subespacios de funciones integrables respecto de una medidavectorialyextensio...

Notation and preliminariesVector measures and spaces of integrable functions

A factorization theoremAn application: extensions of operators on p-convex B. f. s.

Subespacios de funciones integrables respecto de unamedida vectorial y extensi on de operadores

Enrique A. S anchez P erezJ. M. Calabuig O. Delgado

Instituto de Matematica Pura y Aplicada (I.M.P.A.).Departamento de Matematica Aplicada.

Universidad Politecnica de Valencia.Camino de Vera S/N, 46022, Valencia.

X EARCOMallorca, Mayo 2007

Enrique A. S anchez P erez Subespacios y extensiones de operadores

En esta charla mostramos que la extension de un operador definido en un espacio defunciones de Banach es equivalente, bajo ciertos requisitos mınimos, a sufactorizaci on a trav es de un subespacio de funciones integrables respecto de lamedida vectorial mT asociada al operador T . Ademas, este subespacio particulardetermina aquella propiedad del operador que se pretende preservar, en el sentido deque constituye el dominio optimo -el mayor subespacio al que se puede extender-, demanera que la extension conserva esta propiedad. Esto permite un planteamientoabstracto de ciertos aspectos de la teorıa de dominio optimo, que se ha desarrolladode manera importante en los ultimos tiempos, principalmente en la direccion decaracterizar cual es el mayor espacio de funciones de Banach al que se puedenextender algunos operadores de importancia en analisis. Ademas, permite clasificarlas propiedades de los operadores que se conservan en la extensi on en funci ondel subespacio de L1(mT ) que constituye su dominio optimo.

Index of the talk

1 Notation and preliminaries

2 Vector measures and spaces of integrable functions

3 A factorization theorem

4 An application: extensions of operators on p-convex B. f. s.

Index of the talk

FIRST PART

Notation and preliminaries

General notation

III (Ω,Σ,µ) will be a finite measure space,

III E be a Banach space and E ′ its topological dual,

III BE the unit ball of E ,

III P(A) will represent the set of partitions π of A into finite number of disjointmeasurable sets,

III m : Σ → E will be a (countably additive) vector measure,

III The semivariation of m over A ∈ Σ is defined by

‖m‖(A) = supx ′∈BE ′

|〈m,x ′〉|(A) = supx ′∈BE ′

supπ∈P(A)

∑B∈π

|〈m,x ′〉(B)|,

with the usual notation

〈m,x ′〉(B) = 〈m(B),x ′〉 for each B ∈ Σ,

III A ∈ Σ is called m-null if ‖m‖(A) = 0.

General notation

III (Ω,Σ,µ) will be a finite measure space,

III E be a Banach space and E ′ its topological dual,

III BE the unit ball of E ,

III P(A) will represent the set of partitions π of A into finite number of disjointmeasurable sets,

III m : Σ → E will be a (countably additive) vector measure,

III The semivariation of m over A ∈ Σ is defined by

‖m‖(A) = supx ′∈BE ′

|〈m,x ′〉|(A) = supx ′∈BE ′

supπ∈P(A)

∑B∈π

|〈m,x ′〉(B)|,

with the usual notation

〈m,x ′〉(B) = 〈m(B),x ′〉 for each B ∈ Σ,

III A ∈ Σ is called m-null if ‖m‖(A) = 0.

Remark

In this talk we assume that µ and m are mutually absolutely continuous.In particular, µ can be a Rybakov measure; recall that a Rybakov measure for a vectormeasure m is a scalar measure ν defined as the variation of a measure 〈m,x ′〉, wherex ′ ∈ E ′, whenever m is absolutely continuous with respect to ν .A Rybakov measure always exists for every vector measure m (see [5, IX.2.2]).

Banach function (sub)spaces.

Let L0(µ) be the space of all measurable real functions on Ω.

1 We will call Banach function space to any Banach space X (µ) ⊆ L0(µ) with norm‖ · ‖X (µ) satisfying that1) if f ∈ L0(µ), g ∈ X (µ) with |f | ≤ |g| µ-a.e. then f ∈ X (µ) and ‖f‖X (µ) ≤ ‖g‖X (µ).2) For every A ∈ Σ, the characteristic function χA belongs to X (µ).

