geometric properties of banach function spaces of vector measure p-integrable...

Lp (m)-spaces and factorizationsVolterra operators

Applications: Strongly p-th power factorable operators

Geometric properties of Banach function spaces ofvector measure p-integrable functions

Enrique A. Sanchez Perez

Instituto Universitario de Matematica Pura y Aplicada(I.U.M.P.A.),

Universidad Politecnica de Valencia.

Bangalore 2010

Enrique A. Sanchez Perez Geometric properties of Banach function spaces of vector measure p-integrable functions

Let (Ω,Σ) be a measurable space, X a Banach space and m : Σ→ X a vectormeasure. Let 1≤ p < ∞, and consider the space Lp(m) of p-integrable functions withrespect to m. It is well known that these spaces are order continuous p-convex Banachfunction spaces with respect to µ, where µ is a Rybakov measure for m. In fact, eachBanach lattice having these properties can be written (order isomorphically) as anLp(m) space for a positive vector measure m. In this talk we explain further geometricproperties of these spaces, their subspaces and the seminorms that define the weaktopology on Lp(m). The main applications that are shown are related to thefactorization and extension theory of operators in these spaces.

III Each p-convex Banach function space can be represented as a Lp(m), a space ofp-integrable functions with respect to a vector measure m. This is the starting pointof a representation theory for operators acting on spaces of integrable functions thatallows the extension on the results that are known on factorization of operatorsdefined between these spaces. Moreover, it provides a framework in which newproblems on the structure of these function spaces appear.

III Consider a finite measure space (Ω,Σ,µ) and an order continuous Banach functionspace with a weak unit X(µ) over it. If T : X(µ)→ E is an operator with values on aBanach space E , it is always possible to associate to T a vector measure mT ; ifmoreover the operator has the right properties, it can be factorized through Lp(mT ).

III Each p-convex Banach function space can be represented as a Lp(m), a space ofp-integrable functions with respect to a vector measure m. This is the starting pointof a representation theory for operators acting on spaces of integrable functions thatallows the extension on the results that are known on factorization of operatorsdefined between these spaces. Moreover, it provides a framework in which newproblems on the structure of these function spaces appear.

III Consider a finite measure space (Ω,Σ,µ) and an order continuous Banach functionspace with a weak unit X(µ) over it. If T : X(µ)→ E is an operator with values on aBanach space E , it is always possible to associate to T a vector measure mT ; ifmoreover the operator has the right properties, it can be factorized through Lp(mT ).

1 If the operator is p-concave and X(µ) is a p-convex Banach function space, it ispossible to obtain a strong factorization (defined by a multiplication operator)through the space Lp(µ).(Maurey-Rosenthal Theorems). A. Defant.

2 General framework: inequalities for operators and geometry.

3 Rosenthal, Krivine and Maurey (1970’s), Banach lattice properties, p-convexity,p-concavity.

4 Applications: almost everywhere convergence of sequences, unconditionalconvergence of sequences; characterizations of operators belonging to particularoperator ideals...

1 Lp(m)-spaces and factorizations

2 Volterra operators

3 Applications: Strongly p-th power factorable operators

Let (Ω,Σ,µ) be a measure space. X(µ) is a Banach function space over µ if:1) If |f (ω)| ≤ |g(ω)| µ-a.e., where f ∈ L0(µ) and g ∈ X(µ), then f ∈ X(µ) and‖f‖ ≤ ‖g‖.2) For each A ∈Σ of finite measure, χA ∈ X(µ).

If X(µ) is σ -order continuous (for each fn ↓ 0, lımn ‖fn‖= 0), the dual of X(µ)equals its Kothe dual; each continuous functional ϕ ∈ X(µ)∗ an be written as anintegral ϕ(f ) =

∫Ω fg dµ.

