Equivalent norms in polynomial spaces and applications

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Equivalent norms in polynomial spaces and applications Gustavo Araújo a,1 , P. Jiménez-Rodríguez b , Gustavo A. Muñoz-Fernández c,2 , Juan B. Seoane-Sepúlveda d,c,∗,2 a

Unidade Acadêmica de Ciências Exatas e da Natureza, CFP, Universidade Federal de Campina Grande, Cajazeiras, PB, 58900-000, Brazil b Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA c Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain d Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 13–15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 6 November 2015 Available online xxxx Submitted by J.A. Ball Dedicated to our advisor, colleague and friend Richard M. Aron

a b s t r a c t In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy–Littlewood constants for 2-homogeneous polynomials on 2p spaces, 2 < p ≤ ∞. We also provide lower estimates for the Hardy–Littlewood constants for polynomials of higher degrees. © 2016 Elsevier Inc. All rights reserved.

Keywords: Norms Absolutely summing operators Bohnenblust–Hille inequality Hardy–Littlewood inequality

1. Introduction Let α = (α1 , . . . , αn ) ∈ (N ∪ {0})n , and define |α| := α1 + · · · + αn . Let P(m Kn ) be the finite dimensional linear space of all homogeneous polynomials of degree m on Kn (K = R or K = C). If xα stands for the αn n m n 1 monomial xα 1 · · · xn for x = (x1 , . . . , xn ) ∈ K and P ∈ P( K ), then P can be written as P (x) =



aα x α .

(1.1)

|α|=m

* Corresponding author at: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain. E-mail addresses: [email protected] (G. Araújo), [email protected] (P. Jiménez-Rodríguez), [email protected] (G.A. Muñoz-Fernández), [email protected] (J.B. Seoane-Sepúlveda). 1 Supported by PDSE/CAPES 8015/14-7. 2 Supported by the Spanish Ministry of Science and Innovation, grant MTM2012-34341. http://dx.doi.org/10.1016/j.jmaa.2016.03.039 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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If | · | is a norm on Kn , then P  := sup |P (x)|, x∈BX

where BX is the closed unit ball of the Banach space X = (Kn , | · |), defines a norm in P(m Kn ) usually called polynomial norm. The space P(m Kn ) endowed with the polynomial norm induced by X is denoted by P(m X). Equivalent norms within the real and complex settings have been the aim of many researchers since the 20th century (see, e.g. [5,6]). Other norms customarily used in P(m Kn ) besides the polynomial norm are the q norms of the coefficients, i.e., if P is as in (1.1) and q ≥ 1, then

|P |q :=

⎧ ⎨ 

q |α|=m |aα |

 q1

if 1 ≤ q < +∞,

⎩max{|a | : |α| = m} α

if q = +∞,

defines another norm in P(m Kn ). It is interesting to observe that the q norms are equivalent on Kn and that we have the following well known sharp estimates: | · |q ≤ | · |s ≤ n s − q | · |q , 1

1

for 1 ≤ s ≤ q. The polynomial norm P  is most of the times very difficult to compute, whereas the q norm of the coefficients |P |q can be obtained straightforwardly. For this reason it would be convenient to have a good estimate of P  in terms of |P |q . If  · p represents the polynomial norm of P(m np ), this paper is devoted to obtain sharp estimates on  · p (1 ≤ p ≤ +∞) by comparison with the norm | · |q (1 ≤ q ≤ +∞). Actually since all norms in finite dimensional spaces are equivalent, the polynomial norm  · p and the q norm | · |q of the coefficients are equivalent in P(m Rn ) for all 1 ≤ p, q ≤ +∞, and therefore there exist constants k > 0 and K > 0 such that kP p ≤ |P |q ≤ KP p ,

(1.2)

for all P ∈ P(m Kn ). If B|·|q and B·p denote, respectively, the closed unit ball of the spaces (P(m Kn ), | · |q ) and (P(m Kn ),  · p ), then (1.2) shows that the mapping B|·|q  P → P p is bounded by k1 whereas the mapping B·p  P → |P |q is bounded by K. Also, the continuity of P → P p and P → |P |q over (P(m Kn ), | · |q ) and (P(m Kn ),  · p ) respectively, together with the fact that the closed unit balls of the spaces (P(m Kn ), | · |q ) and (P(m Kn ),  · p ) are compact justify, the following definitions: Definition 1.1. If 1 ≤ p, q ≤ +∞ then we define

 km,n,q,p : = max P p : P ∈ B|·|q ,

Km,n,q,p : = max |P |q : P ∈ B·p .   Since km,n,q,p > 0, we can define km,n,q,p := k 1 . Also, we say that P ∈ P(m Kn ) is extremal for km,n,q,p , m,n,q,p  km,n,q,p or Km,n,q,p , if P p = km,n,q,p |P |q , km,n,q,p P p = |P |q or |P |q = Km,n,q,p P p , respectively.

Observe that km,n,q,p is the biggest k fitting in the first inequality in (1.2) whereas Km,n,q,p is the  smallest possible K in the second inequality in (1.2). Also, if a polynomial is extremal for km,n,q,p , km,n,q,p or Km,n,q,p , then its multiples are also extremal.

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Remark 1.2. In this paper we will work only with real polynomials. Actually we will study mainly polynomials   in R2 , i.e., n = 2. Therefore, for the sake of simplicity we will use km,q,p , km,q,p and Km,q,p instead of km,2,q,p , km,2,q,p and Km,2,q,p respectively. In this paper we calculate the exact values of the constants km,q,p and Km,q,p for several choices of m, q and p with m ∈ N and 1 ≤ q, p ≤ +∞, considering all the time real polynomials. It is convenient to notice that during the pair review process of this paper, another manuscript related to the same problem appeared (see [18]). In this work the authors find sharp asymptotic estimates on km,n,q,p and Km,n,q,p for a wide range of q’s and p’s as n goes to infinity. Although the present paper and [18] study the very same problem, they really do not overlap, but complement each other. Notice that we will be working in m + 1 dimensions since P(m R2 )  Rm+1 . The main tool that will be used is the well known Krein–Milman approach. This method is a consequence of the Krein–Milman Theorem, from which it can be proved that for every convex function f : C → R that attains its maximum on a convex set C ⊂ Rm+1 there exists an extreme point e of C such that f (e) = max{f (x) : x ∈ C} (see [9,19,23,26] for instance, for other applications of the Krein–Milman approach). The target function to which the Krein–Milman approach will be applied to calculate Km,q,p is B·p  P → |P |q .

