spaces. It recovers (see Remark 5.5 and Corollary 5.6) some previously known results. Proof. In order to keep thinsg readable we write K = id : m Ψ r → ℓ r (2e) 1/r . We first show that if The general result follows form the fact that B m Ψ r and mon H ∞ (B ℓ r ) are both symmetric (Corollary 3.6). We choose now f ∈ H ∞ (B ℓ r ) and fix ε > 0 so that (1 + ε) 1/r z m Ψ r K < 1. By Theorem 4.3 we can find C r (ε) > 0 so that The choice of ε and fact that m 1 m (2+ 1 r ) → 1 as m → ∞ immediately give that the series converges and complete the proof.
A useful tool when dealing with mon H ∞ (B c 0 ) is that, if a sequence belong to such a set of monomial convergence and we modify finitely many coordinates, then the resulting sequence remains in the set of monomial convergence (see [DGMPG08, Lemma 2] or [DGMSP19, Proposition 10.14]). It is unknown whether or not an analogous result result holds for ℓ r (see the comments regarding this problem in [Sch15, Chapter 10]). We overcome this with the following proposition, a weaker version of this, but enough for our purposes.
Proof. Let a 1 , . . . , a N be positive real numbers such that |z i | < a i for every 1 ≤ i ≤ N and a := N n=1 a r n < ρ.
Given for f ∈ H ∞ (B ℓ r ) and k 1 , . . . , k N ∈ N, we define (following the proof of [DGMPG08, Lemma 2]) Note that f k 1 ,...,k N is well defined on the contracted ball (1 − a) 1/r B ℓ r and, in fact, belongs to H ∞ ((1 − a) 1/r B ℓ r ) (because f ∈ H ∞ (B ℓ r )) and Our next step is to understand the coefficients c α ( f k 1 ,...,k N ) in relation to those of f . For each multiindex α = (α 1 , . . . , α n , 0, . . .) with α n = 0, an application of the Cauchy integral formula yields We have now everything we need to proceed. Note that, since a < ρ, Now using (20) and (21) (recall that |u n | = |z n | for n ≥ N + 1) we have |z n | a n k n < ∞, as we wanted.
Let us make a last observation before we proceed with the proof of Theorem 5.1. Given a Banach sequence space X , for every f ∈ H ∞ (t B X ) and t > 0 the function f t given by This implies t mon H ∞ (B X ) ⊂ mon H ∞ (t B x ) for every Banach sequence space X and every t > 0.
Noting that t B X is the open unit ball of the Banach sequence space (X , t · X ), the previous inclusion This altogether shows for every Banach sequence space X and every t > 0. We are now in conditions of proving Theorem 5.1. where C z * ,r is a positive constant that depends only on z * and r . Choosing m = ⌊log(n + 1)⌋ we get lim sup n→∞ 1 log(n + 1) 1− 1 r n k=1 |z * n | ≤ 1, which gives our claim.
We now face the proof of the lower inclusion In order to keep the notation as simple as possible, let K = 2e id : m Ψ r → ℓ r r . Take z ∈ C N such that Let us observe that On the other hand, for n ≥ N , This altogether gives u m Ψ r < (1 + ε)L. We choose ρ > N k=1 |z k | r such id : m Ψ r → ℓ r r (2e)(L(1 + ε)) r + ρ < 1, and, using (23) we get Lemma 5.3 and equation (22) imply u ∈ mon H ∞ ((1 − ρ) 1/r B ℓ r ) and, then Proposition 5.4 gives z * ∈ mon H ∞ (B ℓ r ). Finally, Corollary 6.7 yields z ∈ mon H ∞ (B ℓ r ) and completes the proof. Thus On the other hand, if z ∈ B ℓ r is such that lim sup n→∞ n k=1 z * k log(n + 1) 1−1/r ≤ 1 then there is a constant c > 0 so that z * n ≤ c log(n + 1) 1−1/r n .
From this we easily get that z ∈ ℓ 1+ε for every ε > 0, and we recover (4) from Theorem 5.1.

LOWER INCLUSIONS FOR THE SET OF MONOMIAL CONVERGENCE OF P ( m ℓ r )
We turn now our attention to the set of monomial convergence of homogeneous polynomials. We fix 1 < r ≤ 2 and m ≥ 2 and define q = (mr ′ ) ′ = mr r (m−1)+1 . As we already pointed out, we know from [BDS, Theorem 5.1] and [DMP09, Example 4.6] that for every ε > 0. Our aim now is to tighten this lower bound. We find a lower inclusion that gets narrower when m gets bigger.