Let (Ω,Σ,µ) be a measure space. X(µ) is a Banach function space over µ if:1) If |f (ω)| ≤ |g(ω)| µ-a.e., where f ∈ L0(µ) and g ∈ X(µ), then f ∈ X(µ) and‖f‖ ≤ ‖g‖.2) For each A ∈Σ of finite measure, χA ∈ X(µ).

If X(µ) is σ -order continuous (for each fn ↓ 0, lımn ‖fn‖= 0), the dual of X(µ)equals its Kothe dual; each continuous functional ϕ ∈ X(µ)∗ an be written as anintegral ϕ(f ) =

∫Ω fg dµ.

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 (

∑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‖(

∑i=1|xi |p)1/p‖.

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 (

∑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‖(

∑i=1|xi |p)1/p‖.

Let X be a Banach space. We consider a countably additive vector measurem : Σ→ X ; i.e. if Ai∞

i=1 is a family of disjoint sets Σ, then

m(∪∞

i=1Ai ) =∞

∑i=1

m(Ai ).

If m is a vector measure, we denote by |m| its variation and by ‖m‖ itssemivariation.

If x ′ ∈ X ∗, then 〈m,x ′〉(A) := 〈m(A),x ′〉, A ∈Σ, defines a measure.

A Rybakov measure for m is a measure as |〈m,x ′〉| that controls m.

If X(µ) is an order continuous Banach function space and T : X(µ)→ E is anoperator, the expression

mT (A) := T (χA), A ∈Σ,

defines a countably additive vector measure.

m(∪∞

i=1Ai ) =∞

∑i=1

m(Ai ).

mT (A) := T (χA), A ∈Σ,

m(∪∞

i=1Ai ) =∞

∑i=1

m(Ai ).

mT (A) := T (χA), A ∈Σ,

A Σ-measurable function f is integrable with respect to a vector measure m : Σ→ E if

1) f is 〈m,x ′〉-integrable for each x ′ ∈ X ∗, and

2) for each A ∈Σ there exists a unique element mf (A) ∈ X such that⟨mf (A),x ′

⟩=∫

Af d〈m,x ′〉, x ′ ∈ X ∗.

This element is usually denoted by∫

A f dm.

⟩=∫

Af d〈m,x ′〉, x ′ ∈ X ∗.

A f dm.

⟩=∫

Af d〈m,x ′〉, x ′ ∈ X ∗.

A f dm.

The space L1(m) of all (classes of a.e. equal) of integrable functions with respectto a vector measure with the norm

‖f‖L1(m) := supx ′∈BX∗

∫|f |d |〈m,x ′〉|, f ∈ L1(m),

is an o.c. Banach function space with weak unit over any Rybakov measure|〈m,x ′〉| de m.The space L1

w (m) is defined as the space of all (classes of) functions satisfying 1),with the same norm.

If 1 < p < ∞, the space Lp(m) is defined in the same way; in this case, the norm isgiven by the expression

‖f‖Lp(m) := supx ′∈BX ′

Ω|f |pd |〈m,x ′〉|)1/p , f ∈ Lp(m). (1)

Lp(m) is a p-convex o.c. Banach function space with weak unit over each Rybakovmeasure for m.

Lpw (m) is defined as the set of (classes of) measurable functions f such that |f |p is

scalarly integrable; the norm is also given by (1).

∫|f |d |〈m,x ′〉|, f ∈ L1(m),

Ω|f |pd |〈m,x ′〉|)1/p , f ∈ Lp(m). (1)

∫|f |d |〈m,x ′〉|, f ∈ L1(m),

Ω|f |pd |〈m,x ′〉|)1/p , f ∈ Lp(m). (1)

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‖pX(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.

X(µ)[p] := |f |p : f ∈ X(µ).

‖g‖X(µ)[p]:= ‖|g|1/p‖pX(µ), g ∈ X(µ)[p],

X(µ)[p] := |f |p : f ∈ X(µ).