(1.3)

 On the other hand, km,q,p can be calculated by applying the Krein–Milman approach to the function

B|·|q  P → P p . According to the previous comments, it seems essential to have a complete description of the sets of extreme points of B|·|q and B·p , denoted from now on as ext(B|·|q ) and ext(B·p ) respectively. As for ext(B|·|q ), it is well known that ⎧ ⎪ if q = 1, ⎪ k : 1 ≤ k ≤ m + 1} ⎨{±e m+1 ext(B|·|q ) = { k=1 k ek : k = ±1} if q = +∞, ⎪ ⎪ ⎩S if 1 < q < +∞, |·|q where {e1 , . . . , em+1 } is the canonical basis of Rm+1 and S|·|q is the unit sphere of (Rm+1 , | · |q ). On the other hand, the set ext(B·p ) has also been studied by several authors in [12,13,20–22] and will be explicitly stated for the sake of completeness whenever it is used. The calculation of the equivalence constants k2,q,p and K2,q,p has been divided into several cases. In Section 2 we will study the cases where p ∈ {1, ∞}, while in Section 3 we will study the cases where p ∈ (1, ∞). The results we have obtained in those two sections are summarized in Table 1. The problems we have just stated in the previous paragraphs are closely related to other questions of interest. For instance, the famous polynomial Bohnenblust–Hille and Hardy–Littlewood constants are defined from the constants Km,n,q,p considered above. The m-th polynomial Bohnenblust–Hille constant is 2m 2m nothing but an upper bound on Km,n, m+1 ,∞ , for n ∈ N. The reason why the specific choice q = m+1 and p = ∞ is of interest rests on the fact that if q ≥ on m and q such that

2m m+1 ,

then there exists a constant Dm,q > 0 depending only

|P |q ≤ Dm,q P ∞ ,

(1.4)

2m for all P ∈ P(m Kn ) and every n ∈ N. Moreover, any constant fitting in (1.4) for q < m+1 depends necessarily on n. This result was proved by Bohnenblust and Hille in 1931 (see [8]). Observe that any plausible choice

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Table 1 In this table () means that we have obtained the value of the constant for p ≥ 2. However we have strong numerical evidence supporting that the same holds for p ≥ 1. Also, the cases in () are still open questions. q

p

k2,q,p

1

1

1

1

∞

1

[1, 2)

∞

3q

1

1 q

−1 −1

K2,q,p √ 2+ 2 √ 1+ 2 See Remark 2.7 1

[2, ∞] [1, 2) [2, ∞] 1

∞ 1 1 (1, ∞)

3 1 1 1

∞

[4/3, ∞)

2p 3

2 p ()

[1, 2)

2

()



[2, ∞]

[2, ∞]

()

2

2q See Remark 2.7 4 See Remark 3.10

2

2



1 1 q 2 1+2 q−2

1 2(q−1) 2 1+2 q−2   max 1 , 2 q

p

for Dm,q in (1.4) must satisfy Dm,q ≥ sup{Km,n,q,∞ : n ∈ N}. The best (in the sense of smallest) possible 2m choice for Dm,q in (1.4) when q = m+1 is called the polynomial Bohnenblust–Hille constant. It is interesting to notice that there exists a considerable difference between the polynomial Bohnenblust–Hille constants for real and complex polynomials. For this reason, the polynomial Bohnenblust–Hille constants are usually denoted by DK,m . Also, if we keep n ∈ N fixed, the best (smallest) Dm (n) > 0 in 2m ≤ Dm (n)P ∞ , |P | m+1 2m for all P ∈ P(m Kn ) is denoted by DK,m (n). Observe that DK,m (n) = Km,n, m+1 ,∞ . The calculation of the Bohnenblust–Hille constants DK,m and DK,m (n) has motivated a large amount of papers (see, for example, [14]), but their exact values are still unknown except for very restricted choices of m’s and n’s. The best lower and upper estimates on DK,m and DK,m (n) known nowadays can be found in [7,9,15,16,24,27]. A similar result to that of Bohnenblust–Hille can be proved for other values of p different from ∞. Indeed, there are constants Cm,p and Dm,p independent from n such that

p |P | p−m ≤ Cm,p P p

|P |

2mp mp+p−2m

≤ Dm,p P p

for m < p ≤ 2m,

(1.5)

for 2m ≤ p ≤ ∞,

(1.6)

2mp 2m for all P ∈ P(m Kn ) and every n ∈ N. Here we put mp+p−2m = m+1 when p = ∞. Moreover, the exponents p 2mp p p−m and mp+p−2m in (1.5) and (1.6) respectively are optimal in the sense that for q < p−m and m < p ≤ 2m 2mp and 2m ≤ p ≤ ∞ on the other hand, any constant H fitting in the on the one hand, or q < mp+p−2m inequality

|P |q ≤ HP p for all P ∈ P(m Kn ) depends necessarily on n. The proof of the previous highly non-trivial results can be found in [1,17]. Let us denote by CK,m,p and DK,m,p the best (smallest) possible constants in (1.5) and (1.6) respectively. These constants are called the polynomial Hardy–Littlewood constants. Notice that the polynomial Bohnenblust–Hille constant DK,m coincides with the Hardy–Littlewood constant DK,m,p when p = ∞. Similarly as in the Bohnenblust–Hille setting, we define CK,m,p (n) and DK,m,p (n) as the best (smallest) value of the constants appearing in (1.5) and (1.6) respectively, for n ∈ N fixed. Observe that

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p 2mp CK,m,p (n) = Km,n, p−m ,p if m < p ≤ 2m and DK,m,p (n) = Km,n, mp+p−2m ,p if 2m ≤ p ≤ ∞. Therefore, we have: ⎧ ⎪ p ⎨CK,m,p ≥ sup Km,n, p−m for m < p ≤ 2m, ,p

n

⎪ mp ⎩DK,m,p ≥ sup Km,n, mp+p−2m ,p n

for 2m ≤ p ≤ ∞.

The calculation of the polynomial Hardy–Littlewood constants CK,m,p , CK,m,p (n), DK,m,p and DK,m,p (n) has been the objective of a steadily increasing number of publications during the last few years. We refer the interested reader to [2,3] and the references therein for a more detailed understanding on this topic. Section 4 is devoted to apply the results from Sections 2 and 3 to obtain sharp Hardy–Littlewood constants for 2-homogeneous polynomials on R2 . Finally, in Section 5 we provide some numerical estimates on the Hardy–Littlewood constants for polynomials of higher degree on R2 . 2. Equivalence constants k2,q,p and K2,q,p for p ∈ {1, ∞} In the following results it will be useful to have a formula for ax2 + by 2 + cxy1 and ax2 + by 2 + cxy∞ . This can be obtained by using a formula for the norm  · R defined in R3 by (a, b, c)R :=

sup |ax2 + bx + c|. x∈[−1,1]

This norm was used by Aron and Klimek in [6], where the authors provided an explicit formula for it. We reproduce it for completeness below: ⎧    b  ⎨ b2 − c  − 1 2, if |b| < 2|a| and ac + 1 < 12  2a 4a (2.1) (a, b, c)R = ⎩|a + c| + |b| otherwise. In fact, the more general norm  · [α,β] will be needed, where (a, b, c)[α,β] = sup |ax2 + bx + c|, x∈[α,β]

for α < β. Observe, however, that the norm  · [α,β] can be derived from  · R . Indeed, since the mapping ϕ(x) =

α+β α−β x+ , 2 2

is a bijection of [−1, 1] onto [α, β], for any quadratic polynomial P we have P [α,β] = P ◦ ϕR , or in other words  

2

2   α−β α−β α2 − β 2 α+β α+β   a+ b, b+c  . (a, b, c)[α,β] =  a, a+ (2.2)   2 2 2 2 2 R

Now, using a combination of (2.1) and (2.2), one can derive easily the following formulas: Theorem 2.1. If a, b, c ∈ R then ax2 + by 2 + cxy1 = max



   a+b+c a−b a+b−c   ,  , ,   4 2 4 R 

   a+b−c b−a a+b+c    , ,   4 2 4 R

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and ax2 + by 2 + cxy∞ = max{(a, c, b)R , (b, c, a)R }. Proof. In order to calculate ax2 +by 2 +cxy1 notice that ax2 +by 2 +cxy attains its norm over the unit sphere of 21 . Actually, by symmetry there is a norm attaining point either in the segment {(x, 1 + x) : x ∈ [−1, 0]} or in the segment {(x, 1 − x) : x ∈ [0, 1]}. Hence

ax2 + by 2 + cxy1 = sup ax2 + b(1 + x)2 + cx(1 + x)[−1,0] , ax2 + b(1 − x)2 + cx(1 − x)[0,1] . The proof of the first formula follows now as a direct application of (2.1) and (2.2). The proof of the second formula is similar (even easier) and is left to the reader. 2 Now we are ready to obtain equivalence constants. We focus first on the calculation of k2,q,p with p ∈ {1, ∞}. The problem is particularly easy when q belongs as well to the set {1, ∞}: Theorem 2.2. For q, p ∈ {1, ∞} we have

k2,q,p

⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎨1 = ⎪ 1 ⎪ ⎪ ⎪ ⎩1 3

if q = p = 1, if q = 1 and p = ∞, if q = ∞ and p = 1, if q = p = ∞.