2 By a Banach function subspace of X (µ) we mean a Banach function spacecontinuously contained in X (µ), with the same order structure (allowing differentnorms).

3 A Banach function space is order continuous if order bounded increasingsequences are convergent in norm.

Remark

In this talk we assume that µ and m are mutually absolutely continuous.In particular, µ can be a Rybakov measure; recall that a Rybakov measure for a vectormeasure m is a scalar measure ν defined as the variation of a measure 〈m,x ′〉, wherex ′ ∈ E ′, whenever m is absolutely continuous with respect to ν .A Rybakov measure always exists for every vector measure m (see [5, IX.2.2]).

Banach function (sub)spaces.

Let L0(µ) be the space of all measurable real functions on Ω.

1 We will call Banach function space to any Banach space X (µ) ⊆ L0(µ) with norm‖ · ‖X (µ) satisfying that1) if f ∈ L0(µ), g ∈ X (µ) with |f | ≤ |g| µ-a.e. then f ∈ X (µ) and ‖f‖X (µ) ≤ ‖g‖X (µ).2) For every A ∈ Σ, the characteristic function χA belongs to X (µ).

2 By a Banach function subspace of X (µ) we mean a Banach function spacecontinuously contained in X (µ), with the same order structure (allowing differentnorms).

3 A Banach function space is order continuous if order bounded increasingsequences are convergent in norm.

L1(m)L1(m)L1(m) spaces.

A function f : Ω → R is said to be integrable with respect to the measure m if1 for each x ′ ∈ E ′ we have that f ∈ L1(〈m,x ′〉),2 for each A ∈ Σ there exists xA ∈ E such that

〈xA,x ′〉 =∫

Afd〈m,x ′〉 for every x ′ ∈ E ′.

The element xA is usually denoted by∫

A f dm. The space of the classes (equalitym-almost everywhere) of these functions is denoted by L1(m). The expression

‖f‖m = supx ′∈BE ′

∫|f |d |〈m,x ′〉| for each f ∈ L1(m),

defines a lattice norm on L1(m) for which L1(m) is an order continuous Banachfunction space.

The indefinite integral mf : Σ → E of a function f ∈ L1(m) is defined by

mf (A) =∫

Afdm, A ∈ Σ.

The Orlicz-Pettis Theorem ensures that mf is again a countably additive vectormeasure. An equivalent norm for L1(m) is given by

|||f |||m = supA∈Σ

∥∥∥∥∫A

∥∥∥∥E

for each f ∈ L1(m).

Moreover |||f |||L1(m) ≤ ‖f‖L1(m) ≤ 2|||f |||L1(m) ([5, Ch. I.1.11]).

Lp(m)Lp(m)Lp(m) spaces.

Let 1 ≤ p < ∞. A measurable function f : Ω → R is said to be p-integrable if |f |p isintegrable with respect to m. The expression.

‖f‖Lp(m) = supx ′∈BE ′

(∫|f |p d |〈m,x ′〉|

for each f ∈ Lp(m),

defines a lattice norm on Lp(m) ([6]). The space Lp(m) is an order continuous Banachfunction subspace of L1(m).

mf (A) =∫

Afdm, A ∈ Σ.

|||f |||m = supA∈Σ

∥∥∥∥∫A

∥∥∥∥E

(∫|f |p d |〈m,x ′〉|

mf (A) =∫

Afdm, A ∈ Σ.

|||f |||m = supA∈Σ

∥∥∥∥∫A

∥∥∥∥E

(∫|f |p d |〈m,x ′〉|

SECOND PART

Vector measures and spaces of integrable functions.

Y (µ)Y (µ)Y (µ) semivariation.

Let m : Σ → E be a (countably additive) vector measure and Y (µ) a Banach functionspace. The Y (µ)-semivariation ‖m‖Y (µ) of m is defined by

‖m‖Y (µ) := sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BY (µ)

L1Y (µ)(m)L1Y (µ)(m)L1Y (µ)(m) spaces.

We define L1Y (µ)(m) as the space of all (classes of m-a.e. equal) functions f of L1(m)

such that the vector measure associated to m, mf , has finite Y (µ)-semivariationendowed with the norm

‖f‖L1Y (µ)(m) := ‖mf ‖Y (µ) = sup

ϕ simpleϕ∈BY (µ)

∥∥∥∥∫f ϕdm

∥∥∥∥E

L1Y (µ)(m) is always a Banach function subspace of L1(m).