‖g‖X(µ)[p]:= ‖|g|1/p‖pX(µ), g ∈ X(µ)[p],

Theorem

Let 1 < p < ∞. If E is a p-convex o. c. Banach lattice with weak unit, then there exists apositive vector measure m with values in E such that Lp(m) and E are latticeisomorphic.

Demostracion.

If E is p-convex, it is always possible to find an equivalent lattice norm of Esatisfying that its p-convexity constant is 1.

There is a probability space (Ω,Σ,µ) such that E is lattice isomorphic andisometric to a Banach function space X(µ) (in particular, X(µ) is p-convex).

The p-power X(µ)[p] of X(µ) is an order continuous Banach function space.

The set function ν : Σ→ X(µ)[p] is a vector measure such that L1(ν) = X(µ)[p].

So, X(µ) =(X(µ)[p]

)[1/p]

= L1(ν)[1/p] = Lp(ν).

Theorem

Demostracion.

So, X(µ) =(X(µ)[p]

)[1/p]

= L1(ν)[1/p] = Lp(ν).

Theorem

Demostracion.

So, X(µ) =(X(µ)[p]

)[1/p]

= L1(ν)[1/p] = Lp(ν).

Theorem

Demostracion.

So, X(µ) =(X(µ)[p]

)[1/p]

= L1(ν)[1/p] = Lp(ν).

Operators factorizing through the p-th power

Let µ be a finite measure. Let X(µ) be an o.c. Banach function space and E a Banachspace. If 1≤ p < ∞, an operator T : X(µ)→ E is p-th power factorable ifT[p] : X(µ)[p]→ E satisfies

T = T[p] i[p], (2)

where i[p] : X(µ)→ X(µ)[p] denotes the inclusion map.

Operators factorizing through the p-th power

Let µ be a finite measure. Let X(µ) be an o.c. Banach function space and E a Banachspace. If 1≤ p < ∞, an operator T : X(µ)→ E is p-th power factorable ifT[p] : X(µ)[p]→ E satisfies

T = T[p] i[p], (2)

where i[p] : X(µ)→ X(µ)[p] denotes the inclusion map.

i.e., the following diagram commutes

X(µ)T - E

HHjX(µ)[p]

We denote by F[p](X(µ),E) the class of all p-th power factorable operators.

We consider µ-determined operators: If ‖mT ‖(A) = 0 then µ(A) = 0 for eachA ∈Σ.

Example: if µ is a finite measure and 1 < p < ∞, then the inclusion mapi : Lp(µ)→ L1(µ) is p-th power factorable.

i.e., the following diagram commutes

X(µ)T - E

HHjX(µ)[p]

We denote by F[p](X(µ),E) the class of all p-th power factorable operators.

We consider µ-determined operators: If ‖mT ‖(A) = 0 then µ(A) = 0 for eachA ∈Σ.

Example: if µ is a finite measure and 1 < p < ∞, then the inclusion mapi : Lp(µ)→ L1(µ) is p-th power factorable.

Theorem

Let 1≤ p < ∞, and let X(µ) be an o.c. Banach function space over a finite measure µ

and E a Banach space. Let T : X(µ)→ E be µ-determined. Then the following areequivalent.

(i) T factorizes through the p-th power.

(ii) There is a constant C > 0 such that

‖T (f )‖E ≤ C‖f‖X(µ)[p]= C

∥∥ |f |1/p∥∥pX(µ)

, f ∈ X(µ). (3)

(iii) X(µ)⊆ Lp(mT ), and the inclusion map is continuous.