Extremal polynomials are given in the following list: p1,1 (x, y) = ±x2 , ±y 2 , p1,∞ (x, y) = ±x2 , ±y 2 , ±xy, p∞,1 (x, y) = ±x2 ± y 2 ± xy,   p∞,∞ (x, y) = ± x2 + y 2 ± xy .   Proof. It will be enough to calculate k2,q,p with p, q ∈ {1, ∞}. Recall that k2,q,p = 1/k2,q,p . According to the Krein–Milman approach, the extreme polynomials in B|·|q (see (1.3)) and Theorem 2.1 we have  k2,1,1 = max{ ± ek 1 : k = 1, 2, 3} = 1 where 1 is attained for ±e1 and ±e2 ,  = max{ ± ek ∞ : k = 1, 2, 3} = 1 where 1 is attained for ±e1 , ±e2 and ±e3 , k2,1,∞  = max{ ± e1 ± e2 ± e3 1 } = 1 where 1 is attained for ±e1 ± e2 ± e3 , k2,∞,1  = max{ ± e1 ± e2 ± e3 ∞ } = 3 where 3 is attained for ±(e1 + e2 ± e3 ). k2,∞,∞

2

The calculation of k2,q,p for p ∈ {1, ∞} and q ∈ (1, ∞) is much more complicated due to the fact that the set of extreme points of B|·|q with q ∈ (1, ∞) is the whole unit sphere S|·|q . Using a heuristic reasoning we obtain what seems to be the exact value of k2,q,p with p ∈ {1, ∞} and q ∈ (1, ∞) together with the corresponding extremal polynomials. Our argument is based on the numerical approach of k2,q,p with p ∈ {1, ∞} and q ∈ (1, ∞): For the calculation of k2,q,1 we have considered a mesh of points P in S|·|1 , and then we have computed (using MATLAB) the quotient |P |q /|P 1 for all the points of the mesh. For several refinements of the mesh and several choices of q in (1, ∞), the quotient was always less than or equal to 1, with equality attained for

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Table 2 Numerical approximation of k2,q,∞ for several values of q and the corresponding extremal polynomials. q

k2,q,∞

Extremal polynomial

1 2 5 10 50 100

1.0000 0.5774 0.4152 0.3720 0.3408 0.3371

±e1 , ±e2 , ±e3 ±(0.5750, −0.5768, −0.5802) ±(0.8050, −0.8050, −0.8021) ±(0.8950, −0.8983, −0.8945) ±(0.9800, −0.9761, −0.9785) ±(0.9900, −0.9905, −0.9862)

1

Fig. 1. Graph of the numerical approximation of k2,q,∞ for q ∈ [1, 15] (black) compared to the graph of 3 q

−1

(gray).

±e1 , ±e2 . This suggests that k2,q,1 = 1, ∀q ∈ (1, ∞) and that the polynomials of the form pq,1 = ±e1 , ±e2 are extremal. As for the calculation of k2,q,∞ (q ∈ (1, +∞)), the reader can find in Table 2 a list with numerical approximations of k2,q,∞ (q ∈ (1, +∞)) for several values of q obtained using MATLAB. It called our attention how close the approximated extreme polynomials are from having the form ±(a, −a, −a) with a ∈ (0, 1). If those polynomials were indeed extremal for k2,q,∞ , then we would have 1

k2,q,∞

1 |(a, −a, −a)|q 3q = 3 q −1 , = = (a, −a, −a)∞ 3

for all q ∈ (1, ∞). This reasoning leads us to conjecture that k2,q,∞ = 3 q −1 , for all q ∈ (1, ∞) and that the multiples of the polynomial pq,∞ (x, y) = x2 − y 2 − xy are extreme for k2,q,∞ . The reader can find in Fig. 1 1 a clarifying graph of our numerical approximations of k2,q,∞ in comparison with 3 q −1 . Observe that both graphs almost coincide. In order to calculate K2,q,p with p = 1 or p = ∞ using the Krein–Milman approach, we need a full description of ext(B·1 ) and ext(B·∞ ), which is provided in the following results: 1

Theorem 2.3. (See Y.S. Choi, S.G. Kim, and H. Ki, [13].) The extreme polynomials of B·1 are of the form (a) P (x, y) = ±x2 ± 2xy ± y 2 , or (b) P (x, y) = ±

4|t|−t2 (x2 2

− y 2 ) + txy, where |t| ∈ (2, 4].

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Theorem 2.4. (See Y.S. Choi, S.G. Kim, [12].) The extreme polynomials of B·∞ are of the form (a) P (x, y) = ±x2 , or (b) P (x, y) = ±y2 , or     (c) P (x, y) = ± tx2 − ty 2 ± 2 t(1 − t)xy , where t ∈ 12 , 1 . Theorem 2.5. For q, p ∈ {1, ∞} we have

K2,q,p

⎧ √ ⎪ 2+2 2 ⎪ ⎪ ⎪ ⎨1 + √2 = ⎪ 4 ⎪ ⎪ ⎪ ⎩ 1

if q = p = 1, if q = 1 and p = ∞, if q = ∞ and p = 1, if q = p = ∞.

Extremal polynomials for each of the above cases are given in the following list: √

√ 2 2 (x − y 2 ) + (2 + 2)xy, 2 √ √ √

2 2+ 2 2 2+ 2 2 x − y ± xy , P1,∞ (x, y) = ± 4 4 2 P1,1 (x, y) = ±

P∞,1 (x, y) = ±4xy,



P∞,∞ (x, y) = ±x , ±y , ± 2

2

1 2 1 2 x − y ± xy . 2 2

Proof. Using the Krein–Milman approach K2,1,1 = sup{|P |1 : P ∈ B·1 } = sup{|P |1 : P ∈ ext(B·1 )}, K2,1,∞ = sup{|P |1 : P ∈ B·∞ } = sup{|P |1 : P ∈ ext(B·∞ )}, K2,∞,1 = sup{|P |∞ : P ∈ B·1 } = sup{|P |∞ : P ∈ ext(B·1 )}, K2,∞,∞ = sup{|P |∞ : P ∈ B·∞ } = sup{|P |∞ : P ∈ ext(B·∞ )}. Now using Theorems 2.3 and 2.4 we have        ± 4|t| − t2 ± 4|t| − t2   ,− , t  : |t| ∈ (2, 4] = max |(±1, ±1, ±2)|1 ,    2 2 1    √ = max 4, 4t − t2 + t : t ∈ (2, 4]) = 2 + 2 2. 

