espacios de funciones integrables respecto de una medida ......iv encuentro de an alisis funcional y...

Espacios de funciones integrables

respecto de una medida vectorial.

Propiedades y aplicaciones.

Fernando Mayoral, mayoral@us.es

Departamento de Matematica Aplicada II

Universidad de Sevilla

IV Encuentro de Analisis Funcional y sus Aplicaciones

Salobrena, 3-5 de Abril de 2008

1 People

2 Introduction

3 Properties of spaces Lp(m) and Lpw (m).

4 Related Spaces.

5 Multiplication operators on spaces Lp(m).

6 Interpolation of Lp(m) spaces.

People

Universidad Politecnica de Valencia

Enrique A. Sanchez Perez

Irene Ferrando Palomares

Jose Rodrıguez Ruız

Antonio Fernandez Carrion

Ricardo del Campo Acosta

Fernando Mayoral Masa

Francisco Naranjo Naranjo

Carmen Saez Agullo

People

Universidad Politecnica de Valencia

Enrique A. Sanchez Perez

Irene Ferrando Palomares

Jose Rodrıguez Ruız

Antonio Fernandez Carrion

Ricardo del Campo Acosta

Fernando Mayoral Masa

Francisco Naranjo Naranjo

Carmen Saez Agullo

PeopleIntroduction

Properties of spaces Lp(m) and Lpw (m).

Related Spaces.Multiplication operators on spaces Lp(m).

Interpolation of Lp(m) spaces.References

Notation.L1w (m) and L1(m).

Lpw (m) and Lp(m).

Introduction

(Ω,Σ) measurable space (Σ is a σ−algebra in Ω).

m : Σ −→ X (countably additive) vector measure in a (real)

Banach space X with dual X ∗.

The semivariation of m. For A ∈ Σ,

‖m‖(A) := sup |〈m, x∗〉|(A) : x∗ ∈ X ∗, ‖x∗‖ ≤ 1

|〈m, x∗〉| is the measure variation of the real measure defined

〈m, x∗〉(A) := 〈m(A), x∗〉.

Espacios de funciones integrables respecto de una medida vectorial. Propiedades y aplicaciones.

PeopleIntroduction

Lpw (m) and Lp(m).

Introduction

‖m‖(A) := sup |〈m, x∗〉|(A) : x∗ ∈ X ∗, ‖x∗‖ ≤ 1

〈m, x∗〉(A) := 〈m(A), x∗〉.

PeopleIntroduction

Lpw (m) and Lp(m).

Introduction

‖m‖(A) := sup |〈m, x∗〉|(A) : x∗ ∈ X ∗, ‖x∗‖ ≤ 1

〈m, x∗〉(A) := 〈m(A), x∗〉.

PeopleIntroduction

Lpw (m) and Lp(m).

Introduction

‖m‖(A) := sup |〈m, x∗〉|(A) : x∗ ∈ X ∗, ‖x∗‖ ≤ 1

〈m, x∗〉(A) := 〈m(A), x∗〉.

PeopleIntroduction

Lpw (m) and Lp(m).

• L1w (m) is the space of all (equivalence classes of) weakly inte-

grable functions with respect to m. That is, of all real measurable

functions f : Ω −→ R such that f is integrable with respect to each

|〈m, x∗〉|, x∗ ∈ X ∗. L1w (m) becomes a Banach lattice when it is

equipped with the natural order m-a.e. and the norm

‖f ‖1 := sup

∫Ω|f | d |〈m, x∗〉| : ‖x∗‖ ≤ 1

, f ∈ L1

w (m).

For each A ∈ Σ there exists an element ζf (A) ∈ X ∗∗ such that

〈x∗, ζf (A)〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

‖f ‖1 := sup

∫Ω|f | d |〈m, x∗〉| : ‖x∗‖ ≤ 1

, f ∈ L1

w (m).

〈x∗, ζf (A)〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

‖f ‖1 := sup

∫Ω|f | d |〈m, x∗〉| : ‖x∗‖ ≤ 1

, f ∈ L1

w (m).