(iv) T factorizes as

X(µ)T - E

HHHHHHj

Lp(mT )

I(p)mT

Theorem

‖T (f )‖E ≤ C‖f‖X(µ)[p]= C

∥∥ |f |1/p∥∥pX(µ)

, f ∈ X(µ). (3)

X(µ)T - E

HHHHHHj

Lp(mT )

I(p)mT

Theorem

‖T (f )‖E ≤ C‖f‖X(µ)[p]= C

∥∥ |f |1/p∥∥pX(µ)

, f ∈ X(µ). (3)

X(µ)T - E

HHHHHHj

Lp(mT )

I(p)mT

Theorem

‖T (f )‖E ≤ C‖f‖X(µ)[p]= C

∥∥ |f |1/p∥∥pX(µ)

, f ∈ X(µ). (3)

X(µ)T - E

HHHHHHj

Lp(mT )

I(p)mT

Theorem

‖T (f )‖E ≤ C‖f‖X(µ)[p]= C

∥∥ |f |1/p∥∥pX(µ)

, f ∈ X(µ). (3)

X(µ)T - E

HHHHHHj

Lp(mT )

I(p)mT

(v) X(µ)[p] ⊆ L1(mT ), and the inclusion is continuous.

(vi) X(µ)⊆ Lpw (mT ) and the inclusion is continuous.

(vii) X(µ)[p] ⊆ L1w (mT ) and the inclusion is continuous.

(viii) For each x ′ ∈ E∗, the Radon-Nikodym derivatived〈mT ,x ′〉

dµbelongs to the Kothe

dual(X(µ)[p]

)′ of X(µ)[p].

Remark: the p-th power is not necessarily a Banach function space; in general it is onlya quasi-normed lattice. For instance L1(µ)[p] = L1/p(µ). Duality?

(v) X(µ)[p] ⊆ L1(mT ), and the inclusion is continuous.

(vi) X(µ)⊆ Lpw (mT ) and the inclusion is continuous.

(vii) X(µ)[p] ⊆ L1w (mT ) and the inclusion is continuous.

(viii) For each x ′ ∈ E∗, the Radon-Nikodym derivatived〈mT ,x ′〉

dµbelongs to the Kothe

dual(X(µ)[p]

)′ of X(µ)[p].

Remark: the p-th power is not necessarily a Banach function space; in general it is onlya quasi-normed lattice. For instance L1(µ)[p] = L1/p(µ). Duality?

Optimal Domain Theorem.

Theorem

Let 1≤ p < ∞ and T ∈ L(X(µ),E

)a µ-determined p-th power factorable operator. Then

Lp(mT ) is the biggest o.c. Banach function space with weak unit over µ to which one Tcan be extended, with the extension still satisfying that is p-th power factorable.

Demostracion.

Suppose that Y (µ) is a Banach function space such that X(µ)⊆ Y (µ), T can beextended to it and the extension is still p-th power factorable.Let T ∈ L

(Y (µ),E

)be such an extension. Then mT = mT . In particular, T is also

µ-determined.Thus, by the characterization theorem for operators factorizing through the p-th power,Y (µ)⊆ Lp(mT ) = Lp(mT ). This proves the theorem

Theorem

Demostracion.

(Y (µ),E

Theorem

Demostracion.

(Y (µ),E

Theorem

Demostracion.

(Y (µ),E

Theorem

Demostracion.

(Y (µ),E

Example: Volterra operators.

Are Volterra operators p-th power factorable?

Let µ be Lebesgue measure on Ω := [0,1] Σ := B([0,1]) be the Borel σ -algebra.For 1≤ r < ∞, consider the Volterra operator Vr : Lr ([0,1])→ Lr ([0,1]) given by

Vr (f )(t) :=∫ t

0f (u)du, t ∈ [0,1], f ∈ Lr ([0,1]).

Vr is µ-determined.