K2,1,1

√ The reader can check using elementary calculus that the previous maximum is attained at t = 2 + 2, which √ √ also provides the extreme polynomials ± 22 (x2 − y 2 ) + (2 + 2)xy for K2,1,1 .        K2,1,∞ = max |(±1, 0, 0)|1 , |(0, ±1, 0)|1 ,  t, −t, ±2 t(1 − t)  : t ∈ (1/2, 1] 1    √ = max 1, 2t + 2 t(1 − t) : t ∈ [1/2, 1]) = 1 + 2. Observe above is attained at t =   √ that the√maximum √ 2+ 2 2 2+ 2 2 2 ± 4 x − 4 y ± 2 xy for K2,1,∞ .

√ 2+ 2 4 ,

from which we obtain the extreme polynomials

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       ± 4|t| − t2 ± 4|t| − t2   ,− , t  : |t| ∈ (2, 4] = max |(±1, ±1, ±2)|∞ ,    2 2 ∞   √ 2 4t − t , t : t ∈ (2, 4]) = 4. = max 2, 2 

K2,∞,1

Since the previous maximum is attained at t = 4, the polynomials ±4xy are extremal for K2,∞,1 .        K2,∞,∞ = max |(±1, 0, 0)|∞ , |(0, ±1, 0)|∞ ,  t, −t, ±2 t(1 − t)  : t ∈ (1/2, 1] ∞    = max 1, t, 2 t(1 − t) : t ∈ [1/2, 1]) = 1. Now the maximum is attained for the polynomials ±x2 , ±y 2 and also at t = 12 , which provides the additional   extreme polynomial ± 12 x2 − 12 y 2 ± xy for K2,∞,∞ . 2 Theorem 2.6. For every q ∈ [1, ∞), let fq,1 : [2, 4] → R and fq,∞ :

1



2, 1

→ R be given by

  q1 q fq,1 (t) = 21−q (4t − t2 ) 2 + tq ,   q1 q fq,∞ (t) = 2tq + 2q (t − t2 ) 2 . Then K2,q,1 = max{fq,1 (t) : t ∈ [2, 4]},    1 ,1 . K2,q,∞ = max fq,∞ (t) : t ∈ 2 1

Actually K2,q,1 = 4 and K2,q,∞ = 2 q for every q ≥ 2, with extremal polynomials given, respectively, by the multiples of Pq,1 (x, y) = ±4xy, Pq,∞ (x, y) = ±(x2 − y 2 ). Proof. Using the Krein–Milman approach, we have that K2,q,1 = sup{|P |q : P ∈ B·1 } = sup{|P |1 : P ∈ ext(B·1 )}, K2,q,∞ = sup{|P |q : P ∈ B·∞ } = sup{|P |1 : P ∈ ext(B·∞ )}. Now using the extreme polynomials introduced in Theorems 2.3 and 2.4, and taking into account that 1 fq,1 (2) = (2 + 2q ) q , we have ⎧ ⎨

K2,q,1

⎫       ± 4|t| − t2 ⎬ ± 4|t| − t2   ,− , t  : |t| ∈ (2, 4] = max |(±1, ±1, ±2)|q ,    ⎩ ⎭ 2 2    1 q q1 1−q 2 q2 q q : t ∈ (2, 4]) = max (2 + 2 ) , 2 (4t − t ) + t = max{fq,1 (t) : t ∈ [2, 4]}.

q

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10

Table 3 Value of max fq,1 (t) and max fq,∞ (t) obtained numerically for some choices of q. q ∈ [1, 2] 1.00 4/3 3/2 1.75 2.00

Maximum of fq,1 (t) √ 2(1 + 2) 4.11346 4.02012 4.00003 4

Maximum of fq,∞ (t) √ 1+ 2 1.83737 1.67869 1.51651 √ 2

Observe that fq,1 (4) = 4 for all q ≥ 1, which implies that K2,q,1 ≥ 4 for all q ≥ 1. If, in addition, q ≥ 2, then " from the fact that | · |2 ≥ | · |q we deduce that fq,1 (t) ≤ f2,1 (t) for all t ∈ [2, 4], that is, f2,1 (t) = 2t + t2 ≤ f2,1 (4) = 4 for all t ∈ [2, 4]. Therefore K2,q,1 = 4 for all q ≥ 2 and the fact that fq,1 attains its maximum at t = 4 accounts for the fact that Pq,1 (x, y) = ±4xy is extremal for K2,q,1 if q ≥ 2. On the other hand        K2,q,∞ = max |(±1, 0, 0)|q , |(0, ±1, 0, )|q ,  t, −t, 2 t(1 − t)  : t ∈ [1/2, 1] 2

q

    q1 q : t ∈ [1/2, 1]) = max 1, 2tq + 2q (t − t2 ) 2 = max{fq,∞ (t) : t ∈ [1/2, 1]}. 1

Now it is obvious that K2,q,∞ ≥ fq,∞ (1) = 2 q . Also, since 4(t − t2 ) ≤ 1 for

1 2

≤ t ≤ 1, if q ≥ 2 we have

q 1 1 q fq,∞ (t) = tq + 4(t − t2 ) 2 ≤ t2 + 2(t − t2 ) = 2t − t2 ≤ 1, 2 2   1 1 for all t ∈ 12 , 1 . This shows that K2,q,∞ ≤ 2 q for q ≥ 2, and therefore K2,q,∞ = 2 q if q ≥ 2. Also, since fq,∞ attains its maximum at t = 1, it follows that Pq,∞ (x, y) = ±(x2 − y 2 ) is extreme for K2,q,∞ whenever q ≥ 2. 2 Remark 2.7. The maximum of the functions fq,1 and fq,∞ can be approximated using numerical calculus with MATLAB for any 1 < q < 2. We present in Table 3 some values obtained by computer for a few q  s. The exact calculation of the maximum of fq,1 and fq,∞ or the point of attainment of the maximum seems to be a far more difficult task that we do not solve in this paper. However, the symbolic calculus tool of MATLAB yields, for very specific choices of q, the exact point of attainment of the maximum expressed explicitly as a combination of radicals. We present here some of the very few examples where MATLAB managed to provide the exact point of attainment of the maximum. For instance, for q = 4/3 the maximum of fq,1 (t) is attained at, precisely, t=

1 9

"

" √ √ 3 3 2 181 + 9 273 + 1448 − 72 273 + 14 ≈ 3.79842.

Also, for q = 3/2 the maximum of fq,1 is attained at t ≈ 3.941955, whose exact value is given by ⎛% (  &  )  √  1 ⎝& 2 3 ' 3 √ + 5 22/3 3 9 + 93 + 24 + t= 6 −10 32/3 15 9 + 93

+

% ⎛ & ( & & ⎜ &6 ⎝10 32/3 3 '

9+

2 √

93

% )  & √  − 5 22/3 3 3 9 + 93 + 204& '

⎞ −10 32/3

⎟ ⎟ "  + 48⎠ + 18⎟ √  ⎠. + 5 22/3 3 3 9 + 93 + 24 6

" 3

9+

2 √ 93

⎞

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11

Finally, for q = 4/3 the maximum of fq,∞ (t) is attained at, precisely, t=

1 36

"

" √ √ 3 3 2 107 + 9 129 + 856 − 72 129 + 16 ≈ 0.86783,

whereas for q = 3/2 the maximum of fq,∞ is attained at %  & 3 √ 1& 40 & 10 9 + 273 − " t= √  +1+  20 ' 32/3 3 3 9 + 273 %  & 3 √ 2 1& 1 9 + 273 49 & + " + &− + + ) 3 √ 2' 50 5 3 3 9 + √273 10 32/3 273 50 10 39+ − 2/3

+ "  40√  3 3 9+ 273

+1

9 ≈ 0.878721. 20

3. Equivalence constants k2,q,p and K2,q,p for p ∈ (1, ∞) The calculation of k2,q,p with p ∈ (1, ∞) in general seems to be a difficult problem due to the lack of a formula for  · p when p ∈ (1, ∞). However we have been able to obtain explicitly the following values for k2,1,p and k2,∞,p . Theorem 3.1. If p ∈ (1, ∞) then

k2,q,p

⎧ ⎨1 2 = ⎩2p 3

if q = 1, if q = ∞ and p ≥ 43 .

