DIFFERENTIAL PIETSCH MEASURES FOR
DOMINATED POLYNOMIALS ON BANACH SPACES
Geraldo Botelho
Faculdade de Matemática
Universidade Federal de Uberlândia
Uberlândia – Brazil
Relations Between Banach Space Theory and Geometric Measure
Theory
University of Warwick, June 08–12, 2015.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
This is a (part of a) joint work with Daniel
Pellegrino (João Pessoa) and Pilar Rueda
(Valencia).
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Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
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Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
P sends weakly p-summable sequences in E to absolutely
p
n -summable sequences in F , that is, if
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Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
P sends weakly p-summable sequences in E to absolutely
p
n -summable sequences in F , that is, if
∞
w
p
if (P(xj ))∞
j=1 ∈ ` (F ) whenever (xj )j=1 ∈ `p (E ).
n
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
P sends weakly p-summable sequences in E to absolutely
p
n -summable sequences in F , that is, if
∞
w
p
if (P(xj ))∞
j=1 ∈ ` (F ) whenever (xj )j=1 ∈ `p (E ).
n
In the linear case (n=1) we recover the classical ideal of absolutely
p-summing linear operators.
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Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
P sends weakly p-summable sequences in E to absolutely
p
n -summable sequences in F , that is, if
∞
w
p
if (P(xj ))∞
j=1 ∈ ` (F ) whenever (xj )j=1 ∈ `p (E ).
n
In the linear case (n=1) we recover the classical ideal of absolutely
p-summing linear operators.
The class of dominated polynomials, which goes back to Pietsch
1983,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Dominated polynomials
A continuous n-homogeneous polynomial P : E −→ F between
Banach spaces is p-dominated if
P sends weakly p-summable sequences in E to absolutely
p
n -summable sequences in F , that is, if
∞
w
p
if (P(xj ))∞
j=1 ∈ ` (F ) whenever (xj )j=1 ∈ `p (E ).
n
In the linear case (n=1) we recover the classical ideal of absolutely
p-summing linear operators.
The class of dominated polynomials, which goes back to Pietsch
1983, is one of the most studied nonlinear generalizations of the
ideal of absolutely summing linear operators.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Pietsch domination theorem
Like absolutely summing linear operators, dominated polynomials
enjoy a Pietsch domination theorem,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Pietsch domination theorem
Like absolutely summing linear operators, dominated polynomials
enjoy a Pietsch domination theorem, which establishes a deep
connection with measure theory:
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Pietsch domination theorem
Like absolutely summing linear operators, dominated polynomials
enjoy a Pietsch domination theorem, which establishes a deep
connection with measure theory:
Theorem. [Matos 1996] A continuous n-homogeneous polynomial
P : E −→ F is p-dominated if and only if there is a constant C > 0
and a regular Borel probability measure on BE 0 endowed with the
weak* topology such that
Z
kP(x)k ≤ C
!n
p
|ϕ(x)|p dµ(ϕ)
for every x ∈ E .
(1)
BE 0
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Differential Pietsch measures for dominated polynomials
Pietsch domination theorem
Like absolutely summing linear operators, dominated polynomials
enjoy a Pietsch domination theorem, which establishes a deep
connection with measure theory:
Theorem. [Matos 1996] A continuous n-homogeneous polynomial
P : E −→ F is p-dominated if and only if there is a constant C > 0
and a regular Borel probability measure on BE 0 endowed with the
weak* topology such that
Z
kP(x)k ≤ C
!n
p
|ϕ(x)|p dµ(ϕ)
for every x ∈ E .
(1)
BE 0
Any such measure µ is called a Pietsch measure for P.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Closed under differentiation
In the study of special classes of homogeneous polynomials, it is
very important for a class to be closed under differentiation,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Closed under differentiation
In the study of special classes of homogeneous polynomials, it is
very important for a class to be closed under differentiation,
meaning that if a polynomial belongs to the class, then all its
derivatives belong to the class as well. More precisely:
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Closed under differentiation
In the study of special classes of homogeneous polynomials, it is
very important for a class to be closed under differentiation,
meaning that if a polynomial belongs to the class, then all its
derivatives belong to the class as well. More precisely:
• Given an n-homogeneous polynomial P : E −→ F , a ∈ E and
k ≤ n, dbk P(a) is the following k-homogeneous polynomial:
dbk P(a) : E −→ F , dbk P(a)(x) =
Geraldo Botelho
n!