The associated vector measure mVr : A 7→ Vr (χA) in Σ coincides with the Volterrameasure νr of order r , i.e.

mVr (A) = Vr (χA) = νr (A), A ∈Σ; (4)

The variation |νr | of νr is finite and is given by d |νr |(t) = (1− t)1/r dt . So,

Lr ([0,1])⊆ L1([0,1])⊆ L1((1− t)1/r dt)⊆ L1(νr ). (5)

Vr (f )(t) :=∫ t

0f (u)du, t ∈ [0,1], f ∈ Lr ([0,1]).

mVr (A) = Vr (χA) = νr (A), A ∈Σ; (4)

Lr ([0,1])⊆ L1([0,1])⊆ L1((1− t)1/r dt)⊆ L1(νr ). (5)

Vr (f )(t) :=∫ t

0f (u)du, t ∈ [0,1], f ∈ Lr ([0,1]).

mVr (A) = Vr (χA) = νr (A), A ∈Σ; (4)

Lr ([0,1])⊆ L1([0,1])⊆ L1((1− t)1/r dt)⊆ L1(νr ). (5)

Vr (f )(t) :=∫ t

0f (u)du, t ∈ [0,1], f ∈ Lr ([0,1]).

mVr (A) = Vr (χA) = νr (A), A ∈Σ; (4)

Lr ([0,1])⊆ L1([0,1])⊆ L1((1− t)1/r dt)⊆ L1(νr ). (5)

Vr (f )(t) :=∫ t

0f (u)du, t ∈ [0,1], f ∈ Lr ([0,1]).

mVr (A) = Vr (χA) = νr (A), A ∈Σ; (4)

Lr ([0,1])⊆ L1([0,1])⊆ L1((1− t)1/r dt)⊆ L1(νr ). (5)

It can be proved that Lq([0,1])⊆ L1([0,1])⊆ L1((1− t)dt) whenever 1≤ q ≤ ∞,and moreover

Lq([0,1]) * L1((1− t)dt) whenever 0 < q < 1, (6)

which can be deduced from the fact that t 7→ t−1 ·χ[0,1](t) belongs to Lq([0,1]) butnot to L1([0,1]).

(i) Let r := 1. If X(µ) := L1([0,1]) we have that L1(|ν1|) = L1(ν1) and so for each1 < p < ∞,

X(µ)[p] = L1/p([0,1]) * L1((1− t)dt)

= L1(ν1) = L1(mV1 )

Then the characterization theorem V1 is not p-th power factorable.

It can be proved that Lq([0,1])⊆ L1([0,1])⊆ L1((1− t)dt) whenever 1≤ q ≤ ∞,and moreover

Lq([0,1]) * L1((1− t)dt) whenever 0 < q < 1, (6)

which can be deduced from the fact that t 7→ t−1 ·χ[0,1](t) belongs to Lq([0,1]) butnot to L1([0,1]).

(i) Let r := 1. If X(µ) := L1([0,1]) we have that L1(|ν1|) = L1(ν1) and so for each1 < p < ∞,

X(µ)[p] = L1/p([0,1]) * L1((1− t)dt)

= L1(ν1) = L1(mV1 )

Then the characterization theorem V1 is not p-th power factorable.

(ii) Let 1 < r < ∞. For 1 < p < ∞, it can be proved that Vr is p-th power factorable ifand only if p ≤ r .

(a) Suposse that Vr is p-th power factorable. Let S : Lr ([0,1])→ L1([0,1]) be theinclusion map. Then S Vr : Lr ([0,1])→ L1([0,1]) is p-th power factorable Theassociated vector measure mSVr de S Vr equals ν1. Then, the characterizationtheorem gives

Lr/p([0,1]) = X(µ)[p] ⊆ L1(mSVr ) = L1(ν1) = L1((1− t)dt).

but only in the case that r/p ≥ 1, i.e. p ≤ r .(b) Suppose now that p ≤ r . Then for X(µ) := Lr ([0,1]) we have

X(µ)[p] = Lr/p([0,1])⊆ L1([0,1])⊆ L1(νr ) = L1(mVr ).

So by the characterization theorem Vr is p-th power factorable.

Lp(mT ) is not the biggest p-convex o.c. B.f.s for extending T .

X(µ)[p] = Lr/p([0,1])⊆ L1([0,1])⊆ L1(νr ) = L1(mVr ).