Moreover, extremal polynomials for k2,1,p and k2,∞,p are given, respectively, by p1,p (x, y) = ±x2 , ±y 2 p∞,p (x, y) = ±(x2 + y 2 + xy). Proof. According to the Krein–Milman approach, using the extreme polynomials in B|·|1 and B|·|∞ (see (1.3)) we have  k2,1,p = max{ ± ek p : k = 1, 2, 3},  = max{ ± e1 ± e2 ± e3 p }. k2,∞,p

Since e1 p = e2 p = 1 and e3 p = 2− p , we conclude that k2,1,p = 1 for all p ∈ (1, ∞) and the multiples  of e1 and e2 are extremal polynomials for k2,1,p . As for k2,∞,p , observe that k2,∞,p = e1 + e2 + e2 p , or in other words   2 1  k2,∞,p = max x2 + (1 − xp ) p + x(1 − xp ) p : x ∈ [0, 1] . (3.1) 2

It remains to prove that if p ≥ 4/3, then the above maximum is attained at x = 2− p . Let f (x) = x2 + (1 − xp )2/p + x(1 − xp )1/p , for x ∈ [0, 1] and p > 4/3. Make the change u = 1 − xp to reduce the problem to finding that the maximum of g(u) = (1 − u)2/p + u2/p + u1/p (1 − u)1/p is attained at u = 12 . If p ≥ 2 then the functions (1 − u)2/p + u2/p and u1/p (1 − u)1/p attain their maximum simultaneously at u = 1/2 and we are done. Let us suppose now that 43 ≤ p < 2. By symmetry we can assume that u ∈ [1/2, 1]. Then 1

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12

0 = g  (u) =

2 2/p−1 2 1 1 u − (1 − u)2/p−1 + u1/p−1 (1 − u)1/p − u1/p (1 − u)1/p−1 , p p p p

if and only if . / . / u1/p−1 2u1/p + (1 − u)1/p = (1 − u)1/p−1 u1/p + 2(1 − u)1/p , that is,

1−u u

1−1/p =

u1/p + 2(1 − u)1/p , 2u1/p + (1 − u)1/p

or equivalently,  1/p 1 + 2 1−u 1−u u =  . −1/p u 2 1−u + 1 u Make the change s = 1−u u , which lies in (1, ∞). Therefore, finding the critical points of g over [1/2, 1] reduces to solving the equation   s = 2 s1/p − s1−1/p + 1.

(3.2)

It turns out that 1 is a solution for (3.2). Now, if there were a second solution for a value s > 1, as an easy consequence of Rolle’s Theorem applied to (3.2) there should be an s, also greater than 1, so that  1=2 . However, if h(x) = 2 we have

1 1/p−1 px

 1 1/p−1 1 −1/p s s . − 1− p p

  / − 1 − p1 x−1/p , then h(1) =

h (x) =

4 p

(3.3)

− 2 ≤ 1 since p ≥

4 3

but, for every x > 0

 2(1 − p)  p1 −2 − p+1 p − x x < 0, p2

since p < 2, which contradicts (3.3). 2   Remark 3.2. We have not been able to find the exact value of k2,∞,p for p ∈ 1, 43 . In any case, from the proof of Theorem 3.1 we still have   2 1  k2,∞,p = max x2 + (1 − xp ) p + x(1 − xp ) p : x ∈ [0, 1] ,   for p ∈ 1, 43 with extremal polynomial x2 + y 2 + xy. Using numerical computations with MATLAB applied  to the previous formula, we have been able to approach k2,∞,p (and therefore k2,∞,p too) for several values 2 p

of p. The reader can find in Fig. 2 a representation of the difference 23 − k2,∞,p for p ∈ [1, 10] obtained 2   p numerically, which suggests that k2,∞,p < 23 for p ∈ 1, 43 and that 4/3 is (or seems to be) the threshhold 2

value p0 of p for which k2,∞,p =

2p 3

for all p ≥ p0 .

The calculation of K2,q,p with p ∈ [2, ∞) is based on the Krein–Milman approach and a characterization of the extreme points of the unit ball of P(2 2p ). From Propositions 2.1 and 2.3 in [22] we have the extreme polynomials of P(2 2p ) for p > 2.

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2

Fig. 2. Representation of the difference

2p 3

13

− k2,∞,p for p ∈ [1, 10].

Theorem 3.3. (See B. Grecu [22].) Let p > 2. A 2-homogeneous polynomial of unit norm P is an extreme point of the unit ball of P(2 2p ) if and only if (i) P (x, y) = ±(x2 − y 2 ). p p (ii) P (x, y) = ax2 + cy 2 where ac ≥ 0 and |a| p−2 + |c|p−2 = 1, or  p p  p−2 p−2 2 2 (iii) P (x, y) = ± αα2 −β + 2αβ α α2 +β xy , with α, β ≥ 0 and αp + β p = 1. +β 2 x − y +β 2 The reader can find several parametrizations of the extreme points of the unit ball of P(2 22 ) (see for instance [12] and [25]). We will use the parametrization given in [25]: Theorem 3.4. The extreme points of the unit ball of P(2 22 ) are given by  ±(ax2 − ay 2 + 2 1 − a2 xy),

with a ∈ [−1, 1] and

±(x2 + y 2 ).

The previous results in combination with the Krein–Milman approach allow us to prove the following: Theorem 3.5. For every q ≥ 1 and p ≥ 2, let fq,p : [0, 1] → R be given by

fq,p (s) :=

⎧ 1 q q q q ⎪ ⎪ ⎨ 2(1 − s) 2 + 2 s 2 ⎪ ⎪ ⎩

1− 1 2|1−2s|q +2q (1−s) p 2

1 sp

1 +(1−s) p

1− 1 s p

q  1

2

(1−s) p +s p

if p = 2,

q

if p = 2.

Then K2,q,p = max{fq,p (t) : t ∈ [0, 1]}. Proof. For p = 2, using the Krein–Milman approach and Theorem 3.4 we have 

1 q



K2,q,2 = max 2 , 2t + 2 (1 − t ) q

q

2

q 2

 q1

  1  : t ∈ [−1, 1] = max 2 q , fq,2 (s) : s ∈ [0, 1] . 1

Observe that in the last equality we have used the substitution s = 1 − tp . Since fq,2 (1) = 2 q , the result follows.