〈x∗, ζf (A)〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

• L1(m) is the space of all (equivalence classes of) integrable

functions with respect to m. That is of all f ∈ L1w (m) such that for

each A ∈ Σ there exists a vector∫A f dm ∈ X which verifies the

barycentric formula

〈∫

Af dm, x∗〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

L1(m) is a closed ideal of L1w (m).

L1(m) is an order-continuous Banach lattice with a weak unit.

The simple functions are dense in L1(m).

If X is weakly sequentially complete =⇒ L1w (m) = L1(m).

PeopleIntroduction

Lpw (m) and Lp(m).

barycentric formula

〈∫

Af dm, x∗〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

barycentric formula

〈∫

Af dm, x∗〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

barycentric formula

〈∫

Af dm, x∗〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

barycentric formula

〈∫

Af dm, x∗〉 =

f d〈m, x∗〉, ∀x∗ ∈ X ∗.

PeopleIntroduction

Lpw (m) and Lp(m).

The spaces Lp(m) and Lpw (m). [6]

1 < p <∞,

Lpw (m) :=

f : |f |p ∈ L1

Lp(m) :=

f : |f |p ∈ L1(m)

The natural norm for both spaces is given by

‖f ‖p := sup

(∫Ω|f |p d

∣∣⟨m, x ′⟩∣∣) 1p

: x∗ ∈ X ∗, ‖x∗‖ ≤ 1

For 1 ≤ p < q <∞,

⊂ · · · Lqw (m) ⊂ Lp

w (m) ⊂ L1w (m) ⊂ L0(m)

∪ ∪ ∪L∞(m) ⊂ · · · Lq(m) ⊂ Lp(m) ⊂ L1(m)

PeopleIntroduction

Lpw (m) and Lp(m).

1 < p <∞,

Lpw (m) :=

f : |f |p ∈ L1

Lp(m) :=

f : |f |p ∈ L1(m)

‖f ‖p := sup

(∫Ω|f |p d

∣∣⟨m, x ′⟩∣∣) 1p

: x∗ ∈ X ∗, ‖x∗‖ ≤ 1

For 1 ≤ p < q <∞,

⊂ · · · Lqw (m) ⊂ Lp

w (m) ⊂ L1w (m) ⊂ L0(m)

∪ ∪ ∪L∞(m) ⊂ · · · Lq(m) ⊂ Lp(m) ⊂ L1(m)

PeopleIntroduction

Lpw (m) and Lp(m).

1 < p <∞,

Lpw (m) :=

f : |f |p ∈ L1

Lp(m) :=

f : |f |p ∈ L1(m)

‖f ‖p := sup

(∫Ω|f |p d

∣∣⟨m, x ′⟩∣∣) 1p

: x∗ ∈ X ∗, ‖x∗‖ ≤ 1

For 1 ≤ p < q <∞,

⊂ · · · Lqw (m) ⊂ Lp

w (m) ⊂ L1w (m) ⊂ L0(m)

∪ ∪ ∪L∞(m) ⊂ · · · Lq(m) ⊂ Lp(m) ⊂ L1(m)

PeopleIntroduction

Properties of Lp(m).Properties of Lp

w (m).Reflexivity.Inclusions.The integration operator.

Properties of spaces Lp(m).

Lp(m), p ≥ 1. [6]

Lp(m) is an order-continuous Banach lattice with weak order

In Lp(m) verifies a Dominated Convergence Theorem: Let (fn)

be a sequence of p−integrable functions which is pointwise

convergent to f and g a p−integrable function such that

|fn| ≤ g for each n (m−a.e.). Then f ∈ Lp(m) and (fn)

converges to f in norm

If E is a p−convex order-continuous Banach lattice with a

weak order unit, then E is lattice isomorphic to some Lp(m).

PeopleIntroduction

Lp(m), p ≥ 1. [6]

PeopleIntroduction

Lp(m), p ≥ 1. [6]

PeopleIntroduction

Lp(m), p ≥ 1. [6]

PeopleIntroduction

Properties of spaces Lpw (m).

Lpw (m), p ≥ 1

Lpw (m) is a Banach lattice with the Fatou property. That is, if

0 ≤ fn ↑ and (fn) is a norm bounded sequence in Lpw (m), then

f := supn fn is well defined, f ∈ Lpw (m) and ‖fn‖p ↑ ‖f ‖p.