A Maurey-Rosenthal type theorem

Definition

Let 0 < q < ∞ and 1≤ p < ∞. We say that a continuous linear operator T : X(µ)→ Efrom a q-B.f.s. X(µ) into a Banach space E is bidual (p,q)-power-concave if thereexists a constant C1 > 0 such that

∑j=1

∥∥T (fj )∥∥q/p

E ≤ C1

∥∥∥ n

∑j=1

∣∣fj ∣∣q/p∥∥∥

b,X(µ)[q]

, f1, ..., fn ∈ X(µ), n ∈ N, (7)

where ‖ ·∥∥∥

b,X(µ)[q]

denotes the norm in the bidual of X(µ)[q].

A Maurey-Rosenthal type theorem

Definition

Let 0 < q < ∞ and 1≤ p < ∞. We say that a continuous linear operator T : X(µ)→ Efrom a q-B.f.s. X(µ) into a Banach space E is bidual (p,q)-power-concave if thereexists a constant C1 > 0 such that

∑j=1

∥∥T (fj )∥∥q/p

E ≤ C1

∥∥∥ n

∑j=1

∣∣fj ∣∣q/p∥∥∥

b,X(µ)[q]

, f1, ..., fn ∈ X(µ), n ∈ N, (7)

where ‖ ·∥∥∥

b,X(µ)[q]

denotes the norm in the bidual of X(µ)[q].

Example of bidual (p,q)-power-concave operator. Let 1≤ p < q and consider twofinite measure spaces (Ω1,Σ1,µ1) and (Ω2,Σ2,µ2). Take a Bochner integrablefunction φ ∈ Lq/p(µ2,Lq′ (µ1)).

Define the operator uφ : Lpq(µ1)→ Lq/p(µ2) by

uφ (f )(w2) := 〈f ,φ(w2)〉=∫

f (w1)(φ(w2)(w1))dµ1(w1) ∈ Lq/p(µ2),

f ∈ Lpq(µ1).

The operator uφ is well-defined and continuous since Lpq(µ1)⊆ Lq(µ1).

Consider a finite set of functions f1, ..., fn ∈ Lpq(µ1). Then( n

∑i=1‖uφ (fi )‖

q/pLq/p(µ2)

)p/q≤

∑i=1|〈fi ,

φ(w2)

‖φ(w2)‖〉|q/p‖φ(w2)‖q/pdµ2(w2)

≤ sup‖h‖

Lq′ (µ1)≤1

∑i=1|〈fi ,h〉|q/p

)p/q· ‖φ‖Lq/p(µ2 ,Lq′ (µ1))

≤∥∥∥( n

∑i=1

∣∣fi ∣∣q/p)p/q∥∥∥

Lq (µ1)· ‖φ‖

=∥∥∥ n

∑i=1

∣∣fi ∣∣q/p∥∥∥p/q

Lpq (µ1)[q]

· ‖φ‖Lq/p(µ2 ,Lq′ (µ1)),

Therefore, uφ is (p,q)-power concave, and since Lpq(µ1) is q-convex, it is also bidual(p,q)-power concave.

In the case the Bochner integrable function is given by a kernel, i.e.φ(w2)(w1) := K (w1,w2), K being a µ1×µ2 measurable function satisfying((

|K (w1,w2)|q′dµ1(w1)

)q/(pq′)dµ2(w2)

)p/q< ∞,

then uφ is a so called Hille-Tamarkin operator. For the case p = 1 and µ1 = µ2, theclass of Hille-Tamarkin operators is a well-known class of classical kernel operatorsthat has been largely studied.

Volterra operators are particular cases of this class.

