MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS Andreas

J. Korean Math. Soc. 41 (2004), No. 1, pp. 209–229

MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS Andreas Defant, Domingo Garc´ıa and Manuel Maestre P Abstract. Let |α|=m sα z α , z ∈ Cn be a unimodular m-homogeneous polynomial in n variables (i.e. |sα | = 1 for all multi indices α), and let R ⊂ Cn be a (bounded complete) Reinhardt Pdomain. We give lower bounds for the maximum modulus supz∈R | |α|=m sα z α |, and upper estimates for the average of these maximum moduli taken over all possible m-homogeneous Bernoulli polynomials (i.e. sα = ±1 for all multi indices α). Examples show that for a fixed degree m our estimates, for rather large classes of domains R, are asymptotically optimal in the dimension n.

0. Introduction P an In his study [5] of Dirichlet series ∞ n=1 ns Harald Bohr considered vertical strips of uniform, but not absolute convergence of such a series. More precisely, he considered those numbers A and P∞ nonnegative an B for which a given Dirichlet series converges conditionally s n=1 n (i.e., absolutely) in the half plane {s ∈ C : Re s > A}, but converges uniformly on the half planes {s ∈ C : Re s ≥ B + ε}, ε > 0. Bohr asked for the maximal possible width d := B − A of such a strip, and proved that d ≤ 12 . Bohnenblust and Hille in [6] were able to show that d = 21 . In the context of nuclearity and absolute bases for spaces of holomorphic functions over infinite dimensional domains, Dineen and Timoney in [14] gave a new proof based on probabilistic methods. Finally, Boas in a beautiful paper [2] produced a proof in which one of the key ingredients is the Kahane-Salem-Zygmund Theorem (see Received December 17, 2002. 2000 Mathematics Subject Classification: Primary 32A05; Secondary 46B07, 46B09, 46G20. Key words and phrases: several complex variables, power series, polynomials, Banach spaces, unconditional basis, Banach-Mazur distance. The second and third authors were supported by MCYT and FEDER Project BFM2002-01423.

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[18, Theorem 4, Chap. 6, pp.70–71]). This probabilistic result, for each degree m and each dimension n, assures the existence of an mP homogenous Bernoulli polynomial P (z) = |α|=m sα z α on Cn for which m+1 √ sup{|P (z)| : |zk | < 1, k = 1, . . . , n} ≤ Cn 2 log m, where C depends neither on n nor on m. A similar argument was used by Boas and Khavinson [3, proof of Theorem 2] in their study of Bohr’s power series theorem in several variables (see also [1], [9] and [13]). The main goal of this paper is to give, in a systematic P way, upper and lower estimates for the maximum modulus supz∈R | |α|=m sα z α |, where P α Bernoulli polyno|α|=m sα z is an arbitrarily given m-homogeneous P mial in n variables (i.e., a polynomial of the form |α|=m sα z α , where the coefficients sα for all α are signs) and R some Reinhardt domain in Cn . In section 2 we obtain for every unimodular m-homogeneous polyP nomial (i.e., every polynomial of the form |α|=m sα z α , z ∈ Cn , where sα ∈ C satisfies |sα | = 1 for all multi indices α) and every P Reinhardt domain R lower bounds for the maximum modulus supz∈R | |α|=m sα z α |. In section 3 we give upper estimates for the average of such maximum moduli taken over all possible m-homogeneous Bernoulli polynomials. As a consequence we prove for certain classes of domains the existence of m-homogeneous Bernoulli polynomials which have maximum moduli as small as possible. Moreover, we apply our results to various concrete classes of domains R in Cn . 1. Preliminaries We shall use standard notation and notions from Banach space theory as presented e.g. in [22] or [28]. If X is a Banach space over the scalars K = R or C, then X ∗ is its topological dual and BX is its open unit ball. We denote the Banach-Mazur distance of two Banach spaces X and Y by d(X, Y ). For all needed background on polynomials defined on Banach spaces we refer to [12] and [16]. P(m X) denotes the space of all m-homogeneous scalar valued polynomials P on a Banach space X, which together with the norm kP k := sup{|P (x)| : kxk ≤ 1} forms a Banach space. A subset U of Cn is called circled if λx ∈ U for all λ ∈ C, |λ| = 1. A complete bounded Reinhardt domain R ⊂ Cn is a bounded domain that is complete and n-circled, i.e. if x = z1 e1 + · · · + zn en ∈ R, then λ(eθ1 i z1 e1 + · · · + eθn i zn en ) ∈ R for all λ ∈ C, |λ| ≤ 1 and all θ1 , . . . , θn ∈ R. A basis (xn ) of a Banach space X is said to be unconditional if there is