[4, Th. 4] Let E be any p−convex Banach lattice with the

σ−Fatou property and possessing a weak unit which belongs

to Ea := x ∈ E : |x | ≥ un ↓ 0implies‖un‖ ↓ 0 (the elements

of E with σ−order continuous norm). Then there exists a

vector measure m such that E is Banach lattice isomorphic to

Lpw (m).

PeopleIntroduction

Lpw (m), p ≥ 1

Lpw (m).

PeopleIntroduction

Lpw (m), p ≥ 1

Lpw (m).

PeopleIntroduction

Properties of spaces Lp(m) and Lpw (m). Reflexivity.

Theorem. [6, Cor. 3.10]

For every p > 1, the following conditions are equivalent:

(a) Lpw (m) is reflexive.

(b) Lpw (m) has order continuous norm.

(c) Lpw (m) is a KB-space (every norm bounded, positive,

increasing sequence is norm convergent).

(d) Lpw (m) = Lp(m) as Banach lattices.

(e) L1w (m) = L1(m) as Banach lattices.

PeopleIntroduction

Remarks.

L1(m) and L1w (m) may be reflexive (=⇒ they are equal).

Lp(m) and Lpw (m), p > 1, may be non reflexive

(⇐⇒ Lp(m) 6= Lpw (m)).

PeopleIntroduction

Remarks.

L1(m) and L1w (m) may be reflexive (=⇒ they are equal).

Lp(m) and Lpw (m), p > 1, may be non reflexive

(⇐⇒ Lp(m) 6= Lpw (m)).

PeopleIntroduction

Properties of spaces Lp(m) and Lpw (m). Inclusions.

Proposition. [6]

For p > q ≥ 1 we have Lpw (m) ⊂ Lq(m) and this inclusion is a

L−weakly compact operator. In particular, it is a weakly compact

operator.

Remarks.

The unit ball of Lpw (m) is weakly relatively compact in each

Lq(m) where it is included.

H ⊂ Lp(m) bounded ⇒ q−uniformly integrable, 1 ≤ q < p,

lim‖m‖(A)→0

‖f χA‖q = 0 uniformly in H.

PeopleIntroduction

Proposition. [6]

operator.

Remarks.

lim‖m‖(A)→0

PeopleIntroduction

Proposition. [6]

operator.

Remarks.

lim‖m‖(A)→0

PeopleIntroduction

The integration operator.

Theorem

For p > 1,

a) The integration map

Im : f ∈ Lpw (m) −→ Im(f ) :=

∫f dm ∈ X is weakly compact.

b) The integration map

Im : f ∈ Lpw (m) −→ Im(f ) :=

∫f dm ∈ X is compact if and

only if the range of m is a relatively compact subset of X .

PeopleIntroduction

The integration operator.

Theorem

For p > 1,

a) The integration map

Im : f ∈ Lpw (m) −→ Im(f ) :=

∫f dm ∈ X is weakly compact.

b) The integration map

Im : f ∈ Lpw (m) −→ Im(f ) :=

∫f dm ∈ X is compact if and

only if the range of m is a relatively compact subset of X .

PeopleIntroduction

Theorem (S. Okada, W. Ricker, L. Rodriguez-Piazza, [9]) The

integration map Im : f ∈ L1(m) −→ Im(f ) :=∫

f dm ∈ X is

compact if and only if m verifies

(i) m has finite variation.

(ii) m has a Radon-Nikodym derivative G = dmd |m| ∈ L1(|m|,X )

with respect to |m|.(iii) m the function G has |m|−essentially relatively compact

range in X .

In this case L1(m) = L1(|m|) and Im(f ) =∫

fG d |m| for all

f ∈ L1(m).

PeopleIntroduction

Related Spaces. [5].

Definition.

Given a norming set Λ ⊂ X ∗ let us consider the space L1,sΛ (ν) of all

f ∈ L0(m) such that

a) f ∈ L1(〈m, x∗〉) for every x∗ ∈ Λ.

b) for each A ∈ Σ there is ξf ,Λ(A) ∈ X such that

〈ξf ,Λ(A), x∗〉 =

fd〈m, x∗〉 for every x∗ ∈ Λ.

