Equivalent norms in polynomial spaces.Infinite Analysis Seminar, Celebrating Richard Aron’s work...
Transcript of Equivalent norms in polynomial spaces.Infinite Analysis Seminar, Celebrating Richard Aron’s work...
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Equivalent norms in polynomial spaces.
Pablo Jimenez Rodrıguez
Infinite Analysis Seminar,Celebrating Richard Aron’s work and impact.
October, 28th, 2016
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
Definition
α = (α1, α2, . . . , αn) = (N ∪ {0})n is a multiindex.
|α| =∑n
i=1 α, xα = xα11 xα2 · . . . · xαn
n .
P(x) =∑|α|=m aαxα is a homogenous polynomial of
degree m.
P(mKn) ={P : Kn → K :
P is a homogeneous polynomial of degree m over Kn}
If | · | is a norm in Kn and P ∈ P(mKn), we may define‖P‖ = sup{|P(x)| : x ∈ B(Kn,|·|)}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
Definition
α = (α1, α2, . . . , αn) = (N ∪ {0})n is a multiindex.
|α| =∑n
i=1 α, xα = xα11 xα2 · . . . · xαn
n .
P(x) =∑|α|=m aαxα is a homogenous polynomial of
degree m.
P(mKn) ={P : Kn → K :
P is a homogeneous polynomial of degree m over Kn}
If | · | is a norm in Kn and P ∈ P(mKn), we may define‖P‖ = sup{|P(x)| : x ∈ B(Kn,|·|)}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
Definition
α = (α1, α2, . . . , αn) = (N ∪ {0})n is a multiindex.
|α| =∑n
i=1 α, xα = xα11 xα2 · . . . · xαn
n .
P(x) =∑|α|=m aαxα is a homogenous polynomial of
degree m.
P(mKn) ={P : Kn → K :
P is a homogeneous polynomial of degree m over Kn}
If | · | is a norm in Kn and P ∈ P(mKn), we may define‖P‖ = sup{|P(x)| : x ∈ B(Kn,|·|)}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
Definition
α = (α1, α2, . . . , αn) = (N ∪ {0})n is a multiindex.
|α| =∑n
i=1 α, xα = xα11 xα2 · . . . · xαn
n .
P(x) =∑|α|=m aαxα is a homogenous polynomial of
degree m.
P(mKn) ={P : Kn → K :
P is a homogeneous polynomial of degree m over Kn}
If | · | is a norm in Kn and P ∈ P(mKn), we may define‖P‖ = sup{|P(x)| : x ∈ B(Kn,|·|)}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
Definition
α = (α1, α2, . . . , αn) = (N ∪ {0})n is a multiindex.
|α| =∑n
i=1 α, xα = xα11 xα2 · . . . · xαn
n .
P(x) =∑|α|=m aαxα is a homogenous polynomial of
degree m.
P(mKn) ={P : Kn → K :
P is a homogeneous polynomial of degree m over Kn}
If | · | is a norm in Kn and P ∈ P(mKn), we may define‖P‖ = sup{|P(x)| : x ∈ B(Kn,|·|)}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
We will be interested in the case where the norm in Kn is chosento be the q−th norm (1 ≤ q ≤ ∞).
Denote ‖P‖q = sup{|P(x)| : ‖x‖q ≤ 1}.Define the p−th norm of P(x) =
∑|α|=m aαxα as
|P|p =
(∑
|α|=m |aα|p)1/p
if 1 ≤ p <∞,max{|aα| : |α| = m} if p =∞.
There exist constants k ,K > 0 so that k‖P‖p ≤ |P|q ≤ K‖P‖p forevery 1 ≤ p, q ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
We will be interested in the case where the norm in Kn is chosento be the q−th norm (1 ≤ q ≤ ∞).Denote ‖P‖q = sup{|P(x)| : ‖x‖q ≤ 1}.
Define the p−th norm of P(x) =∑|α|=m aαxα as
|P|p =
(∑
|α|=m |aα|p)1/p
if 1 ≤ p <∞,max{|aα| : |α| = m} if p =∞.