Maximum moduli of unimodular polynomials

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a constant C ≥ 1 such that K and all s1 , . . . , sn ∈ K P for all µ1 , . . . , µn ∈P with |sk | ≤ 1, we have || nk=1 sk µk xk || ≤ C|| nk=1 µk xk ||; in this case the best constant is denoted by χ((xn )), the unconditional basis constant of (xn ). All unconditional bases considered in this paper are normalized. For X = (Kn , k.k) denote by χmon (P(m X)) the unconditional basis constant of the monomials {z α : |α| = m}. We call a Banach space X = (Kn , k.k) symmetric Pn if the canonical vectors ek form a symmetric basis, i.e. kxk = k k=1 sk xπ(k) ek k for each x ∈ X, each permutation π of {1, . . . , n} and each choice of scalars sk with |sk | = 1. A Banach space X for which `1 ⊂ X ⊂ c0 (with norm 1 inclusions) is said to be a Banach sequence space if the canonical sequence P (ek ) forms a 1-unconditional basis; it is called symmetric if kxk = k ∞ k=1 sk xπ(k) ek k for each x ∈ X, each permutation π of N and each choice of scalars sk with |sk | = 1. Recall finally that a Banach lattice X is said to be 2-concave and 2-convex, respectively, if there is ¡ Pn ¢ ¡ Pn ¢ 2 1/2 ≤ Ck 2 1/2 k, a constant C > 0 such that k=1 kxk k k=1 |xk | ¡ Pn ¢ ¡ Pn ¢ 2 1/2 k ≤ C 2 1/2 for all x , . . . , x ∈ X, reand k 1 n k=1 |xk | k=1 kxk k spectively. Here the best constant is denoted by M(2) (X) and M (2) (X), respectively (see [22]). For the notion of cotype q, 2 ≤ q < ∞, of a Banach space X (the cotype constant is denoted by Cq (X)) and its relation with convexity and concavity we also refer to [22].

2. Lower bounds for the maximum modulus of unimodular homogeneous polynomials on Reinhardt domains Fix a degree m and a dimension n. We call an m-homogeneous polynomial X (2.1) s α z α , z ∈ Cn |α|=m

unimodular whenever all coefficients sα ∈ C satisfy |sα | = 1. If all sα ∈ C are signs +1 and −1, then we speak of an m-homogeneous Bernoulli polynomial. Obviously, µ ¶ X X n+m−1 α sup | sα z | ≤ |sα | = . m z∈B`n ∞

|α|=m

|α|=m

Conversely, since the monomials z α , |α| = m, form an orthogonal system of square integrable functions on the n-dimensional torus Tn in Cn (the

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n-th cartesian product of S`12 endowed with the product measure of the normalized Lebesgue measure σ1 ), we get µ ¶ Z X ¢1/2 n + m − 1 1/2 ¡ = | sα z α |2 dz m Tn |α|=m (2.2) µ ¶ X n+m−1 α ≤ sup | sα z | ≤ . m z∈B`n ∞

Hence, the obvious estimate

1 m m! n

|α|=m

≤

¡n+m−1¢ m

≤ nm gives

X 1 √ nm/2 ≤ sup | sα z α | ≤ nm . m! z∈B`n ∞ |α|=m

(2.3)

Let us improve the lower bound. We want to show that there is a constant cm > 0 such that for each unimodular m-homogeneous polynomial as in (2.1) X m+1 sα z α |. (2.4) cm n 2 ≤ sup | z∈B`n ∞

|α|=m

We are going to obtain this lower estimate as a consequence of the following more general result. Proposition 2.1. Let X = (Cn , k.k) be a Banach space such that the canonical basis (ek )nk=1 is 1-unconditional. Then for each m ¡ ¢m Pn X 1 supz∈BX k=1 |zk | | sα z α |. (2.5) ≤ sup m! χmon (P(m X)) z∈BX |α|=m

Proof. For each unimodular m-homogeneous polynomial X sα z α , z ∈ Cn |α|=m

we have ¡

sup

n X

z∈BX k=1

n X ¢m ¡ ¢m |zk | = sup | zk | z∈BX k=1

= sup | z∈BX

X

|α|=m

m! zα| α1 ! · · · αn !

Maximum moduli of unimodular polynomials

213

X

m! sα z α | α1 ! · · · αn !sα z∈BX |α|=m X ≤ m!χmon (P(m X)) sup | sα z α |. = sup |

z∈BX

|α|=m

By [8, Theorem 3] we know that for each m there is a constant dm > 0 such that m−1

χmon (P(m `n∞ )) ≤ dm n 2 . ¡ ¢m P n As nm = supz∈B`n |zk | , inequality (2.6) together with Propok=1 ∞ sition 2.1 gives as desired (2.4) with cm := (m!dm )−1 . In contrast to (2.6) we know that χmon (P(m `n1 )) for fixed degree m is uniformly bounded from above in the dimension n (see [9, (4.6)]), more precisely (2.6)

χmon (P(m `n1 )) ≤

mm . m!

Hence by Proposition 2.1 we have X 1 sα z α |, (2.7) ≤ sup | m m n z∈B` 1

|α|=m

i.e., for each fixed degree m the maximum modulus of a unimodular m-homogeneous polynomial on B`n1 is uniformly bounded from below in n. More generally, we now prove a theorem which includes (2.4) and (2.7) as particular cases. We will need the following lemma which is related to [3, Theorem 3] and [13, Theorem 3.2]. Lemma 2.2. Given r1 , . . . , rn > 0, consider on Cn the norm kzk := sup{| zrkk | : k = 1, . . . , n} and denote the associated Banach space by `n∞ (r1 , . . . , rn ). Then χmon (P(m `n∞ (r1 , . . . , rn ))) ≤ dm n

m−1 2

,

where the constant dm > 0 is the one from (2.6). Proof. Clearly the open unit ball P of `n∞ (r1 , . . . , rn ) is the polydisc {z ∈ Cn : |zk | < rk , k = 1, . . . , n}. Let |α|=m cα z α be an m-homogeneous polynomial on Cn such that X | cα z α | ≤ 1, for all z ∈ B`n∞ (r1 ,...,rn ) . |α|=m

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Then

X

|

cα (rk zk )α | ≤ 1, for all z ∈ B`n∞ ,

|α|=m

which implies |

X

cα rα z α | ≤ 1, for all z ∈ B`n∞ .