P̌(an−k , x k ),
(n − k)!
Differential Pietsch measures for dominated polynomials
Closed under differentiation
In the study of special classes of homogeneous polynomials, it is
very important for a class to be closed under differentiation,
meaning that if a polynomial belongs to the class, then all its
derivatives belong to the class as well. More precisely:
• Given an n-homogeneous polynomial P : E −→ F , a ∈ E and
k ≤ n, dbk P(a) is the following k-homogeneous polynomial:
dbk P(a) : E −→ F , dbk P(a)(x) =
n!
P̌(an−k , x k ),
(n − k)!
where P̌ is the (unique) symmetric n-linear operator on E n
associated to P,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Closed under differentiation
In the study of special classes of homogeneous polynomials, it is
very important for a class to be closed under differentiation,
meaning that if a polynomial belongs to the class, then all its
derivatives belong to the class as well. More precisely:
• Given an n-homogeneous polynomial P : E −→ F , a ∈ E and
k ≤ n, dbk P(a) is the following k-homogeneous polynomial:
dbk P(a) : E −→ F , dbk P(a)(x) =
n!
P̌(an−k , x k ),
(n − k)!
where P̌ is the (unique) symmetric n-linear operator on E n
associated to P, that is
P̌(x, . . . , x) = P(x) for every x ∈ E .
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Differential Pietsch measures for dominated polynomials
Closed under differentiation
• A class Q of homogeneous polynomials between Banach spaces
is closed under differentiation if
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Differential Pietsch measures for dominated polynomials
Closed under differentiation
• A class Q of homogeneous polynomials between Banach spaces
is closed under differentiation if
P ∈ Q =⇒ dbk P(a) ∈ Q for all a and k.
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Differential Pietsch measures for dominated polynomials
Closed under differentiation
• A class Q of homogeneous polynomials between Banach spaces
is closed under differentiation if
P ∈ Q =⇒ dbk P(a) ∈ Q for all a and k.
The attempts to systematize the study of special classes of
homogeneous polynomials usually require the condition of being
closed under differentiation,
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Differential Pietsch measures for dominated polynomials
Closed under differentiation
• A class Q of homogeneous polynomials between Banach spaces
is closed under differentiation if
P ∈ Q =⇒ dbk P(a) ∈ Q for all a and k.
The attempts to systematize the study of special classes of
homogeneous polynomials usually require the condition of being
closed under differentiation, such as
Holomorphy types (Nachbin 60’s, Dineen 70’s, ...),
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Closed under differentiation
• A class Q of homogeneous polynomials between Banach spaces
is closed under differentiation if
P ∈ Q =⇒ dbk P(a) ∈ Q for all a and k.
The attempts to systematize the study of special classes of
homogeneous polynomials usually require the condition of being
closed under differentiation, such as
Holomorphy types (Nachbin 60’s, Dineen 70’s, ...), coherent
sequences of polynomials (Carando-Dimant-Muro 2000’s,
Pellegrino-Ribeiro 2010’s.)
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Differential Pietsch measures for dominated polynomials
Differential Pietsch measures
B.-Pellegrino (2005): The class Pd,p of p-dominated polynomials is
closed under differentiation.
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Differential Pietsch measures for dominated polynomials
Differential Pietsch measures
B.-Pellegrino (2005): The class Pd,p of p-dominated polynomials is
closed under differentiation.
So, if P is p-dominated (hence has a Pietsch measure), then all its
derivatives dbk P(a) are p-dominated (hence have Pietsch
measures).
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures
B.-Pellegrino (2005): The class Pd,p of p-dominated polynomials is
closed under differentiation.
So, if P is p-dominated (hence has a Pietsch measure), then all its
derivatives dbk P(a) are p-dominated (hence have Pietsch
measures).
Question 1. Is it true that any p-dominated polynomial share a
Pietsch measure with its derivatives?
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures
B.-Pellegrino (2005): The class Pd,p of p-dominated polynomials is
closed under differentiation.
So, if P is p-dominated (hence has a Pietsch measure), then all its
derivatives dbk P(a) are p-dominated (hence have Pietsch
measures).