PeopleIntroduction

Related Spaces

Proposition

Given a norming set Λ ⊂ X ∗,

a) L1(m) ⊂ L1,sΛ (m) ⊂ L1

w (m).

L1,sΛ (m), ‖ ‖1

)is a Banach lattice.

Remark

Topological conditions on Λ in order to obtain

L1,sΛ (m) = L1(m) or L1,s

Λ (m) = L1w (m).

PeopleIntroduction

Related Spaces

Proposition

Given a norming set Λ ⊂ X ∗,

a) L1(m) ⊂ L1,sΛ (m) ⊂ L1

w (m).

L1,sΛ (m), ‖ ‖1

)is a Banach lattice.

Remark

Topological conditions on Λ in order to obtain

L1,sΛ (m) = L1(m) or L1,s

Λ (m) = L1w (m).

PeopleIntroduction

Multiplication operators Lp(m) −→ L1(m), p > 1.Multiplication operators Lp(m) −→ Lp(m), p ≥ 1.

Multiplication operators. [3].

•Let us consider multiplication operators

Mg : f ∈ F −→ Mg (f ) := gf ∈ G (1)

between function spaces F and G where each space is an Lp(m) or

Lpw (m) space.

•Remark. If p, q > 1 are conjugate exponents

L1(m) = Lp(m) · Lq(m) =

= Lpw (m) · Lq(m) = Lp(m) · Lq

w (m).

L1w (m) = Lp

w (m) · Lqw (m).

PeopleIntroduction

Multiplication operators. [3].

•Let us consider multiplication operators

Mg : f ∈ F −→ Mg (f ) := gf ∈ G (1)

between function spaces F and G where each space is an Lp(m) or

Lpw (m) space.

•Remark. If p, q > 1 are conjugate exponents

L1(m) = Lp(m) · Lq(m) =

= Lpw (m) · Lq(m) = Lp(m) · Lq

w (m).

L1w (m) = Lp

w (m) · Lqw (m).

PeopleIntroduction

Multiplication operators from Lp(m) into L1(m), p > 1.

Theorem

Let p, q > 1 be conjugate exponents and g ∈ L0(m).

(1) g ∈ Lqw (m)⇐⇒ Mg : Lp(m) −→ L1(m) is continuous.

⇐⇒ Mg : Lpw (m) −→ L1

w (m) is continuous.

(2) g ∈ Lq(m)⇐⇒ Mg : Lp(m) −→ L1(m) is weakly compact.

⇐⇒ Mg : Lpw (m) −→ L1(m) is continuous.

⇐⇒ Mg : Lpw (m) −→ L1(m) is weakly compact.

PeopleIntroduction

Multiplication operators from Lp(m) into Lp(m), p ≥ 1.

Theorem

Let p ≥ 1, g ∈ L0(m) and mδ the restriction of m to

Gδ = ω ∈ Ω : |g (ω) | ≥ δ for each δ > 0.

(1) Mg : Lp(m) −→ Lp(m) is continuous ⇐⇒ g ∈ L∞(m).

(2) Mg : Lpw (m) −→ Lp(m) is continuous ⇐⇒ g ∈ L∞(m) and

Lp(mδ) = Lpw (mδ) for each δ > 0.

The condition Lp(mδ) = Lpw (mδ) for each δ > 0 does not

imply that Lp(mG ) = Lpw (mG ).

If p > 1 the condition Lp(mδ) = Lpw (mδ) is equivalent to

reflexivity.

PeopleIntroduction

Theorem

reflexivity.

PeopleIntroduction

Theorem

reflexivity.

PeopleIntroduction

Mg : Lp(m) −→ Lp(m) weakly compact

Theorem

Let p ≥ 1, g and mδ as before.

(1) Mg : Lpw (m) −→ Lp

w (m) is weakly compact ⇐⇒ g ∈ L∞(m)

and Lp(mδ) is reflexive for each δ > 0.

(2) When m is an atomless vector measure with σ−finite

variation, Mg : L1(m) −→ L1(m) is weakly compact if and

only if g = 0 m−a.e.

PeopleIntroduction

Mg : Lp(m) −→ Lp(m) weakly compact

Theorem

Let p ≥ 1, g and mδ as before.