|α|=m

Hence, we obtain by (2.6) that X m−1 | |cα |rα z α | ≤ dm n 2 , for all z ∈ B`n∞ , |α|=m

which finally gives as desired that X m−1 | |cα |z α | ≤ dm n 2 , for all z ∈ B`n∞ (r1 ,...,rn ) . |α|=m

The following result is our main lower bound for maximum moduli of unimodular m-homogeneous polynomials. Theorem 2.3. For each m there is a constant P cm > 0 such that for each unimodular m-homogeneous polynomial |α|=m sα z α , z ∈ Cn the following estimates hold. (1) For each Reinhardt domain R ⊂ Cn ¢m ¡ P X supz∈R nk=1 |zk | sα z α |. ≤ sup | cm m−1 z∈R n 2 |α|=m (2) For each Banach space X := (Cn , k.k) for which the e0k s form a 1-unconditional basis ¡ ¢m P X supz∈BX nk=1 |zk | cm sα z α |. ≤ sup | n d(X, `1 )m−1 z∈BX |α|=m

(3) If, additionally, BX ⊂ B`n2 we obtain cm sup

n X

z∈BX k=1

|zk | ≤ sup | z∈BX

X

sα z α |.

|α|=m

Proof. Each Reinhardt domain R is, by definition, a union of open polydiscs. Given u ∈ R, we take r1 , . . . , rn > 0 such that u ∈ {z ∈ Cn :

Maximum moduli of unimodular polynomials

215

|zk | < rk , k = 1, . . . , n} ⊂ R. By Proposition 2.1 and Lemma 2.2 there is a constant cm > 0 for which P ( nk=1 |uk |)m cm m−1 2 ¡ n ¢m P supz∈B`n (r ,...,r ) nk=1 |zk | n ∞ 1 ≤ cm m−1 2 n X X | sα z α | ≤ sup | sα z α |, ≤ sup z∈B`n ∞ (r1 ,...,rn )

|α|=m

z∈R

|α|=m

and hence (1) is proved. Statement (2) follows from Proposition 2.1 and the fact that by [9, Theorem 6.1] there exists a constant dm > 0 such that (2.8)

χmon (P(m X)) ≤ dm d(X, `n1 )m−1 ;

now it is enough to take cm := (m!dm )−1 . The proof of (3) is a consequence of the fact that by [26, Corollary 2] and [9, (5.2)]: n n X X 1 √ sup |zk | ≤ d(X, `n1 ) ≤ sup |zk |. 2 z∈BX k=1 z∈BX k=1

This completes the proof. Let us collect some concrete examples in order to illustrate our results.We start with estimates for mixed Minkowski spaces `up (`vq ) := {(xk )uk=1 : x1 , . . . , xu ∈ Cv } u X k(xk )uk=1 kp,q := ( kxk kpq )1/p .

P

k=1

Example 2.4. Let |α|=m sα z α , z ∈ Cu (Cv ) be a unimodular mhomogeneous polynomial in uv variables (for some order of these variables). Then  m+1 − m m+1 − m p v 2 q  u 2 if 2 ≤ p, q ≤ ∞,       X 1 1 α sup | sα z | ≥ C u1− p v 1− q if 1 ≤ p, q ≤ 2,  v z∈B`u  p (`q ) |α|=m      1− p1 m+1 −m q u v 2 if 1 ≤ p ≤ 2 ≤ q ≤ ∞, C > 0 some constant depending only on m, p and q.

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Andreas Defant, Domingo Garc´ıa and Manuel Maestre

Let us remark that, by taking p = q and v = 1 in this example, we obtain lower estimates for every unimodular m-homogeneous polynomials on `np , 1 ≤ p ≤ ∞:  m+1 − m p  if 2 ≤ p ≤ ∞, n 2 X α (2.9) sup | sα z | ≥ C  z∈B`n  1− p1 p |α|=m n if 1 ≤ p ≤ 2 . The results of the next section will show in which sense these estimates are optimal. Our proof of Example 2.4 needs the following Lemma 2.5. In the three cases 1 ≤ p, q ≤ 2 or 2 ≤ p, q ≤ ∞ or 1 ≤ p ≤ 2 ≤ q ≤ ∞ we have u u v v d(`up (`vq ), `uv 1 ) ≤ d(`p , `1 )d(`q , `1 ).

Proof. For 1 ≤ p, q ≤ 2 we know from [26, Corollary 2] that X 1− 1 1− 1 ek ⊗ el k`u0 (`v0 ) = u p v q = d(`up , `u1 )d(`vq , `v1 ); d(`up (`vq ), `uv ) ≤ k 1 k,l

p

q

see e.g. [28, 37.6]. If 2 ≤ p, q ≤ ∞, then M (2) (`up (`vq )) = 1, and hence it follows from [28, Corollary 41.9] that as desired √ d(`up (`vq ), `uv 1 ) ≤ C uv, C ≥ 0 some absolute constant. Finally, the remaining case: Note first that for every linear bijection T : `vq −→ `v1 u v u v −1 d(`u1 (`vq ), `uv k = kT kkT −1 k, 1 ) = d(`1 ⊗π `q , `1 ⊗π `1 ) ≤ kid ⊗ T kkid ⊗ T v v hence d(`u1 (`vq ), `uv 1 ) ≤ d(`q , `1 ); for 1 ≤ p ≤ 2 ≤ q ≤ ∞ now by factorization u v u v u v uv d(`up (`vq ), `uv 1 ) ≤ d(`p (`q ), `1 (`q ))d(`1 (`q ), `1 )

≤ kid : `up (`vq ) −→ `u1 (`vq )kd(`vq , `v1 ) = u1−1/p d(`vq , `v1 ) ≤ d(`up , `u1 )d(`vq , `v1 ). For the remaining case 1 ≤ q ≤ 2 ≤ p ≤ ∞ the estimate from Lemma 2.5 is false; by a result of Kwapien and Sch¨ utt from [20, Corollary 3.3] n 2 n n n we, e.g., see that d(`∞ (`1 )), `1 ) √³ n (and not, as one could expect from the estimate of the lemma, n). We believe that atPleast in this special case the optimal lower bound for supz∈B`n (`n )) | |α|=m sα z α | comes from Theorem 2.3(1), namely n

m+1 2

∞

.