Y (µ)Y (µ)Y (µ) semivariation.

Let m : Σ → E be a (countably additive) vector measure and Y (µ) a Banach functionspace. The Y (µ)-semivariation ‖m‖Y (µ) of m is defined by

‖m‖Y (µ) := sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BY (µ)

L1Y (µ)(m)L1Y (µ)(m)L1Y (µ)(m) spaces.

We define L1Y (µ)(m) as the space of all (classes of m-a.e. equal) functions f of L1(m)

such that the vector measure associated to m, mf , has finite Y (µ)-semivariationendowed with the norm

‖f‖L1Y (µ)(m) := ‖mf ‖Y (µ) = sup

ϕ simpleϕ∈BY (µ)

∥∥∥∥E

L1Y (µ)(m) is always a Banach function subspace of L1(m).

Example

Some particular subspaces of L1(m) that are nowadays well known can berepresented in terms of the spaces L1

Y (µ)(m).

1 The semivariation of a vector measure can be computed by

‖m‖ = sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BL∞(µ)

= ‖m‖L∞(µ),

where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for anyfunction f ∈ L1(m), ‖mf ‖ = ‖mf ‖L∞(µ) = ‖f‖L1

L∞(µ)(m).

Hence, L1(m) = L1L∞(µ)(m).

2 Let 1 < p < ∞ and 1/p +1/p′ = 1. It is known that for any function f ∈ Lp(m), theexpression

‖mf ‖Lp′ (m) = supϕ simple

ϕ∈BLp′ (m)

∥∥∥∥E

equals the norm ‖f‖Lpw (m) of a function f belonging to Lp

w (m) ( [7] ).

In fact, it can be proved that Lpw (m) = L1

Lp′ (m)(m).

Example

Y (µ)(m).

‖m‖ = sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BL∞(µ)

= ‖m‖L∞(µ),

L∞(µ)(m).

ϕ∈BLp′ (m)

∥∥∥∥E

w (m) ( [7] ).

Lp′ (m)(m).

Example

Y (µ)(m).

‖m‖ = sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BL∞(µ)

= ‖m‖L∞(µ),

L∞(µ)(m).

ϕ∈BLp′ (m)

∥∥∥∥E

w (m) ( [7] ).

Lp′ (m)(m).

Example

Y (µ)(m).

‖m‖ = sup

∥∥∥∥∥ n

∑i=1

αAim(Ai )

∥∥∥∥∥E

∑i=1

αAiχAi

∈ BL∞(µ)

= ‖m‖L∞(µ),

L∞(µ)(m).

ϕ∈BLp′ (m)

∥∥∥∥E

w (m) ( [7] ).

Lp′ (m)(m).

THIRD PART

A factorization theorem.

Let X (µ) be an order continuous Banach function space and let T : X (µ) → E be anoperator. As usual, we define the (countably additive) vector measure

mT : Σ → E by mT (A) := T (χA).

An operator T is µ-determined if the set of ‖mT ‖-null sets equals the set of µ-null sets.Let Y (µ) be a Banach function space over µ containing the set of the simple functions.

Y (µ)Y (µ)Y (µ)-extensible operators.

A µ-determined operator T : X (µ) → E is said to be Y (µ)-extensible if there is aconstant K > 0 such that

‖T (f ϕ)‖ ≤ K‖f‖X (µ)‖ϕ‖Y (µ)

for every function f ∈ X (µ) and every simple function ϕ.

mT : Σ → E by mT (A) := T (χA).

‖T (f ϕ)‖ ≤ K‖f‖X (µ)‖ϕ‖Y (µ)

mT : Σ → E by mT (A) := T (χA).

‖T (f ϕ)‖ ≤ K‖f‖X (µ)‖ϕ‖Y (µ)

Theorem (First part)

Let X (µ) be an order continuous Banach function space with and let Y (µ) be aBanach function space verifying that the simple functions are dense. Given aµ-determined operator T : X (µ) → E the following assertions are equivalent

1 T is Y (µ)-extensible.2 The bounded linear map T can be factored through the space L1

Y (µ)(mT ) as

X (µ)T //

L1Y (µ)(mT )

IY (µ)

where the integration map IY (µ) is also Y (µ)-extensible.

