Subespacios de funciones integrables respecto de una medidavectorialyextensio ́ndeoperadores

1. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Subespacios de funciones integrables respecto de una medida vectorial y extensión de operadores Enrique A. Sánchez Pérez J. M. Calabuig O. Delgado Instituto de Matemática Pura y Aplicada (I.M.P.A.). Departamento de Matemática Aplicada. Universidad Politécnica de Valencia. Camino de Vera S/N, 46022, Valencia. X EARCO Mallorca, Mayo 2007 Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

2. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. En esta charla mostramos que la extensión de un operador definido en un espacio de funciones de Banach es equivalente, bajo ciertos requisitos m´ınimos, a su factorización a través de un subespacio de funciones integrables respecto de la medida vectorial mT asociada al operador T. Además, este subespacio particular determina aquella propiedad del operador que se pretende preservar, en el sentido de que constituye el dominio óptimo -el mayor subespacio al que se puede extender-, de manera que la extensión conserva esta propiedad. Esto permite un planteamiento abstracto de ciertos aspectos de la teor´ıa de dominio óptimo, que se ha desarrollado de manera importante en los últimos tiempos, principalmente en la dirección de caracterizar cuál es el mayor espacio de funciones de Banach al que se pueden extender algunos operadores de importancia en análisis. Además, permite clasificar las propiedades de los operadores que se conservan en la extensión en función del subespacio de L1(mT ) que constituye su dominio óptimo. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

3. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. 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. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

7. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. FIRST PART Notation and preliminaries Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

8. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. General notation (Ω,Σ,µ) will be a finite measure space, E be a Banach space and E its topological dual, BE the unit ball of E, P(A) will represent the set of partitions π of A into finite number of disjoint measurable sets, m : Σ → E will be a (countably additive) vector measure, The semivariation of m over A ∈ Σ is defined by m (A) = sup x ∈BE | m,x |(A) = sup x ∈BE sup π∈P(A) ∑ B∈π | m,x (B)|, with the usual notation m,x (B) = m(B),x for each B ∈ Σ, A ∈ Σ is called m-null if m (A) = 0. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

9. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. General notation (Ω,Σ,µ) will be a finite measure space, E be a Banach space and E its topological dual, BE the unit ball of E, P(A) will represent the set of partitions π of A into finite number of disjoint measurable sets, m : Σ → E will be a (countably additive) vector measure, The semivariation of m over A ∈ Σ is defined by m (A) = sup x ∈BE | m,x |(A) = sup x ∈BE sup π∈P(A) ∑ B∈π | m,x (B)|, with the usual notation m,x (B) = m(B),x for each B ∈ Σ, A ∈ Σ is called m-null if m (A) = 0. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

10. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. 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 vector measure m is a scalar measure ν defined as the variation of a measure m,x , where x ∈ 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 that 1) 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 space continuously contained in X(µ), with the same order structure (allowing different norms). 3 A Banach function space is order continuous if order bounded increasing sequences are convergent in norm. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

11. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. 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 vector measure m is a scalar measure ν defined as the variation of a measure m,x , where x ∈ 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 that 1) 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 space continuously contained in X(µ), with the same order structure (allowing different norms). 3 A Banach function space is order continuous if order bounded increasing sequences are convergent in norm. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

12. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. L1(m)L1(m)L1(m) spaces. A function f : Ω → R is said to be integrable with respect to the measure m if 1 for each x ∈ E we have that f ∈ L1( m,x ), 2 for each A ∈ Σ there exists xA ∈ E such that xA,x = A fd m,x for every x ∈ E . The element xA is usually denoted by A f dm. The space of the classes (equality m-almost everywhere) of these functions is denoted by L1(m). The expression f m = sup x ∈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 Banach function space. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

13. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. The indefinite integral mf : Σ → E of a function f ∈ L1(m) is defined by mf (A) = A fdm, A ∈ Σ. The Orlicz-Pettis Theorem ensures that mf is again a countably additive vector measure. An equivalent norm for L1(m) is given by |||f|||m = sup A∈Σ A fdm 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 is integrable with respect to m. The expression. f Lp(m) = sup x ∈BE |f|p d| m,x | 1 p for each f ∈ Lp (m), defines a lattice norm on Lp(m) ([6]). The space Lp(m) is an order continuous Banach function subspace of L1(m). Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

16. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. SECOND PART Vector measures and spaces of integrable functions. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

17. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Y(µ)Y(µ)Y(µ) semivariation. Let m : Σ → E be a (countably additive) vector measure and Y(µ) a Banach function space. The Y(µ)-semivariation m Y(µ) of m is defined by m Y(µ) := sup n ∑ i=1 αAi m(Ai ) E : n ∑ i=1 αAi χAi ∈ BY(µ) . L1 Y(µ)(m)L1 Y(µ)(m)L1 Y(µ)(m) spaces. We define L1 Y(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)-semivariation endowed with the norm f L1 Y(µ) (m) := mf Y(µ) = sup ϕ simple ϕ∈BY(µ) fϕdm E . L1 Y(µ)(m) is always a Banach function subspace of L1(m). Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

18. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Y(µ)Y(µ)Y(µ) semivariation. Let m : Σ → E be a (countably additive) vector measure and Y(µ) a Banach function space. The Y(µ)-semivariation m Y(µ) of m is defined by m Y(µ) := sup n ∑ i=1 αAi m(Ai ) E : n ∑ i=1 αAi χAi ∈ BY(µ) . L1 Y(µ)(m)L1 Y(µ)(m)L1 Y(µ)(m) spaces. We define L1 Y(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)-semivariation endowed with the norm f L1 Y(µ) (m) := mf Y(µ) = sup ϕ simple ϕ∈BY(µ) fϕdm E . L1 Y(µ)(m) is always a Banach function subspace of L1(m). Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

19. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Example Some particular subspaces of L1(m) that are nowadays well known can be represented 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 αAi m(Ai ) E : n ∑ i=1 αAi χAi ∈ BL∞(µ) = m L∞(µ), where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for any function f ∈ L1(m), mf = mf L∞(µ) = f L1 L∞(µ) (m). Hence, L1(m) = L1 L∞(µ)(m). 2 Let 1 < p < ∞ and 1/p +1/p = 1. It is known that for any function f ∈ Lp(m), the expression mf Lp (m) = sup ϕ simple ϕ∈B Lp (m) fϕdm E equals the norm f L p w (m) of a function f belonging to L p w (m) ( [7] ). In fact, it can be proved that L p w (m) = L1 Lp (m) (m). Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

23. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. THIRD PART A factorization theorem. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

24. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Let X(µ) be an order continuous Banach function space and let T : X(µ) → E be an operator. 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 a constant K > 0 such that T(fϕ) ≤ K f X(µ) ϕ Y(µ) for every function f ∈ X(µ) and every simple function ϕ. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

27. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Theorem (First part) Let X(µ) be an order continuous Banach function space with and let Y(µ) be a Banach function space verifying that the simple functions are dense. Given a µ-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 // i $$ E L1 Y(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 ). Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

28. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Theorem (First part) Let X(µ) be an order continuous Banach function space with and let Y(µ) be a Banach function space verifying that the simple functions are dense. Given a µ-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 // i $$ E L1 Y(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 ). Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

29. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. 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) ⊆ L1 Y(µ)(mT ) with continuous inclusion. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

30. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Example Assume that T is continuous. In this case taking into account that for each f ∈ X(µ) and each ϕ simple function T(fϕ) E ≤ K fϕ 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 taking ϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimal domain 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 the spaces L1 p,µ (m). In this article they prove that the bigger function space (i.e. the optimal domain) to which an Lp-product extensible operator T can be extended preserving this property is exactly the space L1 p,µ (mT ). Actually the notion of Lp-product extensible operator coincides with our definition of Y(µ)-extensible operator with Y(µ) = Lp(µ) for 1 < p < ∞. In this case the space L1 p,µ (mT ) is now L1 Lp(µ)(mT ). Hence Theorem 6 in [1] is a particular case of Theorem 3. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

31. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Example Assume that T is continuous. In this case taking into account that for each f ∈ X(µ) and each ϕ simple function T(fϕ) E ≤ K fϕ 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 taking ϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimal domain 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 the spaces L1 p,µ (m). In this article they prove that the bigger function space (i.e. the optimal domain) to which an Lp-product extensible operator T can be extended preserving this property is exactly the space L1 p,µ (mT ). Actually the notion of Lp-product extensible operator coincides with our definition of Y(µ)-extensible operator with Y(µ) = Lp(µ) for 1 < p < ∞. In this case the space L1 p,µ (mT ) is now L1 Lp(µ)(mT ). Hence Theorem 6 in [1] is a particular case of Theorem 3. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

32. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. FOURTH PART An application: extension of operators on p-convex B. f. s. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

33. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions 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 set x1,...,xn ∈ E, ( n ∑ 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 a constant K such that for each finite family x1,...,xn ∈ E, ( n ∑ 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 and continuous. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

37. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Let X(µ) be an order continuous p-convex Banach function space and E a p-concave Banach lattice. Consider a positive operator T : X(µ) → E. Then it is known that there is a function h belonging to the dual of X(µ)[p] such that T(f) ≤ K |f|p hdµ 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(fϕ) : |f|p hdµ ≤ 1}, ϕ simple. By using a density argument we can generate a function space G(µ) satisfying that fϕ dmT ≤ Kq∗ (ϕ) |f|p hdµ 1/p . Now we can apply the factorization theorem to this inequality. We obtain in this way an extension of T to the subspace L1 G(µ)(mT ) that is optimal in the sense that L1 G(µ)(mT ) is the biggest space to which T can be extended still satisfying the inequality. The subspace L1 G(µ)(mT ) is p-convex. Enrique A. S´anchez P´erez Subespacios y extensiones de operadores

38. Notation and preliminaries Vector measures and spaces of integrable functions A factorization theorem An application: extensions of operators on p-convex B. f. s. Some references Calabuig J.M., Galaz F., Jiménez Fernández E. and Sánchez Pérez E.A., Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series. To appear in Mathematische Zeitschrift. Curbera, G.P. and Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality , Studia Math., 158(2), (2003), 131-152 (see also Corrigenda in the same journal , 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. Fernández, A., Mayoral, F., Naranjo, F., Sáez, C. and Sánchez-Pérez, E.A.: Spaces of p-integrable functions with respect to a vector measure., Positivity, 10(2006), 1-16. Sánchez Pérez E.A., Compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Illinois J. Math. 45,3(2001), 907-923. Enrique A. Sánchez Pérez Subespacios y extensiones de operadores

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