workshop operators and banach lattices welcome …obl/afernandez.pdf · welcome! this is a two-day...
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Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
measure
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
measure
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Interpolation of spaces of integrable functions with respect to a vector measure
Antonio Fernández (Universidad de Sevilla)
interpolation integrable
measure
The team (FQM–133).
The Bartle–Dunford–Schwartz integral.
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
f : ⌦ �! K
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! K
h⌫, x0i (A) := h⌫(A), x0i , A 2 RR ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! K
h⌫, x0i (A) := h⌫(A), x0i , A 2 R
k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]
R ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
A set is said to be null if
f : ⌦ �! K
A 2 Rloc
h⌫, x0i (A) := h⌫(A), x0i , A 2 R
k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]
k⌫k(A) = 0.
R ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
is said to be integrable if:
I) for each x
0 2 X
0,
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
f 2 L
1 (|h⌫, x0i|)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
is said to be integrable if:
I) for each and
II) For every there exists
such that
x
0 2 X
0,
A 2 Rloc
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
f 2 L
1 (|h⌫, x0i|) Z
Afd⌫ 2 X
⌧Z
Afd⌫, x
0�
=
Z
Afd h⌫, x0i , x
0 2 X
0.
The function spaces.
The function spaces.
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
The function spaces.
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
The function spaces.
For
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
1 p < 1
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
Lpw(⌫) :=
�f : ⌦ �! K : |f |p 2 L1
w(⌫) ,
Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)
.
The function spaces.
For
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
kfkp := sup
(✓Z
⌦|f |p d |h⌫, x0i|
◆ 1p
: kx0k 1
).
1 p < 1
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
Lpw(⌫) :=
�f : ⌦ �! K : |f |p 2 L1
w(⌫) ,
Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)
.
Basic properties.
Basic properties.
1) is a Banach lattice with o.c.n:
Lp(⌫)
Basic properties.
1) is a Banach lattice with o.c.n:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
3) If then
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
3) If then
In general,
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).
Lp(⌫) Lpw(⌫).
Basic properties.
4) and can be non reflexive, even if
Lp(⌫) Lpw(⌫)
p > 1.
Basic properties.
4) and can be non reflexive, even if
5) If then
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Basic properties.
4) and can be non reflexive, even if
5) If then
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
Basic properties.
4) and can be non reflexive, even if
5) If then
6) In general,
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
L1(⌫) 6✓ L1(⌫).
Basic properties.
4) and can be non reflexive, even if
5) If then
6) In general,
7) and
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
Lp(m) Lp[0, 1] Lp(⌫) Lp (R)
p > 1.
1 p1 < p2 < 1,
L1(⌫) 6✓ L1(⌫).
Interpolation methods.
Interpolation methods.
Let be a (compatible) couple of Banach
spaces.
(X0, X1)
X0 \X1,! F(X0, X1) ,!X0 +X1
Interpolation methods.
Let be a (compatible) couple of Banach
spaces. An interpolation method
with the interpolation property:
X0 \X1 ,! F(X0, X1) ,! X0 +X1
F
(X0, X1)
Interpolation methods.
Let be a (compatible) couple of Banach
spaces. An interpolation method
with the interpolation property:
X0T�! Y0
X1T�! Y1
9>=
>;) F(X0, X1)
T�! F(Y0, Y1)
X0 \X1 ,! F(X0, X1) ,! X0 +X1
F
(X0, X1)
Compatible couple of Banach spaces.
Compatible couple of Banach spaces.
If and sigma-finite
1 p0 6= p1 < 1 ⌫ : R ! X
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
Compatible couple of Banach spaces.
If and sigma-finite
we can consider the following (compatible) couples
of Banach spaces:
1 p0 6= p1 < 1 ⌫ : R ! X
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1
w (⌫))
(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1
w (⌫))
Compatible couple of Banach spaces.
If and sigma-finite
we can consider the following (compatible) couples
of Banach spaces:
The ambient space:
1 p0 6= p1 < 1 ⌫ : R ! X
L0(⌫).
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1
w (⌫))
(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1
w (⌫))
Complex methods. [Calderón (1964)]
Complex methods. [Calderón (1964)]
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product: X1�✓0 X✓
1 .