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Now assume that p > 2 and apply the Krein–Milman approach and Theorem 3.3. Then ⎧ ⎨ K2,q,p = max

1

⎩

2q ,

q

max p

p |a| p−2

+|c| p−2 =1

|a|

+|c|

ac≥0

q

1

(|a| + |c| ) q ,

   p

q  q1   α − β p q αp−2 + β p−2   max 2 2 + 2αβ α + β2  α2 + β 2 αp +β p =1α,β≥0 ⎧ ⎫ ⎨ 1 ⎬ 1 q q q = max 2 q , max (|a| + |c| ) , max f (s) . q,p p p ac≥0 ⎩ ⎭ s∈[0,1] p−2 p−2 =1

1

Observe that in the last inequality we have considered the change of variable s = β p . Since fq,p (1) = 2 q ≥ p p 1 (|a|q + |c|q ) q for all (a, c) such that |a| p−2 + |c| p−2 = 1, the result follows. 2 Remark 3.6. The statement of Theorem 3.5 appeared first in [4] for p > 2 (although the authors in [4] provide a different parametrization of fq,p than that given here). A proof for the case p > 2 was given later in [11] using ideas taken from [4] and an incomplete description of the extreme points of the unit ball of P(2 2p ) with p > 2. The mapping fq,p is difficult to optimize in general. However, in the following results we manage to obtain the maximum of fq,p for several choices of q, p > 1, which allows us to obtain the exact value of K2,q,p . Theorem 3.7. If q > 1 then ⎧ 2 ⎪ ⎪ ⎪ ⎨ K2,q,2 =

2

1 1+2 q−2

1

if q ≥ 2,

q

⎪ ⎪

1 ⎪ 2 ⎩ 1+2 2(q−1) q−2

if 1 < q < 2.

Moreover, the following polynomials are extremal for K2,q,2 : ⎧ ⎨±(x2 − y 2 )    Pq,2 (x, y) = ⎩± a0 x2 − a0 y 2 + 2 1 − a20 xy

if q ≥ 2, if 1 < q < 2,

 − 12 2(1−q) where a0 = 1 + 2 q−2 . Proof. According to Theorem 3.5 it will be enough to find the maximun of q

q

q gq,2 (s) := fq,2 (s) = 2(1 − s) 2 + 2q s 2 ,

with s ∈ [0, 1]. Observe that gq,2 (s) is convex if q/2 ≥ 1. Hence max{gq,2 (s) : s ∈ [0, 1]} = max{gq,2 (0), gq,2 (1)} = gq,2 (1) = 2q . On the other hand, if q/2 < 1 then gq,2 (s) is strictly concave on [0, 1], and therefore its maximum is attained at the unique critical point of gq,2 (s) in (0, 1), which can be proved to be s0 =

2

2(1−q) q−2

1+2

2(1−q) q−2

. Hence, if 1 < q < 2,

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15

  1 2q 1 + 2 q−2 max{gq,2 (s) : s ∈ [0, 1]} = gq,2 (s0 ) =   q2 . 2(q−1) 1 + 2 q−2 In order to obtain the extremal polynomials given above, just recall from the proof of Theorem 3.5 that the √ relationship between the parameter a in Theorem 3.4 and the variable s is a = 1 − s. 2 The following result has a very technical proof that will be given in detail. It generalizes the results appearing in [11]. Actually, the authors in [11] just prove (leaving the key calculations to the reader) our Case 1 here: Theorem 3.8. If q, p > 2 then 

K2,q,p = 2

max

1 2 q,p



.

Moreover, if fq,p is as in Theorem 3.5 and q, p > 2, then the following polynomials are extremal for K2,q,p : 2

Pq,p (x, y) = ±2 p xy

if q ≥ p/2,

Pq,p (x, y) = ±(x − y ) 2

2

if q < p/2.

     1 2 2 1 max q1 , p Proof. Since K2,q,p ≥ max fq,p (1), fq,p (2− p ) = max 2 q , 2 p = 2 , it is enough to show that 

max

1 2 q,p



K2,q,p ≤ 2 . The proof of the latter inequality will be divided into 4 cases, where we will use the fact that K2,q,p = max{fq,p (t) : t ∈ [0, 1]} proved in Theorem 3.5. In fact, due to the symmetry of fq,p when p = 2, in order to maximize fq,p over [0, 1] we can restrict attention to the interval [0, 1/2]. Case 1: Suppose that 2 < p ≤ 4 (and hence p/2 ≤ q). Observe that fq,p (t) ≤ f2,p (t) since q ≥ 2. We will 2 show that max {f2,p (t) : t ∈ [0, 1]} = 2− p whenever 2 < p ≤ 4. Observe first that  1/2 2 + 4(1 − s)2−a sa + 4(1 − s)a s2−a f2,p (s) = , (1 − s)a + sa where a = 2/p ∈ [ 12 , 1). Then f2,p (s) ≤ 2a for every s in [0, 1/2] is equivalent to 1 + (1 − s)2−a sa + (1 − s)a s2−a ≤ 22a−2 (1 − s)2a + 22a (1 − s)a sa + 22a−2 s2a , 2 that is, 1 ≤ 22a−2 (1 − s)2a + 22a (1 − s)a sa + 22a−2 s2a − (1 − s)2−a sa − (1 − s)a s2−a 2     = (1 − s)a sa 22a − (1 − s)2−2a − s2−2a + 22a−2 s2a + (1 − s)2a .   Now, (1 − s)a sa 22a − (1 − s)2−2a − s2−2a ≥ 0 over [0, 1/2], since the function (1 − s)2−2a + s2−2a attains its maximum over [0, 1/2] at s = 0, where its value is 1, whereas 22a ≥ 2. Also, the function s2a + (1 − s)2a attains its minimum over [0, 1/2] at s = 12 . Then,       (1 − s)a sa 22a − (1 − s)2−2a − s2−2a + 22a−2 s2a + (1 − s)2a ≥ 22a−2 s2a + (1 − s)2a 2a 1 1 2a−2 2 = . ≥2 2 2

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16

Case 2: Assume now that p ≥ 4 and q ≥ p2 . Now, fq,p (s) ≤ 22/p for s ∈ [0, 1/2] if and only if  Since q ≥

p 2

. /q 1/q . / ≤ 22/p (1 − s)2/p + s2/p . 2(1 − 2s)q + 2q (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p then | · |q ≤ | · |p/2 and hence,  . /q 1/q 2(1 − 2s)q + 2q (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p  ≤

. /p/2 2/p p 2 2q (1 − 2s)p/2 + 2p/2 (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p

 ≤

2(1 − 2s)

p/2

+2

p/2

.

(1 − s)

1−1/p 1/p

s

+ (1 − s)

1/p 1−1/p

s

/p/2 2/p .

So it will be enough to show . . /p/2 /p/2 2(1 − 2s)p/2 + 2p/2 (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p ≤ 2 (1 − s)2/p + s2/p , or equivalently . . /p/2 /p/2 2p/2 (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p ≤ 2 (1 − s)2/p + s2/p − 2(1 − 2s)p/2 ,

(3.4)

for 0 ≤ s ≤ 12 . If f (s) and g(s) denote, respectively, the left and right hand side of (3.4), then it is easy to see that f and g are strictly concave and strictly increasing in [0, 1/2]. Hence f and g coincide at most at two points in [0, 1/2]. Since f (0) = g(0) and f (1/2) = g(1/2) the proof follows from the easily verified fact that f (1/4) < g(1/4). Case 3: If we assume now 4 ≤ q < equivalent to 

p 2,

we would like to show that fq,p (s) ≤ 21/q , for t ∈ [0, 1/2], which is

.  q /1/q 1 ≤ (1 − s)2/p + s2/p , |1 − 2s|q + 21− q (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p

for t ∈ [0, 1/2]. Acting as in the previous case and using that 4 ≤ q ≤ p2 , the problem reduces to show that .  /4 . /4 (1 − 2s)4 + 21−2/p (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p ≤ (1 − s)2/p + s2/p , for t ∈ [0, 1/2], or equivalently .