We start with Theorem 6.3, which proves the case m = 2 in the previous theorem and also provides an elementary proof of the fact that ℓ q is contained in mon P ( m ℓ r ). We even get a very good estimate for the sums. We will show later in Remark 6.14 (see also the comments after it) that for m ≥ 3 something more can be achieved. We need first a lemma. Proof. We proceed by induction on m. The statement is trivially satisfied for m = 2 and we assume it holds for m − 1. Fix then k and choose n 1 , . . . , n k ∈ N, all non-zero, such that n 1 + · · · + n k = m. We may assume n 1 ≥ · · · ≥ n k ≥ 1. We consider two possible cases. First, if k < e Finally, if n k > 1 then We may use again the induction hypothesis and the fact that n k ≤ m/k to obtain from (27) From here we conclude as in the previous case.
For each fixed 1 ≤ k ≤ n we have, using (6) and Lemma 6.2 (we write a r = e 1 r −1 −1 2 ) and the fact that This gives the case m = 2 in Theorem 6.1. We face now the problem of getting the result for other m's. The general philosophy is always to try to get a bound as that in (28), where in the right-handside we have some constants that depend on r and m (but not on n, the number of variables), the norm of the polynomial and the norm of z in some space X . This then implies X ⊂ mon P ( m ℓ r ).
What we do is to take the sum as depending on m different variables; that is, for each polynomial P we consider with z (1) , . . . , z (m) ∈ C n and then try to get an estimate that involves the norms of the z ( j ) in (possibly) different spaces. This then gives that the smallest of these spaces is contained in the set of monomial convergence (see Remark 6.10). We do this (giving the proof of Theorem 6.1) in two stages (that we present in the following two subsections). First we give an estimate for the sum that involves both ℓ q,1 and ℓ q,∞ norms (the precise statement is given in Proposition 6.4). Then we interpret this inequality as operators from ℓ q,∞ ×· · ·×ℓ q,∞ ×ℓ q,1 ×ℓ q,∞ ×· · ·×ℓ q,∞ to ℓ 1 (J (m, n)) and use interpolation techniques to improve the ℓ q,1 -norm (by weakening the ℓ q,∞ -norm). This is done in Theorem 6.9.
What happens here is that, since in the estimate in Proposition 6.4 some of the variables have to be decreasing, we cannot use general multilinear interpolation, but interpolation in cones (a more detailed explanation is given in Section 6.2).
6.1. First bound for the sum. As we announced, our first step towards the proof of Theorem 6.1 is Proof. First of all let us note that a simple computation shows that s q ′ − 1 r ′ ≤ − 1 mr ′ < 0 for every 1 ≤ s ≤ m − 1. We now proceed by induction on k. For k = 1 we use (31) and (32) to have Let us suppose now that the statement holds for k − 1 and prove it for k.
For the following next we need the following well known Hardy-Littlewood rearrangement inequality (see for example [HLP52, Section 10.2, Theorem 368]).
Lemma 6.7. Let (a k ) k∈N and (b k ) k∈N two non-increasing sequences of non-negative real numbers.
Then, for every m ∈ N and every injection σ : Lemma 6.8. Let 1 < r ≤ 2, m ≥ 3. Fix q := (mr ′ ) ′ and 1 ≤ k ≤ m −2. For every z (1) , . . . , z (k) ∈ C n we have Proof. We begin by splitting the sum in a convenient way 1≤j 1 ≤···≤j m−1 ≤n |z (1) We fix j k and bound the first block using Lemma 6.6, taking into account that we have now m − k − 1 z's and that 1 With this, and bounding the second block using Lemma 6.5 we get It easy to see that k−1 q ′ + m−k q ′ − 1 r ′ = 1 q − 1. Therefore, using Lemma 6.7 we have As it was the case for the study of holomorphic functions, Lemma 4.5 (in fact (16), which is [BDS, Lemma 3.5]) is a crucial tool for the proof of Proposition 6.4.
Proof of Proposition 6.4. We begin by using Hölder's inequality and (16) (noting that |i| ≤ (m − 1)! for every i ∈ J (m − 1, n)) and (30) to have Observe now that, for each N ∈ N we have N −r /q ≤ 2 r /q x −r /q for every N ≤ x < N + 1. Then The proof now finishes with a straightforward application of Lemma 6.8.

Real interpolation on cones.
What we are going to do now is to look at the inequalities for sums like in (6) from the point of view of multilinear mappings. We fix a polynomial P ∈ P ( m C n ) and consider the mapping C n × · · · × C n → ℓ 1 (J (m, n)), given by Note that, since everything here is finite dimensional, the mapping is well defined. The idea is, then, to consider norms on the domain spaces so that the norm of this mapping is bounded by a term involving the norm of the polynomial and some constant independent of n. Since the inequality that we get in Proposition 6.4 requires some variables to be decreasing we have to restrict ourselves to cones of decreasing sequences.