There exist constants k ,K > 0 so that k‖P‖p ≤ |P|q ≤ K‖P‖p forevery 1 ≤ p, q ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
We will be interested in the case where the norm in Kn is chosento be the q−th norm (1 ≤ q ≤ ∞).Denote ‖P‖q = sup{|P(x)| : ‖x‖q ≤ 1}.Define the p−th norm of P(x) =
∑|α|=m aαxα as
|P|p =
(∑
|α|=m |aα|p)1/p
if 1 ≤ p <∞,max{|aα| : |α| = m} if p =∞.
There exist constants k ,K > 0 so that k‖P‖p ≤ |P|q ≤ K‖P‖p forevery 1 ≤ p, q ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Introductory notation
We will be interested in the case where the norm in Kn is chosento be the q−th norm (1 ≤ q ≤ ∞).Denote ‖P‖q = sup{|P(x)| : ‖x‖q ≤ 1}.Define the p−th norm of P(x) =
∑|α|=m aαxα as
|P|p =
(∑
|α|=m |aα|p)1/p
if 1 ≤ p <∞,max{|aα| : |α| = m} if p =∞.
There exist constants k ,K > 0 so that k‖P‖p ≤ |P|q ≤ K‖P‖p forevery 1 ≤ p, q ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ Dn,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ Dn,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ Dn,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ Dn,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal.
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ Dn,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal.
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ DR,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal.
Theorem
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ DR,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal.
Theorem
The real Bohnenblust-Hille constant is hypercontractive.
lim supm D1/mR,m = 2
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Bohnenblust, Hille, 1931)
There exists a constant Dn,m ≥ 1 such that for every P ∈ P(m`n∞)we have
|P| 2mm+1≤ DR,m‖P‖∞.
Dn,m can be chosen in a way that it is independent on n.
The value for p = 2mm+1 is optimal.
Theorem
The real Bohnenblust-Hille constant is hypercontractive.
lim supm D1/mR,m = 2
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,n,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,n,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,n,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,n,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,n,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,n,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
The Hardy-Littlewood constant.
There are constants Hm,n,p and Lm,n,p so that
|P| pp−m≤ Hm,p‖P‖p for m < p ≤ 2m,
|P| 2mpmp+p−2m
≤ Lm,p‖P‖p for 2m ≤ p ≤ ∞.
The choice of Hm,n,p and Lm,n,p can be made independent of n.The exponents p
m−p for m < p ≤ 2m and 2mpmp+p−2m for
2m ≤ p ≤ ∞ are optimal.
Definition
The smallest constant we can fit in the inequalities above are theHardy-Littlewood constants.
DR,m = Lm,∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
If we work within polynomials of a fixed degree, we define
DR,m(n) := max{ |P| 2m
m+1
‖P‖∞
},
Hm,p(n) := max{ |P| p
p−m
‖P‖p
}, if m ≤ p ≤ 2m,
Lm,p(n) := max{ |P| 2mp
mp+p−2m
‖P‖p
}, if 2m ≤ p ≤ ∞,
for every P ∈ P(mRn).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
If we work within polynomials of a fixed degree, we define
DR,m(n) := max{ |P| 2m
m+1
‖P‖∞
},
Hm,p(n) := max{ |P| p
p−m
‖P‖p
}, if m ≤ p ≤ 2m,
Lm,p(n) := max{ |P| 2mp
mp+p−2m
‖P‖p
}, if 2m ≤ p ≤ ∞,
for every P ∈ P(mRn).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
If we work within polynomials of a fixed degree, we define
DR,m(n) := max{ |P| 2m
m+1
‖P‖∞
},
Hm,p(n) := max{ |P| p
p−m
‖P‖p
}, if m ≤ p ≤ 2m,
Lm,p(n) := max{ |P| 2mp
mp+p−2m
‖P‖p
}, if 2m ≤ p ≤ ∞,
for every P ∈ P(mRn).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
If we work within polynomials of a fixed degree, we define
DR,m(n) := max{ |P| 2m
m+1
‖P‖∞
},
Hm,p(n) := max{ |P| p
p−m
‖P‖p
}, if m ≤ p ≤ 2m,
Lm,p(n) := max{ |P| 2mp
mp+p−2m
‖P‖p
}, if 2m ≤ p ≤ ∞,
for every P ∈ P(mRn).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
If we work within polynomials of a fixed degree, we define
DR,m(n) := max{ |P| 2m
m+1
‖P‖∞
},
Hm,p(n) := max{ |P| p