(1) Mg : Lpw (m) −→ Lp

w (m) is weakly compact ⇐⇒ g ∈ L∞(m)

and Lp(mδ) is reflexive for each δ > 0.

(2) When m is an atomless vector measure with σ−finite

variation, Mg : L1(m) −→ L1(m) is weakly compact if and

only if g = 0 m−a.e.

PeopleIntroduction

Mg : Lp(m) −→ Lp(m) compact

Theorem

Let p ≥ 1, g and mδ as before. Mg : Lpw (m) −→ Lp

w (m) is compact

⇐⇒ g ∈ L∞(m) and dim(Lp(mδ)) <∞ for each δ > 0.

Furthermore, in this case, m (ΣG ) := m(A) : A ∈ Σ,A ⊆ G,1

(‖m‖ (A))1p

g dm : A ∈ Σ+

g dm : A ∈ Σ

are relatively compact subsets of X .

PeopleIntroduction

Mg : Lp(m) −→ Lp(m) compact

Theorem

Let p ≥ 1, g and mδ as before. Mg : Lpw (m) −→ Lp

w (m) is compact

⇐⇒ g ∈ L∞(m) and dim(Lp(mδ)) <∞ for each δ > 0.

Furthermore, in this case, m (ΣG ) := m(A) : A ∈ Σ,A ⊆ G,1

(‖m‖ (A))1p

g dm : A ∈ Σ+

g dm : A ∈ Σ

are relatively compact subsets of X .

PeopleIntroduction

Complex Interpolation. First Method of Calderon.Complex Interpolation. Second Method of Calderon.Complex Interpolation of Lp(m) spaces. Tools.Complex Interpolation of Lp(m)−spaces. Results.

Complex Interpolation. First Method of Calderon. [2]

Let (X0,X1) be a Banach couple of complex Banach spaces:

Denote by F(X0,X1) the space of all functions f : Ω −→ X0 + X1,

(Ω = z ∈ C : 0 < Re(z) < 1) having the following properties:

f is continuous and bounded in the norm of X0 + X1.

f is analytic in Ω in the norm of X0 + X1

t ∈ R −→ f (it) ∈ X0 is continuous and bounded.

t ∈ R −→ f (1 + it) ∈ X1 is continuous and bounded.

F(X0,X1) is a Banach space with the norm defined by

‖f ‖F(X0,X1) := max

supt∈R‖f (it)‖X0 , sup

t∈R‖f (1 + it)‖X1

PeopleIntroduction

Complex Interpolation. First Method of Calderon. [2]

Let (X0,X1) be a Banach couple of complex Banach spaces:

Denote by F(X0,X1) the space of all functions f : Ω −→ X0 + X1,

(Ω = z ∈ C : 0 < Re(z) < 1) having the following properties:

f is continuous and bounded in the norm of X0 + X1.

f is analytic in Ω in the norm of X0 + X1

t ∈ R −→ f (it) ∈ X0 is continuous and bounded.

t ∈ R −→ f (1 + it) ∈ X1 is continuous and bounded.

F(X0,X1) is a Banach space with the norm defined by

‖f ‖F(X0,X1) := max

supt∈R‖f (it)‖X0 , sup

t∈R‖f (1 + it)‖X1

PeopleIntroduction

Complex Interpolation, [X0,X1][θ], 0 ≤ θ ≤ 1,

is the space of all elements x ∈ X0 + X1 that can be

represented as x = f (θ) for some f ∈ F(X0,X1).

is a Banach space with the norm defined by

‖x‖[X0,X1][θ]:= inf

‖f ‖F(X0,X1) : f (θ) = x

X0 ∩ X1 is a dense subspace.

PeopleIntroduction

‖x‖[X0,X1][θ]:= inf

‖f ‖F(X0,X1) : f (θ) = x

PeopleIntroduction

‖x‖[X0,X1][θ]:= inf

‖f ‖F(X0,X1) : f (θ) = x

PeopleIntroduction

Complex Interpolation. Second Method of Calderon.