1

Maximum moduli of unimodular polynomials

217

Proof of the Example 2.4. All three cases are consequences of Theorem 2.3. We know that X 1−1/p 1−1/q sup | < z, ek ⊗ el > | = kid : `up (`vq ) −→ `uv v . 1 k=u v z∈B`u p (`q ) k,l

Hence the first case is a consequence of Theorem 2.3(1), the second one of 2.3(3), and the last of 2.3(2) combined with the preceding lemma. The next example deals with Orlicz spaces `ϕ , and is a considerable extension of (2.9) (take ϕ(t) = tp ). Example 2.6. Let ϕ be an Orlicz function satisfying the ∆2 -condition. Then for each m there is a constant P cm > 0 such that for each unimodular m-homogeneous polynomial |α|=m sα z α in n variables, the following estimates hold. P m+1 (1) cm n 2 ϕ−1 (1/n)m ≤ supz∈B`ϕ | |α|=m sα z α |. P (2) cm nϕ−1 (1/n) ≤ supz∈B`ϕ | |α|=m sα z α |, provided that t2 ≤ Kϕ(t) for all t and some K. Proof. The proof is again a simple consequence of Theorem 2.3, the observations (3.5) and (3.6)P (anticipated from section 3), combined with 1 for the fundamental the well-known equality k nk=1 ek k`ϕ = ϕ−1 (1/n) function; the condition in (2) assures that `ϕ ⊂ `2 . We finish this section with an example for a non-convex domain. Example 2.7. Given S > 1 and n ≥ 2 consider the Reinhardt domain R := {(z1 , . . . , zn ) ∈ Cn : |z1 · · · zn | < 1 , |zk | < S, k = 1, . . . , n}. Then for each m there exists a constant P hm > 0 such that for all unimodular m-homogeneous polynomials |α|=m sα z α in n variables X m+1 hm S m n 2 ≤ sup | sα z α |. z∈R

|α|=m

Proof. We have that (2.10)

sup

n X

z∈R k=1

|zk | = (n − 1)S +

1 S n−1

;

indeed, if we take the compact set K := {x ∈ [0, S]n : x1 · · · xn ≤ r} for 0 < r ≤ 1, then the maximum of the function f (x) := x1 + · · · + r r xn on K is (n − 1)S + S n−1 , attained at (S, . . . , S, S n−1 ) and at any point whose coordinates are a permutation of it. Let x0 ∈ K such that f (x0 ) = supx∈K f (x). As f has no critical points, x0 belongs to

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Andreas Defant, Domingo Garc´ıa and Manuel Maestre

∂K, the boundary of K. On the other hand x0 does not belong to {x ∈ (0, S)n : x1 · · · xn = r}. In this case the Lagrange multiplier method gives only the critical point (r1/n , . . . , r1/n ). But the function r g : (0, ∞) −→ R defined by g(x) := (n − 1)x + xn−1 attains a strict 1/n absolute minimum at x = r and f (r1/n , . . . , r1/n ) = nr1/n = g(r1/n ) < g(S) = (n − 1)S +

r S n−1

= f (S, . . . , S,

r S n−1

).

If x ∈ ∂K and for some k we have xk = 0, then f (x) ≤ (n − 1)S < r f (S, . . . , S, S n−1 ) and x 6= x0 . Finally, the only possibility left is that for some k we have xk = S. Then, by the symmetry of the function f , we can assume xn = S. By induction, we have sup{(x1 , . . . , xn−1 ) ∈ [0, S]n−1 : x1 · · · xn−1 ≤ rS −1 } = (n−2)S+

rS −1 , S n−2

thus f (x1 , . . . , xn−1 , S) ≤ S + sup{(x1 , . . . , xn−1 ) ∈ [0, S]n−1 : x1 · · · xn−1 ≤ rS −1 } r r = (n − 1)S + n−1 = f (S, . . . , S, n−1 ). S S Pn Now, since supz∈R k=1 |zk | coincides with the maximum of f on {x ∈ [0, S]n : x1 · · · xn ≤ 1}, we obtain the equality (2.10). Hence, by Theorem 2.3 (1) there exists a constant cm > 0 such that ¢m ¡ 1 ¡ n − 1 ¢m m m+1 (n − 1)S + S n−1 1 m m+1 cm m S n 2 ≤ cm S n 2 ≤ cm m−1 2 n n 2 X ≤ sup | sα z α |. z∈R

|α|=m

The conclusion follows by taking hm = cm 21m . 3. Expectations of the modulus maximum of homogeneous Bernoulli random polynomials on Reinhardt domains The Kahane-Salem-Zygmund Theorem [18, Theorem 4, Chap. 6, pp.70–71] shows that for the polydisc typically the maximum modulus X sup | sα z α | z∈B`n ∞