p−m
‖P‖p
}, if m ≤ p ≤ 2m,
Lm,p(n) := max{ |P| 2mp
mp+p−2m
‖P‖p
}, if 2m ≤ p ≤ ∞,
for every P ∈ P(mRn).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,p,q := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
Hm,p(n) = Km,n, pp−m
,p, if m < p ≤ 2m,
Lm,p(n) = Km,n, 2mpmp+p−2m
,p if 2m ≤ p ≤ ∞.Hm,p ≥ supn Km,n, pp−m
,p for m < p ≤ 2m,
Lm,p ≥ supn Km,n, 2mpmp+p−2m
,p for 2m ≤ p ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,p,q := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}Hm,p(n) = Km,n, p
p−m,p, if m < p ≤ 2m,
Lm,p(n) = Km,n, 2mpmp+p−2m
,p if 2m ≤ p ≤ ∞.Hm,p ≥ supn Km,n, pp−m
,p for m < p ≤ 2m,
Lm,p ≥ supn Km,n, 2mpmp+p−2m
,p for 2m ≤ p ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,p,q := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}Hm,p(n) = Km,n, p
p−m,p, if m < p ≤ 2m,
Lm,p(n) = Km,n, 2mpmp+p−2m
,p if 2m ≤ p ≤ ∞.
Hm,p ≥ supn Km,n, pp−m
,p for m < p ≤ 2m,
Lm,p ≥ supn Km,n, 2mpmp+p−2m
,p for 2m ≤ p ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,p,q := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}Hm,p(n) = Km,n, p
p−m,p, if m < p ≤ 2m,
Lm,p(n) = Km,n, 2mpmp+p−2m
,p if 2m ≤ p ≤ ∞.Hm,p ≥ supn Km,n, pp−m
,p for m < p ≤ 2m,
Lm,p ≥ supn Km,n, 2mpmp+p−2m
,p for 2m ≤ p ≤ ∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,q,p := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
We will be working on R2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′m,n,q,p := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
Km,n,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
We will be working on R2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′2,q,p := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
K2,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
We will be working on R2.
In the present talk, I will give someexact values and bounds for k ′2,p,q, K2,q,p for certain values of p
and q. Remark that, if k2,q,p = 1k ′2,q,p
, then k2,p,q and K2,p,q are the
best possible constants for which the inequalities
k2,p,q‖P‖p ≤ |P|q ≤ K2,q,p‖P‖phold for every P ∈ P(mR2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′2,q,p := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
K2,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
We will be working on R2. In the present talk, I will give someexact values and bounds for k ′2,p,q, K2,q,p for certain values of p
and q.
Remark that, if k2,q,p = 1k ′2,q,p
, then k2,p,q and K2,p,q are the
best possible constants for which the inequalities
k2,p,q‖P‖p ≤ |P|q ≤ K2,q,p‖P‖phold for every P ∈ P(mR2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Generalizing those ideas.
Definition
Let 1 ≤ p, q ≤ ∞. We define
k ′2,q,p := max{‖P‖p : P ∈ B(P(mRn),|·|q)
},
K2,q,p = max{|P|q : P ∈ B(P(mRn),‖·‖p)
}
We will be working on R2. In the present talk, I will give someexact values and bounds for k ′2,p,q, K2,q,p for certain values of p
and q. Remark that, if k2,q,p = 1k ′2,q,p
, then k2,p,q and K2,p,q are the
best possible constants for which the inequalities
k2,p,q‖P‖p ≤ |P|q ≤ K2,q,p‖P‖phold for every P ∈ P(mR2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
We will make use of a corollary of the Krein-Milmann theorem,according to which the maximum of a continuous convex functionover a convex set is attained over the set of extreme points.
ext(B|·|q) =
{±ek : 1 ≤ k ≤ m + 1 if q = 1,
{∑m+1
k=1 εkek : εk = ±1 if q =∞,S|·|q if 1 < q <∞.}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
We will make use of a corollary of the Krein-Milmann theorem,according to which the maximum of a continuous convex functionover a convex set is attained over the set of extreme points.
ext(B|·|q) =
{±ek : 1 ≤ k ≤ m + 1 if q = 1,
{∑m+1
k=1 εkek : εk = ±1 if q =∞,S|·|q if 1 < q <∞.}
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Richard Aron and M. Klimek)
Let us denote ‖(a, b, c)‖R := supx∈[−1,1] |ax2 + bx + c|. Then,
‖(a, b, c)‖R =
∣∣∣b24a − c
∣∣∣ if |b| < 2|a| and
ca + 1 < 1
2(| b2a | − 1)2,
|a + c |+ |b| otherwise.