Question 1. Is it true that any p-dominated polynomial share a
Pietsch measure with its derivatives?
• A Pietsch measure µ for a p-dominated polynomial is a
differential Pietsch measure if it is a Pietsch measure for all its
derivatives dbk P(a).
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures
B.-Pellegrino (2005): The class Pd,p of p-dominated polynomials is
closed under differentiation.
So, if P is p-dominated (hence has a Pietsch measure), then all its
derivatives dbk P(a) are p-dominated (hence have Pietsch
measures).
Question 1. Is it true that any p-dominated polynomial share a
Pietsch measure with its derivatives?
• A Pietsch measure µ for a p-dominated polynomial is a
differential Pietsch measure if it is a Pietsch measure for all its
derivatives dbk P(a).
Question 1. (Rephrased) Does every p-dominated polynomial
admit a differential Pietsch measure?
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
The factorization theorem
Theorem. An n-homogeneous polynomial P : E −→ F is
p-dominated if and only if there are a Banach space G , a
p-summing linear operator u : E −→ G and a continuous
n-homogeneous polynomial Q : G −→ F such that P = Q ◦ u.
E
u
G
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P
/F
@
Q
Differential Pietsch measures for dominated polynomials
The factorization theorem
Theorem. An n-homogeneous polynomial P : E −→ F is
p-dominated if and only if there are a Banach space G , a
p-summing linear operator u : E −→ G and a continuous
n-homogeneous polynomial Q : G −→ F such that P = Q ◦ u.
E
u
G
P
/F
@
Q
History: Pietsch (1983)
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
The factorization theorem
Theorem. An n-homogeneous polynomial P : E −→ F is
p-dominated if and only if there are a Banach space G , a
p-summing linear operator u : E −→ G and a continuous
n-homogeneous polynomial Q : G −→ F such that P = Q ◦ u.
E
u
G
P
/F
@
Q
History: Pietsch (1983)
Geiss 1987
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
The factorization theorem
Theorem. An n-homogeneous polynomial P : E −→ F is
p-dominated if and only if there are a Banach space G , a
p-summing linear operator u : E −→ G and a continuous
n-homogeneous polynomial Q : G −→ F such that P = Q ◦ u.
E
u
G
P
/F
@
Q
History: Pietsch (1983)
Geiss 1987
D. Pérez-Garcı́a (2002)
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
The factorization theorem
Theorem. An n-homogeneous polynomial P : E −→ F is
p-dominated if and only if there are a Banach space G , a
p-summing linear operator u : E −→ G and a continuous
n-homogeneous polynomial Q : G −→ F such that P = Q ◦ u.
E
u
G
P
/F
@
Q
History: Pietsch (1983)
Geiss 1987
D. Pérez-Garcı́a (2002)
B.-Pellegrino-Rueda (E. Sánchez-Pérez) (2014)
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Proof. Given a p-dominated n-homogeneous polynomial
P : E −→ F , by the factorization theorem, P = Q ◦ u,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Proof. Given a p-dominated n-homogeneous polynomial
P : E −→ F , by the factorization theorem, P = Q ◦ u, where u is a
p-summing linear operator.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Proof. Given a p-dominated n-homogeneous polynomial
P : E −→ F , by the factorization theorem, P = Q ◦ u, where u is a
p-summing linear operator. It is easy to check that
P̌ = Q̌ ◦ (u, . . . , u).
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Proof. Given a p-dominated n-homogeneous polynomial
P : E −→ F , by the factorization theorem, P = Q ◦ u, where u is a
p-summing linear operator. It is easy to check that
P̌ = Q̌ ◦ (u, . . . , u).
Let µ be a Pietsch measure for u with constant C .
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
Theorem. Every p-dominated polynomial admits a differential
Pietsch measure.
Proof. Given a p-dominated n-homogeneous polynomial
P : E −→ F , by the factorization theorem, P = Q ◦ u, where u is a
p-summing linear operator. It is easy to check that
P̌ = Q̌ ◦ (u, . . . , u).
Let µ be a Pietsch measure for u with constant C . Then, for
a ∈ E and k ∈ {1, . . . , n},
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Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
k
b
kd P(a)(x)k = n!
n−k
k P̌(a
, x )
(n − k)!
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Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
k
b
kd P(a)(x)k = =
n!
n−k
k P̌(a
, x )
(n − k)!
n!
