Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb C}^2$ is exactly $\sqrt[4]{\frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${\mathbb R}^2$ of higher degrees.

If · is a norm on K n , then the formula P := sup{|P(x)| : x ∈ B X }, for all P ∈ P( m K n ), where B X is the unit ball of the Banach space X = (K n , · ), defines a norm in P( m K n ) usually called polynomial norm. The space P( m K n ) endowed with the polynomial norm induced by X is denoted by P( m X ).
where n ∞ (R) and n ∞ (C) are, respectively, the real and complex versions of n ∞ . Observe that (1.3) coincides with the first inequality in (1.2) for p = 2m m+1 except for the fact that D m in (1.3) can be chosen in such a way that it is independent from the dimension n. Actually Bohnenblust and Hille showed that 2m m+1 is optimal in (1.3) in the sense that for p < 2m m+1 , any constant D fitting in the inequality |P| p ≤ D P , for all P ∈ P( m n ∞ (K)) depends necessarily on n. The best constants in (1.3) depend considerably on whether we consider the real or the complex version of n ∞ , which motivates the following definition: If we restrict attention to a certain subset E of P( m n ∞ (K)) for some n ∈ N, then we define For simplicity we will often use the notation D K,m (n) instead of D K,m (P( m n ∞ (K))). Note that 1 ≤ D K,m (n) ≤ D K,m , for all m, n ∈ N.
A good idea of the asymptotic growth of the constants D K,m and D K,m (n) is provided by the following definition: The asymptotic hypercontractivity constant of the polynomial BH inequality is Similarly, if we restrict attention to polynomials in n variables then we define It was shown in [2] that the complex polynomial Bohnenblust-Hille inequality is, at most, hypercontractive. In [3] the estimate on D C,m was improved. In fact the authors show that for every B > 1 there exists A > 0 such that D C,m ≤ Ae B √ m log m , from which it follows that H C,∞ (n) = H C,∞ = 1, for all n ∈ N. For the real case, it has been recently proved in [4] that H R,∞ = 2. However, not many exact values of D K,m (n) are known so far. This paper is devoted to calculate, explicitly or numerically some values of these constants.
This paper is arranged in two main sections. In Section 2, we employ some results on the geometry of spaces of polynomials in order to provide the exact value of D C,2 (2). In Section 3, we use a similar technique to find the exact value of D R,2 (2). We also provide lower estimates for D R,m (2) and H R,∞ (2) by means of numerical calculus.

The exact value of D C,2 (2)
Throughout this section we will often identify any two-variable polynomial az 2 +bwz+cw 2 or any one-variable polynomial aλ 2 +bλ+c, for a, b, c ∈ K, with the vector (a, b, c) ∈ K 3 . Also, we use the standard notation az 2 + bwz + cw 2 D for the supremum of |az 2 + bwz + cw 2 | for z, w in the unit disk D of C. Similarly, aλ 2 + bλ + c D stands for the supremum of |aλ 2 + bλ + c| for λ ∈ D. Observe that being the last of the latter equalities due to the maximum modulus principle.
The main result of this section depends upon the following lemma, which is of independent interest. in az 2 + bzw + cw 2 D , we can assume (without loss of generality) that a, c ≥ 0. We can also assume that a ≥ c by swapping z and w. We have: Similarly, if a , b , c are real numbers, then: . On the other hand, so that it is easy to see Also, giving the value λ = 1 (if Re(b) ≥ 0) or λ = −1 (if Re(b) ≤ 0), we can see that Divide both, numerator and denominator, by a 2 in order to convert the problem in having to achieve In the second part of the proof we shall need to employ a couple of real-valued functions that will come in handy to achieve our purpose. Let us first focus our attention on the choice of the constants a , b , c , as before, In this case, choose Then, Hence, we will achieve the desired result if we can guarantee in other words, where k has been chosen so that (a, b, c) 4 and again we guarantee that we achieve what we are searching for if we get in other words, The reader can check using elementary calculus that, if where k is chosen as in the previous case. For λ = sign(Re(b)), we still need to make sure that Next, choose In that case, Hence, we will achieve what we are searching for if we can assure that On in on, we need to prove and (in this case) we will be working with the condition 1 ≤ 2 (x, y) := 3 2 , and, in conclusion, we shall need to guarantee that 2 (x, y)}, for 0 ≤ x ≤ y ≤ 1, which we also leave as an exercise to the reader.
And, with this last case, the proof is complete.
In order to prove that D C,2 (2) = 4 3 2 we will also need the following description of the extreme points of the unit ball of R 3 endowed with the norm (a, b, c) D := sup{|az 2 + bz + c| : |z| ≤ 1} for a, b, c ∈ R. This norm has been studied by Aron and klimek [24], where they denote it by · C . Observe, once again that Theorem 2.2 (Aron and Klimek [24]) Let E R be the real subspace of P( 2 2 ∞ (C)) given by {az 2 + bwz + cw 2 : (a, b, c) where ext(B E R ) is the set of extreme points of the unit ball of E R , namely B E R and G = {(s, t) ∈ R 2 : |s| + |t| < 1 and |s + t| ≤ (s + t) 2 } ∪ {±(1, 0), ±(0, 1)}.
Theorem 2.3 The optimal complex polynomial Bohnenblust-Hille constant for polynomials in E R , which we denote by D C,2 (E R ), is given by D C,2 (E R ) = 4 3 2 . Moreover, Proof Using convexity we have If (s, t) = |s| Observe that the fact that D C,2 (2) ≥ 4 3 2 was already proved in [18]. The reader can find a sketch of the graph of on the part of G contained in the second quadrant in Figure 1.