3 There is a constant K > 0 verifying that for every f ∈ L1(m)

‖(mT )f ‖Y (µ) ≤ K‖f‖X (µ)

where (mT )f : Σ → E is the vector measure given by (mT )f (A) =∫

A fdmT .4 X (µ) ·Y (µ) ⊆ L1(mT ).

Theorem (First part)

Let X (µ) be an order continuous Banach function space with and let Y (µ) be aBanach function space verifying that the simple functions are dense. Given aµ-determined operator T : X (µ) → E the following assertions are equivalent

1 T is Y (µ)-extensible.2 The bounded linear map T can be factored through the space L1

Y (µ)(mT ) as

X (µ)T //

L1Y (µ)(mT )

IY (µ)

where the integration map IY (µ) is also Y (µ)-extensible.

3 There is a constant K > 0 verifying that for every f ∈ L1(m)

‖(mT )f ‖Y (µ) ≤ K‖f‖X (µ)

where (mT )f : Σ → E is the vector measure given by (mT )f (A) =∫

A fdmT .4 X (µ) ·Y (µ) ⊆ L1(mT ).

Theorem (Second part, optimal domain)

Moreover, if Z (µ) is any other order continuous Banach function space such that

X (µ) ⊆ Z (µ) with continuous inclusion,

T can be extended to Z (µ) and this extension is Y (µ)-extensible,

then Z (µ) ⊆ L1Y (µ)(mT ) with continuous inclusion.

Example

Assume that T is continuous. In this case taking into account that for each f ∈ X (µ)and each ϕ simple function

‖T (f ϕ)‖E ≤ K‖f ϕ‖X (µ) ≤ K‖f‖X (µ)‖ϕ‖L∞(µ),

we obtain that T is L∞(µ)-extensible. Reciprocally if T is L∞(µ)-extensible then takingϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimaldomain is L1

L∞(µ)(m) = L1(m) so this result includes the case studied by Curbera G.P.and Ricker W. in [2].

Example ([1])

In [1] the authors introduce the notion of Lp-product extensible operator and define thespaces L1

p,µ (m). In this article they prove that the bigger function space (i.e. the optimaldomain) to which an Lp-product extensible operator T can be extended preserving thisproperty is exactly the space L1

p,µ (mT ). Actually the notion of Lp-product extensibleoperator coincides with our definition of Y (µ)-extensible operator with Y (µ) = Lp(µ) for1 < p < ∞. In this case the space L1

p,µ (mT ) is now L1Lp(µ)(mT ). Hence Theorem 6 in [1]

is a particular case of Theorem 3.

Example

Assume that T is continuous. In this case taking into account that for each f ∈ X (µ)and each ϕ simple function

‖T (f ϕ)‖E ≤ K‖f ϕ‖X (µ) ≤ K‖f‖X (µ)‖ϕ‖L∞(µ),

we obtain that T is L∞(µ)-extensible. Reciprocally if T is L∞(µ)-extensible then takingϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimaldomain is L1

L∞(µ)(m) = L1(m) so this result includes the case studied by Curbera G.P.and Ricker W. in [2].

Example ([1])

In [1] the authors introduce the notion of Lp-product extensible operator and define thespaces L1

p,µ (m). In this article they prove that the bigger function space (i.e. the optimaldomain) to which an Lp-product extensible operator T can be extended preserving thisproperty is exactly the space L1

p,µ (mT ). Actually the notion of Lp-product extensibleoperator coincides with our definition of Y (µ)-extensible operator with Y (µ) = Lp(µ) for1 < p < ∞. In this case the space L1

p,µ (mT ) is now L1Lp(µ)(mT ). Hence Theorem 6 in [1]

is a particular case of Theorem 3.

FOURTH PART

An application: extension of operators on p-convex B. f. s.

A Banach lattice E is p-convex if there is a constant K such that for each finite setx1, ...,xn ∈ E ,

∑i=1

|xi |p)1/p‖ ≤ K (n

∑i=1

‖xi‖p)1/p.