|α|=m

Maximum moduli of unimodular polynomials

219

of an arbitrarily given unimodular m-homogeneous polynomial is as m+1 small as possible, namely n 2 (see (2.4)). To obtain polynomials on `np with “small” norms Boas in [1, Theorem 4] proves the existence of symmetric complex m-linear forms on (`np )m with “small” norms. His proof, as explicitly stated, is made by a careful inspection of results due to Mantero-Tonge (see [23, Theorem 1.1] and also [24, Proposition 4]) which in turn were inspired by the Kahane-Salem-Zygmund Theorem. This section could be considered as an extension of the Mantero-Tonge results. It is worth to mention that [23, Theorem 1.1] was used by Dineen and Timoney in [13] and [14]. Let us clarify what we mean by “ small” norm. Fix a degree m, a family εα : Ω → {−1, 1}, |α| = m of independent Bernoulli random variables on a probability space (Ω, µ) (each εα takes the values +1 and −1 with equal probability 1/2), and cα ∈ C. Then we call X cα εα z α , z ∈ Cn , |α|=m

an m-homogeneous Bernoulli random polynomial. According to [18, Chapter 6, Theorem 3] there is an absolute constant C > 0 such that X ¢ ¡ m+1 1 εα z α | ≥ Cn 2 (log m)1/2 ≤ 2 n . µ sup | m e z∈B`n ∞

|α|=m

Hence with “ high probability” thePmaximum modulus of every m-hoα n mogeneous Bernoulli polynomial |α|=m sα z , z ∈ C , can be estim+1

mated from above by a constant times (log m)1/2 n 2 ; in particular, there exists a set of signs sα , |α| = m such that X m+1 sup | sα z α | ≤ C(log m)1/2 n 2 . z∈B`n ∞

|α|=m

To see from another point of view that (2.4) for fixed m is optimal in the dimension n, integrate (2.4) in order to obtain the following lower estimate for the expectation of the modulus maximum of an m-homogeneous Bernoulli random polynomial with respect to the polydisc B`n∞ : Z X m+1 2 ≤ sup | εα z α |dµ ; cm n Ω z∈B`n ∞ |α|=m

by [9, Corollary 6.5] we know that this result is optimal. More generally, given a Reinhardt domain R in Cn satisfying one of the assumptions of

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Andreas Defant, Domingo Garc´ıa and Manuel Maestre

Theorem 2.3, integration gives a lower bound for the averages Z X sup | εα z α |dµ , Ω z∈R |α|=m

and a modification of [9, Theorem 3.1] will show that for rich classes of domains R the lower bounds obtained in this way, are optimal. Recall that a real valued random variable X on a probability space (Ω, µ) is said to be Gaussian whenever it has mean zero, is square integrable and its Fourier transform satisfies EeitX = e

−kXk2 2 t2 2

1

, where t ∈ R and kXk2 = (EX 2 ) 2 ;

X is a standard Gaussian random variable if it is measurable and if for every Borel subset B of R we have Z t2 1 µ{ω ∈ Ω : g(ω) ∈ B} = √ e− 2 dt . 2π B The following result is our main technical tool. Theorem 3.1. Let U be a bounded circled set in Cn , and (gα )|α|=m and (gk )1≤k≤n two families of independent standard Gaussian random variables on a probability space (Ω, µ). Then for each choice of scalars cα , |α| = m Z X cα gα z α |dµ sup | z∈U

|α|=m

r ¾ ½ Z n n X X α! 2 m−1 2 ≤ Cm sup |cα | sup( |zk | ) sup | gk zk |dµ, m! z∈U z∈U |α|=m k=1

where 0 ≤ Cm ≤ 2

3 m− 12 2

k=1

3 2

m .

Since the proof is a simple modification of [9, Theorem 3.1] we only sketch some relevant details. In fact, we prove a reformulation which needs some more notation. For natural numbers m, n we define M(m, n) := {1, . . . , n}m and J (m, n) := {j = (j1 , . . . , jm ) ∈ {1, . . . , n}m : j1 ≤ · · · ≤ jm }. Moreover, for j ∈ J (m, n) let |j| be the cardinality of the set of all i ∈ M(m, n) for which there is a permutation τ of {1, . . . , n} such that iτ (k) = jk for 1 ≤ k ≤ m. Finally, if e∗k denotes the kth coefficient functional on Cn and j ∈ J (m, n), then we write e∗j (z) := e∗j1 (z) · · · e∗jm (z), z ∈ Cn . Now observe that if α ∈ (N ∪ {0})n is a multi index such that |α| = m, then z α = z1α1 · · · znαn = e∗j (z) for all z ∈ Cn ,

Maximum moduli of unimodular polynomials α1 −times

αn −times

where j = (1, . . . . . . , 1, . . . , n, . . . . . . , n), and that in this case |j| = Hence, the following inequality is a reformulation of Theorem 3.1: Z X sup | cj gj e∗j (z)|dµ z∈U

(3.1) ≤ Cm

221 m! α! .

j∈J (m,n)

Z n n X X m−1 |cj | p sup( sup | |zk |2 ) 2 gk zk |dµ. |j| z∈U k=1 z∈U k=1 j∈J (m,n) sup

The crucial ingredient of the proof of Theorem 3.1 is Slepian’s lemma. Given N real-valued random variables Xk : Ω → R, (X1 , . . . , XN ) : Ω → RN is said to be P a Gaussian random vector provided that each real linear N combination k=1 αk Xk is Gaussian. For example, if each Xk itself is a real linear combination of standard Gaussians, then they form a Gaussian random vector. The following comparison theorem originates in the work of Slepian [25] (see Fernique [15], and also [17, Remark 1.5] and [21, Corollary 3.14]): let (X1 , . . . , XN ) and (Y1 , . . . , YN ) be Gaussian random vectors such that E|Yi − Yj |2 ≤ E|Xi − Xj |2 for each pair (i, j). Then (3.2)