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
3) If and have the Fatou property:
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
[X0, X1][✓] ,! X1�✓0 X✓
1 = [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
X0 X1
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
3) If and have the Fatou property:
4)
[Lp0(µ), Lp1(µ)][✓]=(Lp0(µ))1�✓ (Lp1(µ))✓
=[Lp0(µ), Lp1(µ)][✓]= Lp(µ).
1 p0, p1 1
1
p:=
1� ✓
p0+
✓
p1
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
[X0, X1][✓] ,! X1�✓0 X✓
1 = [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
X0 X1
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
Complex methods: the problem.
For sigma-finite, and
describe
0 < ✓ < 1 p0 6= p1 1⌫ : R ! X
[Lp0(⌫), Lp1(⌫)][✓] , [Lp0(⌫), Lp1(⌫)][✓] ,
[Lp0(⌫), Lp1w (⌫)][✓] , [Lp0(⌫), Lp1
w (⌫)][✓] ,
[Lp0w (⌫), Lp1(⌫)][✓] , [Lp0
w (⌫), Lp1(⌫)][✓] ,
[Lp0w (⌫), Lp1
w (⌫)][✓] , [Lp0w (⌫), Lp1
w (⌫)][✓] .
Complex method [.,.]θ (sigma-algebra).
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1
1
p:=
1� ✓
p0+
✓
p1
m : ⌃ ! X
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
Complex method [.,.]θ (sigma-algebra).
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
1
p:=
1� ✓
p0+
✓
p1
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
Complex methods (delta-ring).
Complex methods (delta-ring).
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)(R)
(R)(R)
(R)(⌃)
(⌃)(⌃)
(⌃)
Complex methods (delta-ring).
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)
(R)
(R)(⌃)
(⌃)(⌃)
(⌃)
Complex methods (delta-ring). Example.
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
It is not difficult to show that
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
It is not difficult to show that
Then, for 1 p0, p1 < 1,
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = [`1, `1][✓] = `1 = Lpw(⌫),
[Lp0(⌫), Lp1(⌫)][✓] = [c0, c0][✓] = c0 = Lp(⌫).
Intermediate spaces (sigma-algebra).
Intermediate spaces (sigma-algebra).
If and where
then
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Lp0(m) + Lp1(m)
Lp0w (m) + Lp1(m)
Lp0(m) + Lp1w (m)
Lp0w (m) + Lp1
w (m)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Lp0(m) + Lp1(m)
Lp0w (m) + Lp1(m)
Lp0(m) + Lp1w (m)
Lp0w (m) + Lp1
w (m)
Gain of integrability (sigma-algebra).
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
r < s, Lsw(m) ✓ Lr(m).
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f 2 Lsw(m) ✓ Lr
w(m)f
⌦
r < s, Lsw(m) ✓ Lr(m).
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f�[fn] 2 Lr(m)
f 2 Lsw(m) ✓ Lr
w(m)f
⌦
n f�[fn]
r < s, Lsw(m) ✓ Lr(m).
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f�[fn] 2 Lr(m)
f 2 Lsw(m) ✓ Lr
w(m)
��f � f�[fn]
��Lr
w(m)! 0
f
⌦
n f�[fn]
r < s, Lsw(m) ✓ Lr(m).
Intermediate spaces (delta-ring).
Intermediate spaces (delta-ring).
If then
1 p0 < p < p1 < 1,
(i) Lp0w (⌫) \ Lp1
w (⌫) ✓ Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫).
(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).
(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1
w (⌫).
(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0
w (⌫) + Lp1(⌫).
Intermediate spaces (delta-ring).
If then
Nevertheless,
1 p0 < p < p1 < 1,
(i) Lp0w (⌫) \ Lp1
w (⌫) ✓ Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫).
(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).
(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1
w (⌫).
(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0
w (⌫) + Lp1(⌫).
Lp0w (⌫) \ Lp1
w (⌫) 6✓ Lp(⌫),
Lpw(⌫) 6✓ Lp0(⌫) + Lp1(⌫).
Locally strongly additive measure.
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
2) is not
locally strongly additive.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
2) is not
locally strongly additive.
3) Every is (locally) strongly additive.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
m : ⌃ ! X
k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0
Intermediate spaces (delta-ring).