Let T : E → F be an operator, where F is a lattice. T is p-concave if there is aconstant K such that for each finite family x1, ...,xn ∈ E ,

∑i=1

‖T (xi )‖p)1/p ≤ K‖(n

∑i=1

|xi |p)1/p‖.

If X (µ) a Banach function space we define its p-th power X (µ)[p] as the space

X (µ)[p] := |f |p : f ∈ X (µ).

If X (µ) is p-convex, X (µ)[p] is a Banach function space over µ with the quasi norm

‖g‖X (µ)[p]:= ‖|g|1/p‖p

X (µ), g ∈ X (µ)[p],

that in this case is equivalent to a norm.

If µ is a finite measure, then the inclusion X (µ) ⊆ X (µ)[p] is well-defined andcontinuous.

∑i=1

|xi |p)1/p‖ ≤ K (n

∑i=1

‖xi‖p)1/p.

∑i=1

‖T (xi )‖p)1/p ≤ K‖(n

∑i=1

|xi |p)1/p‖.

X (µ)[p] := |f |p : f ∈ X (µ).

‖g‖X (µ)[p]:= ‖|g|1/p‖p

X (µ), g ∈ X (µ)[p],

∑i=1

|xi |p)1/p‖ ≤ K (n

∑i=1

‖xi‖p)1/p.

∑i=1

‖T (xi )‖p)1/p ≤ K‖(n

∑i=1

|xi |p)1/p‖.

X (µ)[p] := |f |p : f ∈ X (µ).

‖g‖X (µ)[p]:= ‖|g|1/p‖p

X (µ), g ∈ X (µ)[p],

∑i=1

|xi |p)1/p‖ ≤ K (n

∑i=1

‖xi‖p)1/p.

∑i=1

‖T (xi )‖p)1/p ≤ K‖(n

∑i=1

|xi |p)1/p‖.

X (µ)[p] := |f |p : f ∈ X (µ).

‖g‖X (µ)[p]:= ‖|g|1/p‖p

X (µ), g ∈ X (µ)[p],

Let X (µ) be an order continuous p-convex Banach function space and E ap-concave Banach lattice. Consider a positive operator T : X (µ) → E . Then it isknown that there is a function h belonging to the dual of X (µ)[p] such that

‖T (f )‖ ≤ K(∫

|f |phdµ)1/p

, f ∈ X (µ).

We can assume also that h > 0 µ-a.e.

We can use this inequality to generate a norm function by

q∗(ϕ) := sup‖T (f ϕ)‖ :∫|f |phdµ ≤ 1, ϕ simple.

By using a density argument we can generate a function space G(µ) satisfyingthat

‖∫

f ϕ dmT ‖ ≤ Kq∗(ϕ)(∫

|f |phdµ)1/p

Now we can apply the factorization theorem to this inequality. We obtain in thisway an extension of T to the subspace L1

G(µ)(mT ) that is optimal in the sense that

L1G(µ)(mT ) is the biggest space to which T can be extended still satisfying the

inequality.

The subspace L1G(µ)(mT ) is p-convex.

Some references

Calabuig J.M., Galaz F., Jimenez Fernandez E. and Sanchez Perez E.A., Strong factorizationof operators on spaces of vector measure integrable functions and unconditional convergenceof series. To appear in Mathematische Zeitschrift.

Curbera, G.P. and Ricker, W.J.: Optimal domains for the kernel operator associated withSobolev’s inequality , Studia Math., 158(2), (2003), 131-152 (see also Corrigenda in the samejournal , 170, (2005), 217-218).

Delgado, O.: Optimal domains for kernel operators on [0,∞)× [0,∞), Studia Math. 174 (2006),131-145.

Delgado, O. and Soria, J.: Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007),119-133.

Diestel J. and Uhl J.J., Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc.,Providence, RI, 1977.

Fernandez, A., Mayoral, F., Naranjo, F., Saez, C. and Sanchez-Perez, E.A.: Spaces ofp-integrable functions with respect to a vector measure., Positivity, 10(2006), 1-16.

Sanchez Perez E.A., Compactness arguments for spaces of p-integrable functions withrespect to a vector measure and factorization of operators through Lebesgue-Bochnerspaces. Illinois J. Math. 45,3(2001), 907-923.

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￼￼￼￼￼￼￼subespacios de funciones integrables respecto de una medidavectorialyextensio...

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