E max Yi ≤ E max Xi . i

i

Proof of inequality (3.1). Without loss p of generality we may assume that all coefficients cj are real, and |cj | ≤ |j| for all j ∈ J (m, n). The aim is to estimate instead of the average Z X sup | cj gj e∗j (z)|dµ , z∈U

j∈J (m,n)

for each finite set D ⊂ U , the average Z X (3.3) sup cj gj Re e∗j (z)dµ. z∈D

j∈J (m,n)

Indeed, if P is an m-homogeneous polynomial on Cn , then it is very easy to prove, by the fact that U is circled and P is homogeneous, that kP k = supz∈U Re P (z). Since, by hypothesis, cj ∈ R and gj : Ω −→ R for all j ∈ J (m, n), for each w ∈ Ω X X sup | cj gj (w)e∗j (z)| = sup cj gj (w)Re e∗j (z). z∈U

j∈J (m,n)

z∈U

j∈J (m,n)

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Moreover, by a compactness argument for each ε > 0 there is a finite subset D ⊂ U such that Z X sup cj gj (w)Re e∗j (z)dµ z∈U

Z ≤

j∈J (m,n)

X

sup z∈D

cj gj (w)Re e∗j (z)dµ + ε.

j∈J (m,n)

m−1

m If now {αl }2l=1 is an ordering of {α : α = (αu )m u=1 ∈ {0, 1} such that n m − |α| is even}, then for every j ∈ J (m, n) and z ∈ C ,

Re e∗j (z)

=

m−1 2X

l,m εl al,1 j1 (z) · · · ajm (z),

l=1 ∗ αu ∗ 1−αu and ε := (−1)(m−|α |)/2 , 1 ≤ where al,u l k (z) := Re ek (z) Im ek (z) l ≤ 2m−1 , 1 ≤ u ≤ m and 1 ≤ k ≤ n. Define the following two Gaussian random processes: l

Yz :=

X

l

l

cj gj (w)Re e∗j (z)

j∈J (m,n)

=

X

cj gj

j∈J (m,n) 2m−1

Xz :=

m X n XX

m−1 2X

l,m n εl al,1 j1 (z) · · · ajm (z), z ∈ C ,

l=1 n gl,u,k al,u k (z), z ∈ C ,

l=1 u=1 k=1

where the gj , gl,u,k and gk all stand for independent standard Gaussian P random variables. Note term nk=1 gl,u,k al,u k (z) Pnthat for fixed∗l and u, the P coincides either with k=1 gl,u,k Re ek (z) or with nk=1 gl,u,k Im e∗k (z) for all z ∈ Cn . Fix now a finite set D ⊂ U , and show exactly as in the proof of [9, Theorem 3.1] that for each pair x, y in D Z ( |Yx − Yy |2 dµ)1/2 Z n X √ √ 2 m−1 m−1 2 ≤ 2 m sup( |zk | ) ( |Xx − Xy |2 dµ)1/2 . z∈U k=1

Maximum moduli of unimodular polynomials

223

By Slepian’s lemma (3.2) we finally obtain Z X sup cj gj Re e∗j (z)dµ z∈D

j∈J (m,n)

m−1 m Z 2X n n X XX √ √ 2 m−1 m−1 2 2 m sup( sup ≤ |zk | ) gl,u,k al,u k (z)dµ

z∈U k=1

z∈D l=1 u=1 k=1

n 2X X m Z n X √ ¯X ¯ √ 2 m−1 m−1 ¯ 2 ≤ 2 m sup( |zk | ) sup gl,u,k zk ¯dµ m−1

z∈U k=1

=

l=1 u=1 n X

√ √ 2m−1 m 2m−1 m sup(

z∈U k=1

|zk |2 )

m−1 2

Z

z∈D k=1 n X

¯ sup ¯

z∈D k=1

¯ gk zk ¯dµ.

Together with the consideration from (3.3) this completes the proof. Estimating Bernoulli by Gaussian averages, we add an explicit estimate of the expectation of the maximum modulus of an m-homogeneous Bernoulli random polynomial. Corollary 3.2. Let (εα )|α|=m and (εk )1≤k≤n be a families of independent standard Bernoulli random variables on a probability space (Ω, µ), and let cα , |α| = m be scalars. (1) For each bounded circled set U in Cn we have Z X cα εα z α |dµ sup | z∈U

|α|=m

r ¾ ½ n n X X m−1 α! |cα | sup( |zk |2 ) 2 sup |zk |. m! z∈U z∈U |α|=m

p ≤ log n Cm sup

k=1

k=1

(2) Let X = (Cn , k.k) be a Banach space for which the ek ’s form an 1-unconditional basis, and let 2 ≤ q < ∞. Then we have Z X sup | cα εα z α |dµ z∈BX

|α|=m

r ¾ ½ n n X X m−1 α! |cα | sup ( |zk |2 ) 2 sup |zk |. m! z∈BX z∈BX |α|=m

√ ≤ c qCq (X ∗ ) Cm sup

k=1

Here c > 0 is an absolute constant and Cm as above. Proof. We use the following facts:

k=1

224

Andreas Defant, Domingo Garc´ıa and Manuel Maestre

(i) Gauss averages in a Banach space always dominate Bernoulli averages (see e.g. [28, (4.2), p.15]), (ii) given a Banach space Y and y1 , . . . , yn ∈ Y , then Z X Z X n n p k gk yk kY dµ ≤ log n k εk yk kY dµ k=1