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1 p0 < p1 < 1.
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
⌫
1 p0 < p1 < 1.
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
⌫
1 p0 < p1 < 1.
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
2) for every (some)
⌫
1 p0 < p1 < 1.
p0 < p < p1.
Lp0w (⌫) \ Lp1
w (⌫) ✓ Lp(⌫)
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
2) for every (some)
3) for every (some)
⌫
1 p0 < p1 < 1.
p0 < p < p1.
p0 < p < p1.
Lp0w (⌫) \ Lp1
w (⌫) ✓ Lp(⌫)
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫)
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
Gain of integrability (delta-ring).
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
f�[fn]
n
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
��f�[f>n]
��L
p0w (⌫)
! 0
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
⌫ p0 < p < p1,
f
⌦
n
1
n
���f�[f< 1n ]
���L
p1w (⌫)
! 0
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
���f�[f< 1n ]
���L
p1w (⌫)
! 0
f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
��f�[f>n]
��L
p0w (⌫)
! 0
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
Complex methods: equalities (R).
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
0 < ✓ < 1 p0 < p1 < 1.
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
⌫
0 < ✓ < 1 p0 < p1 < 1.
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
2)
⌫
0 < ✓ < 1 p0 < p1 < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = Lp(⌫).
1
p:=
1� ✓
p0+
✓
p1
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
2)
3)
⌫
0 < ✓ < 1 p0 < p1 < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = Lp(⌫).
[Lp0(⌫), Lp1(⌫)][✓] = [Lp0w (⌫), Lp1(⌫)][✓]
= [Lp0(⌫), Lp1w (⌫)][✓]
= Lpw(⌫).1
p:=
1� ✓
p0+
✓
p1
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
1
p:=
1� ✓
p0+
✓
p1
Lp(⌫) [Lp0w (⌫), Lp1
w (⌫)][✓] Lpw(⌫)
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
1
p:=
1� ✓
p0+
✓
p1
Lp(⌫) [Lp0(⌫), Lp1(⌫)][✓] Lpw(⌫)
Lp(⌫) [Lp0w (⌫), Lp1
w (⌫)][✓] Lpw(⌫)
The real method (.,.)θ,q. [Lions & Peetre (1964)]
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
(X0, X1)
0 < ✓ < 1 q 1,
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
is the space of
those such that
is finite.
f 2 X0 +X1
(X0, X1)
0 < ✓ < 1 q 1, (X0, X1)✓,q
kfk✓,q :=
8>><
>>:
Z 1
0
�t�✓K(t, f)
�q dt
t
� 1q
, q < 1
supt>0
t�✓K(t, f), q = 1
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
is the space of
those such that
is finite.
f 2 X0 +X1
(X0, X1)
0 < ✓ < 1 q 1, (X0, X1)✓,q
kfk✓,q :=
8>><
>>:
Z 1
0
�t�✓K(t, f)
�q dt
t
� 1q
, q < 1
supt>0
t�✓K(t, f), q = 1
K(t, f) := inf {kf0kX0 + tkf1kX1 :
f0 2 X0, f1 2 X1, f = f0 + f1} .
The real method: problem/classical results.
The real method: problem/classical results.
For we want
to describe with 0 < ✓ < 1 q 1.(X0, X1)✓,q ,
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
Lorentz spaces of a vector measure.
Lorentz spaces of a vector measure.
Let and consider
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
⌃m�! X
x
0�! R, x
0 2 X
0,
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
⌃m�! X
x
0�! R, x
0 2 X
0,
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f : ⌦ �! R
⌃m�! X
x
0�! R, x
0 2 X
0,
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
⌃m�! X
x
0�! R, x
0 2 X
0,
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
kfkp,q
:= supn
kfkL
p,q(|hm,x
0i|) : kx0k 1
o
.
⌃m�! X
x
0�! R, x
0 2 X
0,
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
kfkp,q
:= supn
kfkL
p,q(|hm,x
0i|) : kx0k 1
o
.
⌃m�! X
x
0�! R, x
0 2 X
0,
�L1w(m), L1(m)
�✓,q
S(⌃)Lp,q
w (m)✓ Lp,q
w (m).
Distribution and decreasing rearrangement.
Distribution and decreasing rearrangement.