·Q̌ u(a)n−k , u(x)k (n − k)!
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Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
k
b
kd P(a)(x)k = n!
n−k
k P̌(a
, x )
(n − k)!
=
n!
·Q̌ u(a)n−k , u(x)k (n − k)!
≤
n!
·kQ̌k·ku(a)kn−k ·ku(x)kk
(n − k)!
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
k
b
kd P(a)(x)k = n!
n−k
k P̌(a
, x )
(n − k)!
=
n!
·Q̌ u(a)n−k , u(x)k (n − k)!
≤
n!
·kQ̌k·ku(a)kn−k ·ku(x)kk
(n − k)!
n!
≤
· kQ̌k · kukn−k · kakn−k · C k ·
(n − k)!
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Z
!k
p
|ϕ(x)|p dµ(ϕ)
,
BE 0
Differential Pietsch measures for dominated polynomials
Differential Pietsch measures always exist
k
b
kd P(a)(x)k = n!
n−k
k P̌(a
, x )
(n − k)!
=
n!
·Q̌ u(a)n−k , u(x)k (n − k)!
≤
n!
·kQ̌k·ku(a)kn−k ·ku(x)kk
(n − k)!
n!
≤
· kQ̌k · kukn−k · kakn−k · C k ·
(n − k)!
Z
!k
p
|ϕ(x)|p dµ(ϕ)
,
BE 0
proving that µ is a Pietsch measure for dbk P(a).
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
e : E −→ C (BE 0 ) , e(x)(ϕ) = ϕ(x).
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
e : E −→ C (BE 0 ) , e(x)(ϕ) = ϕ(x).
Given 0 < p < +∞ and a regular Borel probability measure µ on
BE 0 ,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
e : E −→ C (BE 0 ) , e(x)(ϕ) = ϕ(x).
Given 0 < p < +∞ and a regular Borel probability measure µ on
BE 0 , let
jp : C (BE 0 ) −→ Lp (µ)
be the canonical map.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
e : E −→ C (BE 0 ) , e(x)(ϕ) = ϕ(x).
Given 0 < p < +∞ and a regular Borel probability measure µ on
BE 0 , let
jp : C (BE 0 ) −→ Lp (µ)
be the canonical map. By jpe we denote the restriction of jp to
e(E ).
e
jp
jpe : E −→ C (BE 0 ) −→ Lp (µ)
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
Given a Banach space E , consider the linear isometry
e : E −→ C (BE 0 ) , e(x)(ϕ) = ϕ(x).
Given 0 < p < +∞ and a regular Borel probability measure µ on
BE 0 , let
jp : C (BE 0 ) −→ Lp (µ)
be the canonical map. By jpe we denote the restriction of jp to
e(E ).
e
jp
jpe : E −→ C (BE 0 ) −→ Lp (µ)
The problems with the proof of the factorization property are
related to the (non)-injectivity of jpe .
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective if and only if
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective if and only if µ is a
differential Pietsch measure for some injective p-dominated
n-homogeneous polynomial defined on E .
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective if and only if µ is a
differential Pietsch measure for some injective p-dominated
n-homogeneous polynomial defined on E .
As to the existence of differential Pietsch measures for which jp is
injective we have:
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective if and only if µ is a
differential Pietsch measure for some injective p-dominated
n-homogeneous polynomial defined on E .
As to the existence of differential Pietsch measures for which jp is
injective we have:
Proposition. If (BE 0 , w ∗ ) is separable,
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
Application
As to the injectivity of jpe we have:
Proposition. Let n ∈ N odd, p ≥ n, E be a real Banach space and
µ ∈ W (BE 0 , w ∗ ) be given. Then jpe is injective if and only if µ is a
differential Pietsch measure for some injective p-dominated
n-homogeneous polynomial defined on E .
As to the existence of differential Pietsch measures for which jp is
injective we have:
Proposition. If (BE 0 , w ∗ ) is separable, then any p-dominated
n-homogeneous polynomial on E admits a differential Pietsch
measure for which the canonical map jq is injective for every
1 ≤ q < ∞.
Geraldo Botelho
Differential Pietsch measures for dominated polynomials
THANK YOU VERY MUCH!
Geraldo Botelho
Differential Pietsch measures for dominated polynomials