If α < β are real numbers and‖(a, b, c)‖[α,β] := supx∈[α,β] |ax2 + bx + c |, then
‖(a, b, c)‖[α,β] =
∥∥∥∥∥((
α− β2
)2
a,α2 − β2
2a +
α− β2
b,
(α + β
2
)2
a +α + β
2b + c
)∥∥∥∥∥R
.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Richard Aron and M. Klimek)
Let us denote ‖(a, b, c)‖R := supx∈[−1,1] |ax2 + bx + c|. Then,
‖(a, b, c)‖R =
∣∣∣b24a − c
∣∣∣ if |b| < 2|a| and
ca + 1 < 1
2(| b2a | − 1)2,
|a + c |+ |b| otherwise.
If α < β are real numbers and‖(a, b, c)‖[α,β] := supx∈[α,β] |ax2 + bx + c |, then
‖(a, b, c)‖[α,β] =
∥∥∥∥∥((
α− β2
)2
a,α2 − β2
2a +
α− β2
b,
(α + β
2
)2
a +α + β
2b + c
)∥∥∥∥∥R
.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
‖ax2 + by2 + cxy‖1 = max
{∥∥∥∥(a + b + c
4,a− b
2,a + b − c
4
)∥∥∥∥R,∥∥∥∥(a + b − c
4,b − a
2,a + b + c
4
)∥∥∥∥R
}
‖ax2 + by2 + cxy‖∞ = max{‖(a, c , b)‖R, ‖(b, c, a)‖R.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
‖ax2 + by2 + cxy‖1 = max
{∥∥∥∥(a + b + c
4,a− b
2,a + b − c
4
)∥∥∥∥R,∥∥∥∥(a + b − c
4,b − a
2,a + b + c
4
)∥∥∥∥R
}
‖ax2 + by2 + cxy‖∞ = max{‖(a, c , b)‖R, ‖(b, c, a)‖R.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (Y.S. Choi, S.G. Kim, H. Ki)
The extreme polynomials of B(P(2R2),‖·‖1) are of the form
P(x , y) = ±x2 ± y2 ± 2xy ,
P(x , y) = ±√
4|t|−t22 (x2 − y2) + txy , where |t| ∈ (2, 4].
Theorem (Y.S. Choi, S.G. Kim)
The extreme polynomials of B(P(2R2),‖·‖∞) are of the form
P(x , y) = ±x2,
P(x , y) = ±y2,
±(tx2 − ty2 ± 2
√t(1− t)xy
), where t ∈ [12 , 1].
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (B. Grecu)
Let p > 2. The extreme points of the unit ball of P(2`2p) are
P(x , y) = ±(x2 − y2),
P(x , y) = ax2 + cy2, ac ≥ 0, |a|p
p−2 + |c|p
p−2 = 1,
P(x , y) = ±(αp−βp
α2+β2 (x2 − y2) + 2αβ αp−2+βp−2
α2+β2 xy), α, β ≥
0, αp + βp = 1.
Theorem (G.A. Munoz-Fernandez, D. Pellegrino, J.B.Seoane-Sepulveda, A. Weber)
The extreme points of the unit ball of P(2`22) are
±(ax2 − ay2 + 2
√1− a2xy
), a ∈ [−1, 1],
±(x2 + y2).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Aron and Klimek.)
For p, q ∈ {1,∞} we have
k2,q,p =
1 if q = p = 1,
1 if q = 1, p =∞,1 if q =∞, p = 1,13 if q = p =∞.
Extremal polynomials are given:
p1,1(x , y) = ±x2, ±y2,p1,∞(x , y) = ±x2, ±y2,±xy ,p∞,1(x , y) = ±x2 ± y2 ± xy ,
p∞,∞(x , y) = ±(x2 + y2 ± xy).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Aron and Klimek.)