Let G(X0,X1) be the space of all functions g : Ω −→ X0 + X1,

having the following properties:

‖g(z)‖X0+X1 ≤ c(1 + |z |) for some c > 0 and all z ∈ Ω.

f is continuous in Ω and analytic in Ω, in the norm of X0 + X1

For all t1, t2 ∈ R, g(j + it1)− g(j + it2) ∈ Xj , j = 0, 1 and

‖g‖G(X0,X1) := maxj=0,1

supt1 6=t2

∥∥∥∥g(j + it1)− g(j + it2)

t1 − t2

∥∥∥∥Xj

The quotient space of G(X0,X1) with the constants is a Banach

space with the quotient norm defined by ‖g‖G(X0,X1).

PeopleIntroduction

Complex Interpolation. Second Method of Calderon.

Let G(X0,X1) be the space of all functions g : Ω −→ X0 + X1,

having the following properties:

‖g(z)‖X0+X1 ≤ c(1 + |z |) for some c > 0 and all z ∈ Ω.

f is continuous in Ω and analytic in Ω, in the norm of X0 + X1

For all t1, t2 ∈ R, g(j + it1)− g(j + it2) ∈ Xj , j = 0, 1 and

‖g‖G(X0,X1) := maxj=0,1

supt1 6=t2

∥∥∥∥g(j + it1)− g(j + it2)

t1 − t2

∥∥∥∥Xj

The quotient space of G(X0,X1) with the constants is a Banach

space with the quotient norm defined by ‖g‖G(X0,X1).

PeopleIntroduction

Complex Interpolation, [X0,X1][θ], 0 < θ < 1,

represented as x = g ′(θ) for some g ∈ G(X0,X1).

‖x‖[X0,X1][θ] := inf‖g‖G(X0,X1) : g ′(θ) = x , g ∈ G

[X0,X1][θ] is a closed subspace of [X0,X1][θ] (see [1]).

[X0,X1][θ] is a subset of the Gagliardo completion of

[X0,X1][θ]. The limits in X0 + X1 of the bounded sequences of

[X0,X1][θ].

PeopleIntroduction

[X0,X1][θ].

PeopleIntroduction

[X0,X1][θ].

PeopleIntroduction

[X0,X1][θ].

PeopleIntroduction

[X0,X1][θ].

PeopleIntroduction

[X0,X1][θ].

PeopleIntroduction

Complex Interpolation of Lp(m) spaces.

Given the Banach interpolation couple (X0,X1) where

X0 = Lp0(m) or Lp0w (m) and X1 = Lp1(m) or Lp1

w (m), 1 ≤ p0 6= p1

we obtain [X0,X1][θ] and [X0,X1][θ], through their relationship with

the Calderon-Lozanovskii’s construction: X 1−θ0 X θ

1 , (over

Banach lattices of measurable functions)x : |x | ≤ λ|x0|1−θ|x1|θ, x0 ∈ B1(X0), x1 ∈ B1(X1)

endowed with the norm ‖x‖θ = inf λ > 0 : ...may not be an interpolation space of the pair (X0,X1).

PeopleIntroduction

w (m), 1 ≤ p0 6= p1

1 , (over

PeopleIntroduction

w (m), 1 ≤ p0 6= p1

1 , (over

PeopleIntroduction

w (m), 1 ≤ p0 6= p1

1 , (over

PeopleIntroduction

the Gustavsson-Peetre’s method: 〈X0,X1, θ〉, 0 < θ < 1. Thespace of elements x ∈ X0 + X1 such that there exists asequence (xk)∈Z of elements xk ∈ X0 ∩ X1 such that

x =∑k∈Z

xk (convergence in X0 + X1),∑k∈Z

2kθxk is weakly unconditionally Cauchy in X0,∑

2k(θ−1)xk is weakly unconditionally Cauchy in X1.

〈X0,X1, θ〉 is a Banach space with the norm defined by

maxj=0,1

∥∥∥∥∥∑

2k(θ−j)xk

∥∥∥∥∥Xj

: F ⊂ Z finite, |λk | ≤ 1

PeopleIntroduction

x =∑k∈Z

maxj=0,1

∥∥∥∥∥∑

2k(θ−j)xk

∥∥∥∥∥Xj

PeopleIntroduction

x =∑k∈Z

maxj=0,1

∥∥∥∥∥∑

2k(θ−j)xk

∥∥∥∥∥Xj

PeopleIntroduction

x =∑k∈Z

maxj=0,1

∥∥∥∥∥∑

2k(θ−j)xk

∥∥∥∥∥Xj

PeopleIntroduction

x =∑k∈Z

maxj=0,1

∥∥∥∥∥∑

2k(θ−j)xk

∥∥∥∥∥Xj

PeopleIntroduction

Relations

For a Banach couple (see [8, Th. 5 and Section 7])

〈X0,X1, θ〉 ⊂ [X0,X1][θ].