k=1

(see e.g. [28, (4.4) p.15]), (iii) there is a constant c > 0 such that, given a Banach space Y of cotype q, for each choice of y1 , . . . , yn ∈ Y , we have Z X Z X n n √ k gk yk kY dµ ≤ c qCq (Y ) k εk yk kY dµ k=1

k=1

(see e.g. [28, (4.3) p.15]), where (gk ), respectively (²k ), is a family of independent standard Gaussian, respectively Bernoulli, random variables on a probability space. Now from (i), (ii) and Theorem 3.1 we obtain Z X sup | cα εα z α |dµ z∈U

|α|=m

r ¾ ½ Z n n X X α! 2 m−1 |zk | ) 2 |cα | sup( εk zk |dµ. sup | m! z∈U z∈U |α|=m

p ≤ log nCm sup

k=1

But (3.4)

Z sup |

n X

z∈U k=1

Z εk zk |dµ ≤

sup

n X

z∈U k=1

k=1

|zk |dµ = sup

n X

z∈U k=1

|zk | ,

which proves (3.2). (3.2) follows from (i), (iii), Theorem 3.1 and (3.4). ∞ Given m ∈ N, if (an (m))∞ n=1 and (bn (m))n=1 are scalar sequences we m write an (m) ³ bn (m) whenever there are some cm , dm > 0 such that cm an (m) ≤ bn (m) ≤ dm an (m) for all n. Again we illustrate our results by several examples. The first example is the counterpart of Example 2.4.

Example 3.3.

 m+1 − m m+1 − m p v 2 q if  u 2 2 ≤ p, q ≤ ∞,      Z  X 1 1 m α sup | εα z |dµ ³ u1− p v 1− q if 1 < p, q ≤ 2,  v z∈B`u  p (`q ) |α|=m      1− p1 m+1 −m q u v 2 if 1 < p ≤ 2 ≤ q ≤ ∞.

Maximum moduli of unimodular polynomials

225

For v = 1 and p = q this result was proved in [9, Corollary 6.5]. If we allow p and q to equal 1, then in the second two cases the proof will show that we have to admit an additional log term. For 1 < q ≤ 2 ≤ p ≤ ∞ the upper estimate also holds, for the lower we don’t know (see the remark after Lemma 2.5). Proof of Example 3.3. The lower bound is immediate from Theorem 2.3. Upper bound: Since `p0 (`q0 ) in all considered cases has finite cotype, by Corollary 3.2(2) and (3.5), we get Z X εα z α |dµ sup | v z∈B`u p (`q )

≤ Dm kid :

|α|=m

`up (`vq )

m−1 −→ `uv k 2 k

X

e∗k ⊗ e∗l k`u0 (`v0 )

k,l

p

q

≤ Dm kid : `up −→ `u2 km−1 kid : `vq −→ `v2 km−1 u1−1/p v 1−1/q , which by H¨older’s inequality is the desired result. We go on with the counterpart of Example 2.7. Example 3.4. Given S > 1 and n ≥ 2 consider the Reinhardt domain R := {(z1 , . . . , zn ) ∈ Cn : |z1 · · · zn | < 1, |zk | < S, k = 1, . . . , n}. Then the following asymptotic estimate holds: Z X m m+1 εα z α |dµ ³ n 2 . sup | z∈R

|α|=m

Proof. The lower bound is a consequence of Example 2.7. The upper estimate follows from Z Z X X α sup | εα z |dµ ≤ sup | εα z α |dµ , z∈R

|α|=m

{z : |zk | 0 such that for every n ¡ ¢m Pn 1 supz∈BXn k=1 |zk | m−1 Dm n 2 Z n n X X X α 2 m−1 2 sup ≤ sup | εα z |dµ ≤ Dm sup ( |zk |. |zk | ) z∈BXn

z∈BXn k=1

|α|=m

z∈BXn k=1

(2) If X is symmetric and 2-convex, then ¡ ¢m Pn Z X m supz∈BXn k=1 |zk | α sup | εα z |dµ ³ . m−1 z∈BXn n 2 |α|=m (3) If X ⊂ `2 , then Z n X X m |zk |. εα z α |dµ ³ sup sup | z∈BXn

|α|=m

z∈BXn k=1

This result is an improvement of [9, Corollary 6.5]. Proof. The lower bound in (1) is a consequence of Theorem 2.3 (1). Since X ∗ has non-trivial concavity, which implies that it has cotype q for some 2 ≤ q < ∞ , the upper bound in (1) is a consequence of Corollary 3.2 (2). In (2) it remains to show the upper bound: We may assume that M (2) (X) = 1, hence P n X supz∈BXn nk=1 |zk | 2 1/2 √ |zk | ) = sup ( n z∈BXn k=1

(see [27, Proposition 2.2] and [11, Proposition 3.5]). Hence, the conclusion in (2) is a consequence of the upper bound in (1). Finally, the upper bound in (3) follows from the upper bound in (1), and the lower bound in (3) by Theorem 2.3 (3). In the symmetric case the above threeP statements can be reformulated in terms of the fundamental function k nk=1 ek kXn , n ∈ N of X: Remark 3.6. Let X be a symmetric Banach sequence space with non-trivial convexity, and Xn := span{ek : 1 ≤ k ≤ n}, n ∈ N. (1’) For each m there is a constant Dm such that for every n m+1

1 n 2 Pn Dm k k=1 ek km Xn Z X n ≤ sup | εα z α |dµ ≤ Dm kid : Xn −→ `n2 km−1 Pn . k k=1 ek kXn z∈BXn |α|=m

Maximum moduli of unimodular polynomials

(2’) If X is 2-convex, then Z X m sup | εα z α |dµ ³ z∈BXn

|α|=m

(3’) If X ⊂ `2 , then Z X m sup | εα z α |dµ ³ z∈BXn

|α|=m

k

k

n Pn

227

m+1 2

m . k=1 ek kXn

Pn

n

k=1 ek kXn

.