For p, q ∈ {1,∞} we have
k2,q,p =
1 if q = p = 1,
1 if q = 1, p =∞,1 if q =∞, p = 1,13 if q = p =∞.
Extremal polynomials are given:
p1,1(x , y) = ±x2, ±y2,p1,∞(x , y) = ±x2, ±y2,±xy ,p∞,1(x , y) = ±x2 ± y2 ± xy ,
p∞,∞(x , y) = ±(x2 + y2 ± xy).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
For k2,q,1 and k2,q,∞ (q ∈ (1,∞)), we used numericalapproximation.
k2,q,1 = 1 for every q ∈ (1,∞), and ±e1, ±e2 are extremal.k2,q,∞ = 31/q−1 for every q ∈ (1,∞), and ±(a,−a,−a) areextremal.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1/3
0.4
0,5
0,6
0,7
0,8
0,9
1
k2,q,∞
31q−1
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
For k2,q,1 and k2,q,∞ (q ∈ (1,∞)), we used numericalapproximation.k2,q,1 = 1 for every q ∈ (1,∞), and ±e1, ±e2 are extremal.
k2,q,∞ = 31/q−1 for every q ∈ (1,∞), and ±(a,−a,−a) areextremal.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1/3
0.4
0,5
0,6
0,7
0,8
0,9
1
k2,q,∞
31q−1
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
For k2,q,1 and k2,q,∞ (q ∈ (1,∞)), we used numericalapproximation.k2,q,1 = 1 for every q ∈ (1,∞), and ±e1, ±e2 are extremal.k2,q,∞ = 31/q−1 for every q ∈ (1,∞), and ±(a,−a,−a) areextremal.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1/3
0.4
0,5
0,6
0,7
0,8
0,9
1
k2,q,∞
31q−1
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
For k2,q,1 and k2,q,∞ (q ∈ (1,∞)), we used numericalapproximation.k2,q,1 = 1 for every q ∈ (1,∞), and ±e1, ±e2 are extremal.k2,q,∞ = 31/q−1 for every q ∈ (1,∞), and ±(a,−a,−a) areextremal.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1/3
0.4
0,5
0,6
0,7
0,8
0,9
1
k2,q,∞
31q−1
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let p ∈ (1,∞). Then,
k2,q,p =
{1 if q = 1,22/p
3 if q =∞ and p ≥ 43 .
Extremal polynomials are given:
p1,p(x , y) = ±x2,±y2,p∞,p(x , y) = ±(x2 + y2 + xy).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let p ∈ (1,∞). Then,
k2,q,p =
{1 if q = 1,22/p
3 if q =∞ and p ≥ 43 .
Extremal polynomials are given:
p1,p(x , y) = ±x2,±y2,p∞,p(x , y) = ±(x2 + y2 + xy).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let p ∈ (1,∞). Then,
k2,q,p =
{1 if q = 1,22/p
3 if q =∞ and p ≥ 43 .
Extremal polynomials are given:
p1,p(x , y) = ±x2,±y2,p∞,p(x , y) = ±(x2 + y2 + xy).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Choi, Kim and Ki)
For q, p ∈ {1,∞},
K2,q,p =
2 + 2
√2 if q = p = 1,
1 +√
2 if q = 1, p =∞,4 if q =∞, p = 1,
1 if q = p =∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Choi, Kim and Ki)
For q, p ∈ {1,∞},
K2,q,p =
2 + 2
√2 if q = p = 1,
1 +√
2 if q = 1, p =∞,4 if q =∞, p = 1,
1 if q = p =∞.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Choi, Kim and Ki)
Extremal polynomials are given:
P1,1(x , y) = ±√
2
2(x2 − y2) + (2 +
√2)xy ,
P1,∞(x , y) = ±
(2 +√
2
4x2 − 2 +
√2
4y2 ±
√2
2xy
),
P∞,1(x , y) = ±4xy ,
P∞,∞(x , y) = ±x2,±y2,±(
1
2x2 − 1
2y2 ± xy
).
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem (From the results proved by Choi, Kim and Ki)
Define
fq,1(t) =(
21−q(4t − t2)q/2 + tq)1/q
, t ∈ [2, 4],
fq,∞(t) =(
2tq + 2q(t − t2)q/2)1/q
, t ∈ [2, 4].