For a Banach couple of lattices of measurable functions (norm

one inclusions)

[X0,X1][θ] ⊂ X 1−θ0 X θ

1 ⊂ [X0,X1][θ] .

If X0 or X1 has order continuous norm,

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

If X 1−θ0 X θ

1 has the Fatou property,

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

PeopleIntroduction

Relations

〈X0,X1, θ〉 ⊂ [X0,X1][θ].

one inclusions)

[X0,X1][θ] ⊂ X 1−θ0 X θ

1 ⊂ [X0,X1][θ] .

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

If X 1−θ0 X θ

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

PeopleIntroduction

Relations

〈X0,X1, θ〉 ⊂ [X0,X1][θ].

one inclusions)

[X0,X1][θ] ⊂ X 1−θ0 X θ

1 ⊂ [X0,X1][θ] .

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

If X 1−θ0 X θ

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

PeopleIntroduction

Relations

〈X0,X1, θ〉 ⊂ [X0,X1][θ].

one inclusions)

[X0,X1][θ] ⊂ X 1−θ0 X θ

1 ⊂ [X0,X1][θ] .

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

If X 1−θ0 X θ

[X0,X1][θ] = X 1−θ0 X θ

1 (isometry).

PeopleIntroduction

Complex Interpolation of Lp(m)−spaces. [7].

Theorem

Let 1 ≤ p0 6= p1 <∞, 0 < θ < 1 be and p(θ) defined by1

p(θ)=

1− θp0

p1. Then:

(1) [Lp0(m), Lp1(m)][θ] =[Lp0

w (m), Lp1(m)]

w (m), Lp1w (m)

= Lp(θ)(m).

(2) [Lp0(m), Lp1(m)][θ] =[Lp0

w (m), Lp1(m)][θ]

w (m), Lp1w (m)

][θ]= L

p(θ)w (m).

PeopleIntroduction

[1] J. Bergh, On the relation between the two complex methods of

interpolation, Indiana Univ. Math. J. 28 (1979), no. 5, 775–778.

MR542336 (80f:46062)

[2] A. P. Calderon, Intermediate spaces and interpolation, the complex

method, Studia Math. 24 (1964), 113-190. MR0167830

[3] R. del Campo, A. Fernandez, I. Ferrando, F. Mayoral, and F. Naranjo,

Multiplication operators on spaces of integrable functions with respect to a

vector measure, J. Math. Anal. Appl. doi:10.1016/j.jmaa.2008.01.080 to

appear, -.

[4] G. P. Curbera and W. J. Ricker, The Fatou property in p–convex Banach

lattices, J. Math. Anal. Appl. 328 (2007), 287-294. MR2285548

[5] A. Fernandez, F. Mayoral, F. Naranjo, and J. Rodrıguez, Norming sets and

integration with respect to Vector Measures, Submited.

[6] A. Fernandez, F. Mayoral, F. Naranjo, C. Saez, and E. A. Sanchez–Perez,

Spaces of p–integrable functions with respect to a vector measure,

Positivity 10 (2006), 1-16. MR2223581Espacios de funciones integrables respecto de una medida vectorial. Propiedades y aplicaciones.

PeopleIntroduction

[7] A. Fernandez, F. Mayoral, F. Naranjo, and E. A. Sanchez–Perez, Complex

Interpolation of spaces of integrable functions with respect to a Vector

Measure, Preprint (2008).

[8] S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal.

44 (1981), no. 1, 50–73. MR638294 (83j:46085)

[9] S. Okada, W. J. Ricker, and L. Rodrıguez–Piazza, Compactness of the

integration operator associated with a vector measure, Studia Math. 150

(2002), 133–149. MR1892725

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