Distribution function. measurable:
f : ⌦ �! Rkmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)
kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})
= supn
|hm,x
0i|f (t) : kx0k 1
o
.
Distribution and decreasing rearrangement.
Distribution function. measurable:
Decreasing rearrangement function.
kmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)
kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})
= supn
|hm,x
0i|f (t) : kx0k 1
o
.
f
⇤(s) := inf {t � 0 : kmkf
(t) s}= �kmkf
(s)
= sup {f⇤x
0(s) : kx0k 1}
f : ⌦ �! R
Lorentz spaces of the semivariation.
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! RLp,q(kmk)
kfkLp,q(kmk) < 1,
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! R
1 p, q < 1
Lp,q(kmk)
kfkLp,q(kmk) :=
Z 1
0
⇣s
1p f⇤(s)
⌘q ds
s
� 1q
,
kfkLp,q(kmk) < 1,
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! R
1 p, q < 1
1 p < 1, q = 1
Lp,q(kmk)
kfkLp,q(kmk) :=
Z 1
0
⇣s
1p f⇤(s)
⌘q ds
s
� 1q
,
kfkLp,1(kmk) := supn
s1p f⇤(s) : s > 0
o
.
kfkLp,q(kmk) < 1,
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
Lp,q(kmk)kf + gkLp,q(kmk) C(p, q)
�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
3) Inclusions.
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
Lorentz spaces of the semivariation.
✓✓
✓✓
Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
...
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
L1(m)✓
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
L1(m)✓
✓
✓✓
✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
L1,1(kmk)
L1,1(kmk)
...
...
......
...
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
L1(m)✓
✓
✓✓
✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
L1,1(kmk)
L1,1(kmk)
...
...
......
... Lp(m)
Lpw(m)
1 p1 < p2 < 1
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
3) Inclusions.
4) For
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
1 p < 1,
Lp,p(kmk) Lp(m) Lpw(m) Lp,1(kmk).
Estimates for the K−functional.
Estimates for the K−functional.
If and then
f 2 L1w(m) t > 0,
Estimates for the K−functional.
If and then
a)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
Estimates for the K−functional.
If and then
a)
b)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
K�t, f, L1
w(m), L1(m)�
Z t
0f⇤(s)ds.
Estimates for the K−functional.
If and then
a)
b)
c)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
K�t, f, L1
w(m), L1(m)�
Z t
0f⇤(s)ds.
f 2 L1(m) 6)Z t
0f⇤(s)ds < 1,
6)Z 1
0f⇤(s)ds < 1.
The real method. [Mayoral, Naranjo & AF, 2011]
The real method. [Mayoral, Naranjo & AF, 2011]
1) For
with equivalence of norm−quasinorm.
0 < ✓ < 1 q 1,�L1(m), L1(m)
�✓,q
=�L1w(m), L1(m)
�✓,q
= L1
1�✓ ,q(kmk),
The real method. [Mayoral, Naranjo & AF, 2011]
1) For
with equivalence of norm−quasinorm.
2) For
0 < ✓ < 1 q 1,
1 p0 6= p1 1; 0 < ✓ < 1 q 1,
1
p=
1� ✓
p0+
✓
p1.
�L1(m), L1(m)
�✓,q
=�L1w(m), L1(m)
�✓,q
= L1
1�✓ ,q(kmk),
(Lp0(m), Lp1(m))✓,q = (Lp0w (m), Lp1
w (m))✓,q
= Lp,q(kmk),
The real method: consequences.
The real method: consequences.
Consequence 1. If the space is
normable.
Lp,q(kmk)p > 1,
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
[Beauzamy 1978] If and the
inclusion is weakly compact,
then is reflexive.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)0 < ✓ < 1 < q < 1
X0 \X1 ✓ X0 +X1
(X0, X1)✓,q
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
[Beauzamy 1978] If and the
inclusion is weakly compact,
then is reflexive.
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then is
weakly compact.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)0 < ✓ < 1 < q < 1
X0 \X1 ✓ X0 +X1
(X0, X1)✓,q
1 p0 < p1 1, Lp1w (m) ✓ Lp0(m)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Interpolation of spaces of integrable functions with respect to a vector measure
Antonio Fernández (Universidad de Sevilla)