 /4 . /4 ≤ (1 − s)2/p + s2/p − (1 − 2s)4 , 21−2/p (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p

(3.5)

for t ∈ [0, 1/2]. The proof of (3.5) follows from the geometric argument used in the previous case, that is, if f (s) and g(s) denote now the left and right hand side of (3.5) respectively, then f and g coincide at most at two points of [0, 1/2] since both are strictly concave and strictly increasing. Since we have again that f (0) = g(0) and f (1/2) = g(1/2) the proof follows from the easily verified fact that f (1/4) < g(1/4). Case 4: Suppose finally that 2 < q ≤ 4. Observe that  0 1 1 2 + k , 2 + k−1 . (2, 4] = 2 2 k≥0

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17

  1 If we take q ∈ 2 + 21k , 2 + 2k−1 we can assume that q ≤ p2 since the case q > p2 has already been covered by the previous cases. We want to show that fq,p (s) ≤ 21/q , for t ∈ [0, 1/2], which is equivalent to 

.  q /1/q 1 ≤ (1 − s)2/p + s2/p , |1 − 2s|q + 21− q (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p

for t ∈ [0, 1/2]. Now since q ≥ 2 + 

1 , 2k

we have

.  /q 1/q 1 |1 − 2s|q + 21− q (1 − s)1−1/p s1/p + (1 − s)1/p s1−1/p 2+

≤

|1 − 2s|

Hence, since 1 − 2+

|1 − 2s|

1 q

1 2k

1 2k

.

1− q1

+ 2



(1 − s)

1−1/p 1/p

s

+ (1 − s)

1/p 1−1/p

/2+

s

1 2k

2k /(2k+1 +1) .

≤ 2 − p2 , we obtain the desired result if we can prove that .

2 1− p

+ 2



(1 − s)

1−1/p 1/p

s

+ (1 − s)

1/p 1−1/p

/2+

s

1 2k

. / k k+1 +1) 2 2 2 /(2 ≤ (1 − s) p + s p ,

for t ∈ [0, 1/2]. The latter inequality follows from the same geometric argument used in Case 2 and Case 3. 2 As an almost immediate consequence of the previous results we have: 2

Theorem 3.9. If p ≥ 2 then K2,∞,p = 2 p . Moreover, the polynomials P∞,2 (x, y) = x2 − y 2 , 2

P∞,p (x, y) = 2 p xy,

p > 2,

are extreme for K2,∞,p , p ≥ 2. Proof. Let us fix p ≥ 2. Observe first that | · |∞ ≤ | · |q for all q ≥ 2 (actually for all q ≥ 1). Hence, if we 2 take q large enough we have, according to Theorems 3.7 and 3.8, that K2,∞,p ≤ K2,q,p = 2 p . Equality is attained considering the extreme polynomials P∞,p (x, y) with p ≥ 2 given above. 2 Remark 3.10. If p > 2, applying once more the Krein–Milman approach together with Theorem 3.3, we prove that K2,1,p = sup{f1,p (s) : s ∈ [0, 1/2]}, where

f1,p (s) =

. / 1 1 1 1 2(1 − 2s) + 2 (1 − s)1− p s p + (1 − s) p s1− p 2

2

(1 − s) p + s p

,

for all s ∈ [0, 1/2]. Although the above supremum can always be approximated using numerical calculus, the problem of finding its exact value is a quite different thing. We have been unable to provide a general formula for the supremum of f1,p (p > 2) nor for its point of attainment, however, for a few specific choices of p this √ 3−2 2 task can be done. For instance, if p = 4, f1,4 attains its maximum over [0, 1/2] at s0 = 6 ≈ 0.02859, √ and hence K2,1,4 = f1,4 (s0 ) = 6.

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4. Application to the calculation of the polynomial Hardy–Littlewood constants The Krein–Milman approach allows us to calculate Hardy–Littlewood constants for the case of 2-homogeneous polynomials on R2 . Let us take into consideration the following general lower bound for m ≥ 2 and 2m ≤ p < ∞ (see [3]): DR,m,p ≥

 √ m m2 p+10m−p−6m2 −4 16 4mp 2 ≥2 .

As a consequence of Theorem 3.8 we have: 2

p p Theorem 4.1. For 2 < p ≤ 4, we have DR,2,p (2) = K2, p−2 ,p = 2 .

Observe that the previous result was obtained by Cavalcante et al. in [11]. Moreover, we can also prove the following interesting result for p = 4. √ Theorem 4.2. We have DR,2,4 (2) = CR,2,4 (2) = K2,2,4 = 2 and, surprisingly, all the extreme polynomials given in Theorem 3.3 of the types (i) and (iii) are also extremal. More explicitly, for the polynomials P (x, y) = ±



α2 − β 2



  x2 − y 2 + 2αβxy , with α, β ≥ 0 and α4 + β 4 = 1,

or P (x, y) = ±(x2 − y 2 ), we obtain |P |2 = 1   Proof. We just need to notice that 2 

√

2P 4 .

21 2   √   1 2 2 4 4  2α 1 − α = 2, for all α ∈ [0, 1]. 1  + 2 4 α +(1−α ) 2 2α4 −1

2

For the rest of plausible values of p we can apply Theorem 3.5: Theorem 4.3. For 4 ≤ p ≤ ∞, 3p−4  4p  4p . / 3p−4 4p 4p 1 1 1 1 1− 1− 2|1 − 2s| 3p−4 + 2 3p−4 (1 − s) p s p + (1 − s) p s p

DR,2,p (2) = K2,

4p 3p−4 ,p

= max

2

2

(1 − s) p + s p

s∈[0,1/2]

.

Remark 4.4. In general, we have not been able to optimize the function appearing in Theorem 4.3. However we provide a table with some numerical calculations. Notice that, as p increases, both the value of the constant K2, 4p ,p and the extremal polynomials approach to the results obtained in [24] for DR,2 (2), 3p−4 namely K2, 43 ,∞ = DR,2 (2) ≈ 1.83737 with extremal polynomial (0.86783, −0.83783, 0.67733). The reader can find a sketch of DR,2,p (2) = K2, 4p ,p as a function of p ≥ 4 in Fig. 3. 3p−4

p 4 5 6 8 12 50 250

DR,2,p (2) = K2, 4p ,p 3p−4 √ 2 1.48488 1.53632 1.60525 1.67869 1.79786 1.82939

Extremal polynomial All extreme polynomials ±(0.90089, −0.90089, 0.56316) ±(0.89489, −0.89489, 0.59895) ±(0.88660, −0.88660, 0.63172) ±(0.87870, −0.87870, 0.65288) ±(0.86992, −0.86992, 0.67276) ±(0.86823, −0.86823, 0.67647)

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Fig. 3. Graph of DR,2,p (2) = K2,

4p 3p−4

,p

19

as a function of p ≥ 2.