To be more precise, if we denote ℓ d q,s := {z ∈ ℓ q,s : |z| = z * } for 1 ≤ s ≤ ∞, Proposition 6.4 tells us that there is a constant C m,r > 1 (independent of P and n) such that, for every 1 ≤ k ≤ m − 1, the mapping given by (33) satisfies (35) T k ≤ C m,r P P ( m ℓ n r ) . All these mappings have the same defining formula (which is m-linear), so it is tempting to apply multilinear interpolation. But, since we need to restrict ourselves to the cone of non-increasing sequences in the last m −k variables, we are not able to directly apply the classical multilinear interpolation results, but interpolation in cones.
For the general theory of interpolation we follow (and refer the reader to) [BL76]. Since (as we have already explained) we have to consider linear operators on cones, we use the K -method of interpolation for operators on the cone of non-increasing sequences, as presented in [CM96]. Then the main result of this section, from which Theorem 6.1, follows is the following. Theorem 6.9. Let 1 < r ≤ 2 and m ≥ 3. Define q := (mr ′ ) ′ and There exists a constant C m,r ≥ 1 such that, for every P ∈ P ( m C n ) the m-linear mapping T : (ℓ n q,s ) d × · · · × (ℓ n q,s ) d given by satisfies T ≤ C m,r P P ( m ℓ n r ) .
Remark 6.10. If we take z (1) = . . . = z (m) = z and observe that z ℓ q,∞ ≤ z ℓ q,s , Theorem 6.9 gives 1≤j 1 ≤···≤j m ≤n |c j (P )z * j 1 · · · z * j m | ≤ C m,r z m ℓ q,s P P ( m ℓ n r ) for every P ∈ P ( m C n ) and z ∈ C n . A standard argument shows that z * ∈ mon P ( m ℓ r ) for every z ∈ ℓ q,s and, then, Corollary 3.6 implies ℓ q,s ⊂ mon P ( m ℓ r ). This gives Theorem 6.1.
Before we proceed, let us fix some notation. Given a Banach function lattice X (in particular a sequence space or a finite dimensional Banach space, on which we are mainly interested), we write X d for the cone of non-increasing functions in X . If Y is any Banach space and S : X → Y is a linear operator we can restrict it to the cone and denote Clearly neither is X d a vector space, nor is S a norm. We will later use an analogous notation for m-linear mappings. We are now ready to state our main tool to interpolate in cones. It is a direct corollary of [CM96, Theorem 1-(b)] (recall that we are using the notation as introduced there).
Since q ′ = mr ′ = ⌊log(n + 1)⌋r ′ , the last expression is ≫ log(n) 1 r . This shows that there exists no constant C > 0 such that for every n and m and all P ∈ P ( m C n ) we have j∈J (m,n) |c j (P )z j | ≤ C m z m ℓ q,log m P P ( m ℓ n r ) .
On the other hand, applying carefully the ideas developed in this section, it is possible to obtain hypercontractive inequalities in some cases. = C m e r ′ w ℓ q,∞ z m−3 ℓ q,∞ z (m−2) ℓ q,∞ z (m−1) ℓ q,1 P P ( m ℓ n r ) .
Thus, proceeding as in Lemma 6.12 we may construct an operator which is bounded from ℓ d q,∞ to ℓ q,1 ′ and also from ℓ d q,1 to ℓ q,∞ ′ . Applying the K -interpolation method restricted to the cone of non-increasing sequences to this operator we can conclude that for any z = z * , j∈J (m,n) |c j (P )z j | ≤ (1 + r ′ )C m e r ′ z m−2 ℓ q,∞ z 2 ℓ q,2 P P ( m ℓ n r ) ≤ C (1 + ε) m P P ( m ℓ n r ) z m ℓ n q . Therefore, by (11), we have proved our claim.
With some extra work it can proved, in a similar way, that given any s ≥ 1 and ε > 0, there exist some m 0 and some C > 0 such that for every n ∈ N, all m ≥ m 0 and every polynomial P ∈ P ( m C n ) we In [GMMb] the exact asymptotic asymptotic growth of the mixed-(r, s) unconditional constant as n tends to infinity was computed for many values of p and q's. To achieve this the authors proved that χ r,s (P ( m C n )) ∼ χ r,s (z j ) j∈J (m,n) .
We complete the result given in [GMMb,Theorem 3.4] by providing the exact asymptotic asymptotic growth for the remaining cases. In this way we have the behaviour of χ p,q (P ( m C n )) for every 1 ≤ p, q ≤ ∞.

Mixed Bohr radius. Let K (B ℓ n
For m = 2 we cannot show that the sequence is 1 n σ 2 n is a multiplier for mon P ( 2 ℓ r ) but using the fact that ℓ q ⊂ monP ( 2 ℓ r ), Theorem 6.1, it is easy to see that we have the inclusion 1 p σ 2 log(p) ε · ℓ r ⊂ mon P 2 ℓ r , for every ε > 0 extending [BDS,Theorem 5.3.] (see also (45)). We leave the details for the reader.