The exact value of D R,2 (2) and lower bounds for D R,m (2)
In [4] it is proved that the asymptotic hypercontractivity constant of the real polynomial BH inequality is exactly 2. Is it true that H R,∞ (2) = 2? The results presented here suggest that, perhaps H R,∞ (2) < 2. In this section, as we did in the previous one, we will also identify polynomials with the vector of its coefficients (Figure 2). Remark 3.1 Throughout this section we will compute several times norms of polynomials on the real line numerically. This is done by using Matlab. In particular, if P(x) is a real Another Matlab predefined function, namely conv.m, is used in order to multiply polynomials. This is done to obtain Figure 4.

The exact calculation of D R,2 (2)
The value of the constant D R,2 (2) can be obtained using the geometry of the unit ball of P( 2 ∞ (R)) described in [25]. We state the result we need for completeness: Theorem 3.2 (Choi and Kim [25]) The set ext(B P( 2 2 ∞ (R)) ) of extreme points of the unit ball of P( 2 2 ∞ (R)) is given by As a consequence of the previous result, we obtain the following: We have that D R,2 (2) = f (t 0 ) ≈ , Moreover, the following normalized polynomials are extreme for this problem: Proof Let We just have to notice that due to the convexity of the p -norms and Theorem 3.2 we have : a ∈ ext(B P( 2 2 ∞ R) )} = sup t∈ [1/2,1] f (t).
Some calculations will show that the last supremum is attained at t = t 0 , concluding the proof.
Now, if a n is the vector of the coefficients of P n 2 for each n ∈ N, then we know that Since P 2 = 1, then (3.1) with n = 300 (see also Figure 4) proves that

Educated guess for the exact calculation of D R,3 (2)
To the authors'knowledge the calculation of P is, in general, far from being easy. However there is a way to compute P for specific cases. For instance Grecu et al. prove in [26,Lemma 3.12] the following formula: Lemma 3.4 If for every a, b ∈ R we define P a,b (x, y) = ax 3 + bx 2 y + bx y 2 + ay 3 then From Lemma 3.4, we have the following sharp polynomial Bohnenblust-Hille type constant: Theorem 3.5 Let P a,b (x, y) = ax 3 + bx 2 y + bx y 2 + ay 3 for a, b ∈ R and consider the subset of P( 3 The authors have numerical evidence to state that Moreover, one polynomial for which D R,3 (2) would be attained is where b 1 ≈ −1.6692 is as in Lemma 3.4. It can be proved from Lemma 3.4 that P 3 ≈ 1.33848, up to 5 decimal places. If a n is the vector of the coefficients of P 3 (x, y) n and we use the fact that

Numerical calculation of D R,5 (2)
Let us define the polynomial In any case, we have It is interesting to observe that we can improve numerically the estimate H ∞,R (2) ≥ 8 √ 27 ≈ 1.50980 (see [4,Theorem 4.2]) by considering polynomials of the form P n 5 . Indeed, if a n is the vector of the coefficients of P n 5 for each n ∈ N, then we know that

Educated guess for the exact calculation of D R,6 (2)
The authors have numerical evidence pointing to the fact that an extreme polynomial in the Bohnenblust-Hille inequality for polynomials in P( 6 2 ∞ (R)) may be of the form Q a,b (x, y) = ax 5 y + bx 3 y 3 + ax y 5 .
This motivates a deeper study of this type of polynomials, which we do in the following result.
In any case we do have that D R,6 (2) ≥ 10.7809.
As we did in the previous cases, it would be interesting to know if we can improve numerically our best lower bound on H R,∞ by considering powers of P 6 (x, y) = Q 1,λ 0 (x, y) = x 5 y + λ 0 x 3 y 3 + x y 5 , with λ 0 as in Theorem 3.6 (λ 0 ≈ −2.2654). If a n is the vector of the coefficients of P n 6 for each n ∈ N, then we know that D R,6n (2) ≥ |a n | 12n 6n+1 P 6 n .

Numerical calculation of D R,7 (2)
Let us define the polynomial P 7 (x, y) = −ax 7 + bx 6 y + cx 5 y 2 − dx 4 y 3 − dx 3 y 4 + cx 2 y 5 + bx y 6  If a n is the vector of the coefficients of P n 7 for each n ∈ N, then we know that Moreover, if we put n = 86 in (3.5) we obtain suggesting that H R,∞ (2) ≥ 1.61725.

Numerical calculation of D R,8 (2)
Let us define the polynomial If a n is the vector of the coefficients of P n 8 for each n ∈ N, then we know that

Numerical calculation of D R,10 (2)
In this case our numerical estimates show that there exists an extreme polynomial in the Bohnenblust-Hille polynomial inequality in P( 10 2 ∞ (R)) of the form P 10 (x, y) = ax 9 y + bx 7 y 3 + x 5 y 5 + bx 3 y 7 + ax y 9 , with If a n is the vector of the coefficients of P n 10 for each n ∈ N, then we know that We have sketched in Figure 4 a summary of the numerical results obtained in this section.

Disclosure statement
No potential conflict of interest was reported by the authors.