Proof. The proof easily follows from the following observations: n X sup ( |zk |2 )1/2 = kid : Xn −→ `n2 k (3.5)

z∈BXn k=1 n X

n X

z∈BXn k=1

k=1

sup

|zk | = k

e∗k kXn∗ = kid : Xn −→ `n1 k,

and the well known fact that for symmetric X for all n n n X X ek kXn e∗k kXn∗ k (3.6) n=k k=1

k=1

(see [22, Proposition 3.a.6, p.118, vol. I]). In our Corollaries 3.2 and 3.5 and Examples 3.3 and 3.4 we have proved upper bounds for the average of the maximum moduli of mhomogenous Bernoulli random polynomials. This shows the existence of m-homogeneous Bernoulli polynomials which have norms being less or equal to the bounds given in these results. Moreover, in section 2 for some cases (see e.g. the Examples 2.4, 2.6 and 2.7) we have given lower bounds for m-homogeneous Bernoulli polynomials which coincide (up to a constant) with the upper bounds obtained for the expectations in 3.3 and 3.4. Hence, we have proved the existence of m-homogeneous Bernoulli polynomials which have norms as small as possible (up to a constant). Again we illustrate this by an example. Example 3.7. Let ϕ be an Orlicz function satisfying the ∆2 -condition. Then for each m there is a constant Dm > 0 such that for every n there are signs sα , |α| = m for which P m+1 (1) supz∈B`ϕ | |α|=m sα z α | ≤ Dm n 2 ϕ−1 (1/n)m , provided ϕ(λt) ≤ Kλ2 ϕ(t) for some K > 1, P all 0 ≤ λ,αt ≤ 1 and −1 (2) supz∈B`ϕ | |α|=m sα z |≤ Dm nϕ (1/n), provided that t2≤ Kϕ(t) for all t and some K.

228

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By Example 2.6 we know that the estimates in the statements (1) and (2) are the “ smallest” possible. P As above this result easily follows from the fact that k nk=1 ek k`ϕ = 1 ; the conditions in (1) and (2) make sure that `ϕ is 2-convex and ϕ−1 (1/n) contained in `2 , respectively (see [19]). Similar results can be obtained for various other types of Banach sequence spaces, e.g. Lorentz spaces. References [1] H. Boas, Majorant series, Several complex variables, J. Korean Math. Soc. 37 (2000), no. 2, 321–337. [2] , The football player and the infinite series, Notices Amer. Math. Soc. 44 (11) (1997), 1430–1435. [3] H. P. Boas and D. Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125, 10 (1997), 2975–2979. [4] H. Bohr, A theorem concerning power series, Proc. London Math. Soc. 13 (1914), no. 2, 1–5. ¨ [5] , Uber die Bedeutung der Potenzreihen unendlich vieler Variabeln in der P Theorie der Dirichletschen Reihen an /ns , Nachrichten von der K¨ oniglichen Gesellschaft der Wissenschafen zu G¨ ottingen, Mathematisch-Physikalische Klasse (1913), 441–488. [6] H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931), 600–622. [7] S. Chevet, S´eries de variable al´eatories Gaussiens ` a valeur dans E⊗F , Application aux produits d’espaces de Wiener abstraits, S´eminaire Maurey-Schwartz, Exp. 19. Ecole Polytechnique, Paliseau, (1977). [8] A. Defant, J. C. D´ıaz, D. Garc´ıa and M. Maestre, Unconditional basis and GordonLewis constants for spaces of polynomials, J. Funct. Anal. 181 (2001), 119–145. [9] A. Defant, D. Garc´ıa and M. Maestre, Bohr’s power series theorem and local Banach space theory, J. reine angew. Math. 557 (2003), 173–197. [10] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North–Holland Math. Studies 176, 1993. [11] A. Defant, M. MastyÃlo and C. Michels, Summing inclusion maps between symmetric sequence spaces, a survey, Recent progress in functional analysis, (Valencia, 2000), 43–60, North-Holland Math. Stud., 189, North-Holland, Amsterdam, 2001. [12] S. Dineen, Complex Analysis on Infinite Dimensional Banach Spaces, SpringerVerlag. Springer Monographs in Mathematics, Springer-Verlag, London, 1999. [13] S. Dineen and R. M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234. [14] , On a problem of H. Bohr, Bull. Soc. Roy. Sci. Li`ege 60 (1991), no. 6, 401–404. [15] X. Fernique, Des resultats nouveaux sur les processus gaussiens, C. R. Acad. Sci. Paris, Ser. A-B 278 (1974), 363–365. [16] K. Floret, Natural norms on symmetric tensor products of normed spaces, Note Mat. 17 (1997), 153–188.

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Andreas Defant Fachbereich Mathematik, Universitaet D–26111, Oldenburg, Germany E-mail : [email protected] Domingo Garc´ıa Departamento de An´alisis Matem´atico Universidad de Valencia Doctor Moliner 50 46100 Burjasot Valencia, Spain E-mail : [email protected] Manuel Maestre Departamento de An´alisis Matem´atico Universidad de Valencia Doctor Moliner 50 46100 Burjasot Valencia, Spain E-mail : [email protected]