5. Lower bounds on the Hardy–Littlewood constants for higher degrees In this section we provide a lower bound on DR,2m,4m (2) = CR,2m,4m (2) by considering powers of the extreme polynomials that appear in Theorem 3.3. Observe that if P is as in Theorem 3.3 (iii) with p = 4m, then P m 4m = P m 4m = 1, and hence DR,2m,4m (2) = CR,2m,4m (2) ≥

|P m |2 = |P m |2 . P m 4m

Using MATLAB in order to compute |P m |2 for large values of m with P as in Theorem 3.3 (iii), we obtain the following estimates: Degree

m

DR,2m,4m (2) = CR,2m,4m (2) ≥

4 8 20 100 400 600 800

2 4 10 50 200 300 400

1.29374 1.38808 1.468720 1.5303100 1.5465100 1.5487600 1.5498800

Observe that the above values improve the estimates on DR,2m,4m (2) = CR,2m,4m (2) obtained in [10] by considering only powers of polynomials of degree 2. The authors in [10] use powers of polynomials of degree greater than 2 in order to obtain a better (bigger) lower estimate on DR,m,2m (2) = CR,m,2m (2). Their strategy consists of considering powers of polynomials of degrees ranging from 2 to 10 enjoying the same symmetry as the extremal polynomials for the Bohnenblust–Hille constants DR,m (2) (m = 2, 3, . . . , 10) appearing in [24]. However, extremal polynomials for DR,m,2m (2) = CR,m,2m (2) may not have the same symmetries as the extremal polynomials for DR,m (2). As a matter of fact, and just to give a numerical hint on the previous comment, if we define P (x, y) = ax5 + bx4 y + cx3 y 2 + dx2 y 3 + exy 4 + f y 5 with

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a = 0.000007233947, b = 0.607036736710, c = −0.000044725373, d = −0.982210559287, e = 0.0000283144953, f = 0.1875854561207, then we obtain DR,5,10 (2) = CR,5,10 (2) ≥

|P |2 ≥ 6.236014. P 10

However, in [10] the authors obtain DR,5,10 (2) = CR,5,10 (2) ≥ 6.191704 using polynomials with the symmetry ax5 − bx4 y − cx3 y 2 + cx2 y 3 + bxy 4 − ay 5 . Notice that if a = 0.19462, b = 0.66008, c = 0.97833 then P5 (x, y) = ax5 − bx4 y − cx3 y 2 + cx2 y 3 + bxy 4 − ay 5 is a good approximation to an extremal polynomial for DR,5 (2) (see [24, Section 3.3]). References [1] N. Albuquerque, F. Bayart, D. Pellegrino, J.B. Seoane-Sepúlveda, Optimal Hardy–Littlewood type inequalities for polynomials and multilinear operators, Israel J. Math. 211 (1) (2016) 197–220, http://dx.doi.org/10.1007/s11856-015-1264-7. [2] G. Araújo, D. Pellegrino, D. da Silva e Silva, On the upper bounds for the constants of the Hardy–Littlewood, J. Funct. Anal. 267 (2014) 1878–1888. [3] G. Araújo, P. Jiménez-Rodriguez, G.A. Muñoz-Fernandez, D. Núñez-Alarcón, D. Pellegrino, J.B. Seoane-Sepúlveda, D.M. Serrano-Rodríguez, On the polynomial Hardy–Littlewood inequality, Arch. Math. 104 (2015) 259–270. [4] G. Araújo, P. Jiménez, G. Muñoz-Fernández, J. Seoane-Sepúlveda, Estimates on the norm of polynomials and applications, arXiv:1507.01431. [5] R.M. Aron, B. Beauzamy, P.H. Enflo, Polynomials in many variables: real vs complex norms, J. Approx. Theory 74 (2) (1993) 181–198. [6] R.M. Aron, M. Klimek, Supremum norms for quadratic polynomials, Arch. Math. (Basel) 76 (2001) 73–80.  [7] F. Bayart, D. Pellegrino, J.B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to (log n)/n, Adv. Math. 264 (2014) 726–746. [8] H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (3) (1931) 600–622. [9] J.R. Campos, P. Jiménez-Rodríguez, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, On the real polynomial Bohnenblust–Hille inequality, Linear Algebra Appl. 465 (2015) 391–400. [10] W. Cavalcante, D. Nuñez-Alarcón, D. Pellegrino, New lower bounds for the constants in the real polynomial Hardy– Littlewood inequality, arXiv:1506.00159 [math.FA].

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[11] W. Cavalcante, D. Núñez-Alarcón, D. Pellegrino, The optimal Hardy–Littlewood constants for 2-homogeneous polynomials on p (R2 ) for 2 < p < 4 are 22/p , arXiv:1507.02984v2 [math.FA]. [12] Y.S. Choi, S.G. Kim, The unit ball of P(2 l22 ), Arch. Math. (Basel) 71 (6) (1998) 472–480. [13] Y.S. Choi, S.G. Kim, H. Ki, Extreme polynomials and multilinear forms on l1 , J. Math. Anal. Appl. 228 (2) (1998) 467–482. [14] A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes, K. Seip, The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (1) (2011) 485–497. [15] A. Defant, D. García, M. Maestre, Bohr’s power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003) 173–197. [16] A. Defant, P. Sevilla-Peris, The Bohnenblust–Hille cycle of ideas from a modern point of view, Funct. Approx. Comment. Math. 50 (1) (2014) 55–127. [17] V. Dimant, P. Sevilla-Peris, Summation of coefficients of polynomials on p spaces, arXiv:1309.6063v1 [math.FA]. [18] D. Galicer, M. Mansilla, S. Muro, The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials, arXiv:1602.01735 [math.FA]. [19] J.L. Gámez-Merino, G.A. Muñoz-Fernández, V.M. Sánchez, J.B. Seoane-Sepúlveda, Inequalities for polynomials on the unit square via the Krein–Milman theorem, J. Convex Anal. 20 (1) (2013) 125–142. [20] B.C. Grecu, Geometry of three-homogeneous polynomials on real Hilbert spaces, J. Math. Anal. Appl. 246 (1) (2000) 217–229. [21] B.C. Grecu, Extreme 2-homogeneous polynomials on Hilbert spaces, Quaest. Math. 25 (4) (2002) 421–435. [22] B.C. Grecu, Geometry of 2-homogeneous polynomials on p spaces, 1 < p < ∞, J. Math. Anal. Appl. 273 (2) (2002) 262–282. [23] B.C. Grecu, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, Unconditional constants and polynomial inequalities, J. Approx. Theory 161 (2) (2009) 706–722. [24] P. Jiménez-Rodríguez, G.A. Muñoz-Fernández, M. Murillo-Arcila, J.B. Seoane-Sepúlveda, Sharp values for the constants in the polynomial Bohnenblust–Hille inequality, Linear Multilinear Algebra (2016), http://dx.doi.org/10.1080/03081087. 2015.1115810, in press. [25] G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, A. Weber, Supremum norms for 2-homogeneous polynomials on circle sectors, J. Convex Anal. 21 (3) (2014) 745–764. [26] G.A. Muñoz-Fernández, Y. Sarantopoulos, J.B. Seoane-Sepúlveda, An application of the Krein–Milman theorem to Bernstein and Markov inequalities, J. Convex Anal. 15 (2) (2008) 299–312. [27] J.B. Seoane-Sepúlveda, Sharp generalizations of the multilinear Bohnenblust–Hille inequality, Oberwolfach Rep. 11 (1) (2014) 378–381.