In the case the Bochner integrable function is given by a kernel, i.e.φ(w2)(w1) := K (w1,w2), K being a µ1×µ2 measurable function satisfying((

|K (w1,w2)|q′dµ1(w1)

)q/(pq′)dµ2(w2)

)p/q< ∞,

then uφ is a so called Hille-Tamarkin operator. For the case p = 1 and µ1 = µ2, theclass of Hille-Tamarkin operators is a well-known class of classical kernel operatorsthat has been largely studied.

Volterra operators are particular cases of this class.

Theorem

Let X(µ) be a σ -order continuous q-B.f.s. over a finite measure space (Ω,Σ,µ). Let Ebe a Banach space and an operator T ∈L (X(µ),E) be µ-determined. For any1≤ p < ∞ and 0 < q < ∞, the following assertions are equivalent.

(i) There exists C > 0 such that( n

∑j=1

∥∥T (fj )∥∥q/p

)1/q≤ C

∥∥∥ n

∑j=1

∣∣fj ∣∣q/p∥∥∥1/q

b,X(µ)[q]

for all n ∈ N and f1, ..., fn ∈ X(µ); namely, T is bidual (p,q)-power-concave.

(ii) There exists a function g ∈(X(µ)[q]

)′ with g > 0 (µ-a.e.) such that

∥∥T (f )∥∥

E ≤(∫

Ω|f |q/p g dµ

< ∞, f ∈ X(µ). (8)

(iii) The inclusions X(µ)⊆ Lq(g dµ)⊆ Lp(mT ) hold and are continuous for someg ∈ L0(µ) with g > 0 (µ-a.e.).

(iv) The inclusions X(µ)[p] ⊆ Lq/p(gdµ)⊆ L1(mT ) hold and are continuous for someg ∈ L0(µ) with g > 0 (µ-a.e.).

(v) T ∈F[p](X(µ),E) and there exist an operator S ∈L (Lq/p(µ),E) and a functiong > 0 (µ-a.e.) satisfying gp/q ∈M

(X(µ)[p], Lq/p(µ)

)such that T[p] = S Mgp/q .

That is, the following diagram commutes:

X(µ)[p]

-Mgp/q

Lq/p(µ),

(vi) T ∈F[p]

(X(µ),E

)and the inclusion map J(p)

T : X(µ)→ Lp(mT ) is bidualq-concave.

(vii) T ∈F[p]

(X(µ),E

)and the inclusion map β[p] : X(µ)[p]→ L1(mT ) is bidual

(q/p)-concave.

(viii) T ∈F[p]

(X(µ),E

)and T[p] : X(µ)[p]→ E is bidual (q/p)-concave.

If T ∈L (X(µ),E) satisfies any one of (i)–(viii), then the following diagramcommutes:

X(µ)[p]

?- -Lq/p(gdµ)

- Lq(gdµ) -

L1(mT ) - E

Lp(mT )

with each arrow indicating the respective inclusion map.

Definition

A family of R-valued functions Ψ is called concave if, for every finite set of functionsψ1, ...,ψn ⊆Ψ with n ∈ N and non-negative scalars c1, ...,cn satisfying ∑

nj=1 cj = 1,

there exists ψ ∈Ψ such thatn

∑j=1

cj ψj ≤ ψ.

The following result is known as Ky Fan’s Lemma.

Let W be a compact convex subset of a Hausdorff topological vector space and let Ψbe a concave collection of lower semi-continuous, convex, real functions on W. Letc ∈ R. Suppose, for every ψ ∈Ψ, that there exists xψ ∈W with ψ(xψ )≤ c. Then thereexists x ∈W such that ψ(x)≤ c for all ψ ∈Ψ.

Some references

A. Defant and E.A. Sanchez Perez, Domination of operators on Banach functionspaces. Math. Proc. Cambridge P. Soc., 146, 57-66(2009).

S. Okada, W. J. Ricker and E. A. Sanchez-Perez, Optimal domain and integralextension of operators —Acting in function spaces—, Operator Theory: Advancesand Applications, vol. 180, Birkhauser Verlag, Basel, 2008.

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