For every q ∈ [1,∞),
K2,q,1 = max{fq,1(t) : t ∈ [2, 4]},
K2,q,∞ = max{fq,∞(t) : t ∈ [1
2, 1]}.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.
Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
For q = 43 , the maximum of fq,1(t) is attained at
For q = 32 , the maximum of fq,1(t) is attained at
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
For q = 43 , the maximum of fq,1(t) is attained at
For q = 32 , the maximum of fq,1(t) is attained at
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
For q = 43 , the maximum of fq,1(t) is attained at
For q = 32 , the maximum of fq,1(t) is attained at
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
For q = 43 , the maxima of fq,∞(t) is attained at
For q = 32 , the maximum of fq,1(t) is attained at
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
K2,q,1 = 4 and K2,q,∞ = 21/q for every q ≥ 2.Extremal polynomials are given:
Pq,1(x , y) = ±4xy ,
Pq,∞(x , y) = ±(x2 − y2), for q ≥ 2.
For q = 43 , the maxima of fq,∞(t) is attained at
For q = 32 , the maxima of fq,∞(t) is attained at
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
If q > 1, then
K2,q,2 =
2 if q ≥ 2,
2
(1+2
1q−2
)1/q
(1+2
2(q−1)q−2
)1/q if 1 < q < 2.
Extremal polynomials are given:
Pq,2(x , y) = ±(x2 − y2) q ≥ 2,
Pq,2(x , y) = ±(a0x
2 − a0y2 + 2
√1− a20xy
), 1 < q < 2,
where a0 =
(1 + 2
2(q−1)q−2
)−1/2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
If q > 1, then
K2,q,2 =
2 if q ≥ 2,
2
(1+2
1q−2
)1/q
(1+2
2(q−1)q−2
)1/q if 1 < q < 2.
Extremal polynomials are given:
Pq,2(x , y) = ±(x2 − y2) q ≥ 2,
Pq,2(x , y) = ±(a0x
2 − a0y2 + 2
√1− a20xy
), 1 < q < 2,
where a0 =
(1 + 2
2(q−1)q−2
)−1/2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
If q > 1, then
K2,q,2 =
2 if q ≥ 2,
2
(1+2
1q−2
)1/q
(1+2
2(q−1)q−2
)1/q if 1 < q < 2.
Extremal polynomials are given:
Pq,2(x , y) = ±(x2 − y2) q ≥ 2,
Pq,2(x , y) = ±(a0x
2 − a0y2 + 2
√1− a20xy
), 1 < q < 2,
where a0 =
(1 + 2
2(q−1)q−2
)−1/2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q, p > 2. Then,
K2,q,p = 2max{ 1q, 2p}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q, p > 2. Then,
K2,q,p = 2max{ 1q, 2p}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q, p > 2. Then,
K2,q,p = 2max{ 1q, 2p}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q, p > 2. Then,
K2,q,p = 2max{ 1q, 2p}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
The theorem above still holds for the case q =∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q, p > 2. Then,
K2,q,p = 2max{ 1q, 2p}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
The theorem above still holds for the case q =∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q,p > 2. Then,
K2,∞,p = 22pmax{1
q,}.
Extremal polynomials are given:
Pq,p(x , y) = ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
The theorem above still holds for the case q =∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q,p > 2. Then,
K2,∞,p = 22p .
Extremal polynomials are given:
Pq,p(x , y)= ±22/pxy , q ≥ p
2,
Pq,p(x , y) = ±(x2 − y2), q <p
2.
The theorem above still holds for the case q =∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Theorem
Let q,p > 2. Then,
K2,∞,p = 22p .
Extremal polynomials are given:
P∞,p(x , y)= ±22/pxy , ∞ ≥ p
2,
Pq,p(x , y)= ±(x2 − y2), q <p
2.
The theorem above still holds for the case q =∞
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Pablo Jimenez Rodrıguez Equivalent norms
The beginning: the Bohnenblust-Hille inequality.Preparing the ground
Values for k2,q,p and K2,q,p .
Thank you for your attention!!
Pablo Jimenez Rodrıguez Equivalent norms