On conditional expectations in L^p(mu;L^q(nu;X))
aa r X i v : . [ m a t h . F A ] M a y ON CONDITIONAL EXPECTATIONS IN L p ( µ ; L q ( ν ; X )) QI L ¨U AND JAN VAN NEERVEN
Abstract.
Let ( A, A , µ ) and ( B, B , ν ) be probability spaces, let F be a sub- σ -algebra of the product σ -algebra A × B , let X be a Banach space andlet 1 < p, q < ∞ . We obtain necessary and sufficient conditions in orderthat the conditional expectation with respect to F defines a bounded linearoperator from L p ( µ ; L q ( ν ; X )) onto L p F ( µ ; L q ( ν ; X )), the closed subspace in L p ( µ ; L q ( ν ; X )) of all functions having a strongly F -measurable representa-tive. Introduction
Let ( A, A , µ ) and ( B, B , ν ) be probability spaces, F a sub- σ -algebra of theproduct σ -algebra A × B in A × B , and X a Banach space. For 1 p, q ∞ wedefine L p F ( µ ; L q ( ν ; X )) to be the closed subspace in L p ( µ ; L q ( ν ; X )) consisting ofthose functions which have a strongly F -measurable representative. It is easy tosee (e.g., by using [6, Corollary 1.7]) that L p F ( µ ; L q ( ν ; X )) = L p ( µ ; L q ( ν ; X )) ∩ L F ( µ × ν ; X ) . Furthermore, L p F ( µ ; L q ( ν ; X )) is closed in L p ( µ ; L q ( ν ; X )). Indeed, if f n → f in L p ( µ ; L q ( ν ; X )) with each f n in L p F ( µ ; L q ( ν ; X )), then also f n → f in L ( µ × ν ; X ),and therefore f ∈ L F ( µ × ν ; X ). The reader is referred to [2, 6] for the basictheory of the Lebesgue-Bochner spaces and conditional expectations in these spaces.The same reference contains some standard results concerning the Radon-Nikod´ymproperty that will be needed later on.The aim of this paper is to provide a necessary and sufficient condition in or-der that conditional expectation E ( ·| F ) restrict to a bounded linear operator on L p ( µ ; L q ( ν ; X )) when 1 < p, q < ∞ . We also show that E ( ·| F ) need not to becontractive. An example is given which shows that this result does not extend tothe pair p = ∞ , q = 2.Characterisations of conditional expectation operators on general classes of Ba-nach function spaces E (and their vector-valued counterparts) have been given byvarious authors (see, e.g., [4] and the references therein), but these works usually assume that a bounded operator T : E → E is given and investigate under whatcircumstances it is a conditional expectation operator. We have not been able to Date : August 11, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Conditional expectations in L p ( µ ; L q ( ν ; X )), dual of L p F ( µ ; L q ( ν ; X )),Radon-Nikod´ym property.Qi L¨u is supported by the NSF of China under grants 11471231, the Fundamental ResearchFunds for the Central Universities in China under grant 2015SCU04A02, and Grant MTM2014-52347 of the MICINN, Spain. find any paper addressing the problem of establishing sufficient conditions for con-ditional expectation operators to act in concrete Banach function spaces such asthe mixed-norm L p ( L q )-spaces investigated here.2. Results
Throughout this section, ( A, A , µ ) and ( B, B , ν ) are probability spaces. If 1 p, q ∞ , their conjugates 1 p ′ , q ′ ∞ are defined by p + p ′ = 1 and q + q ′ = 1.It is clear that every f ∈ L p F ( µ ; L q ( ν )) induces a functional φ f ∈ ( L p ′ F ( µ ; L q ′ ( ν ))) ∗ in a canonical way, and the resulting mapping f φ f is contractive. The first mainresult of this note reads as follows. Theorem 2.1.
Let < p ∞ and < q ∞ . If f φ f establishes anisomorphism of Banach spaces L p F ( µ ; L q ( ν )) ≃ ( L p ′ F ( µ ; L q ′ ( ν ))) ∗ , then for any Banach space X the conditional expectation operator E ( ·| F ) on L ( µ × ν ; X ) restricts to a bounded projection on L p ( µ ; L q ( ν ; X )) .Proof. We will show that E ( f | F ) ∈ L p ( µ ; L q ( ν ; X )) for all f ∈ L p ( µ ; L q ( ν ; X )).A standard closed graph argument then gives the boundedness of E ( ·| F ) as anoperator in L p ( µ ; L q ( ν ; X )).Since k E ( f | F ) k X E ( k f k X | F ) µ × ν -almost everywhere, it suffices to provethat E ( g | F ) ∈ L p ( µ ; L q ( ν )) for all g ∈ L p ( µ ; L q ( ν )). To prove the latter, considerthe inclusion mapping I : L p ′ F ( µ ; L q ′ ( ν )) → L p ′ ( µ ; L q ′ ( ν )) . Every g ∈ L p ( µ ; L q ( ν )) defines an element of ( L p ′ ( µ ; L q ′ ( ν ))) ∗ in the natural wayand we have, for all F ∈ F , h F , I ∗ g i = h I F , g i = Z F g d µ × ν. The implicit use of Fubini’s theorem to rewrite the double integral over A and B asan integral over A × B in the second equality is justified by non-negativity, writing g = g + − g − and considering these functions separately. On the other hand, viewing g and F as elements of L ( µ × ν ) and L ∞ ( µ × ν ) respectively, we have Z F g d µ × ν = Z F E ( g | F ) d µ × ν = h F , E ( g | F ) i . We conclude that h F , I ∗ g i = h E ( g | F ) , F i , where on the left the duality is between L p ′ ( µ ; L q ′ ( ν )) and its dual, and on the right between L ( µ × ν ) and L ∞ ( µ × ν ).Passing to linear combinations of indicators, it follows thatsup φ |h φ, I ∗ g i| = sup φ |h E ( g | F ) , φ i| = k E ( g | F ) k < ∞ , where both suprema run over the simple functions φ in L ∞ F ( µ × ν ) of norm L ∞ , F ( µ × ν ), it follows that I ∗ g defines an element of( L ∞ , F ( µ × ν )) ∗ . This identification is one-to-one: for if h φ, I ∗ g i = 0 for all simple F -measurable functions φ , then h φ, I ∗ g i = 0 for all φ ∈ L p ′ F ( µ ; L q ′ ( ν )), noting thatthe simple F -measurable functions are dense in L p ′ F ( µ ; L q ′ ( ν )) (here we use that p ′ and q ′ are finite). N CONDITIONAL EXPECTATIONS IN L p ( µ ; L q ( ν ; X )) 3 As an element of ( L ∞ , F ( µ × ν )) ∗ , I ∗ g equals the function E ( g | F ), viewed as an ele-ment in the same space. Since the embedding of L F ( µ × ν ) into ( L ∞ , F ( µ × ν )) ∗ is iso-metric, it follows that I ∗ g = E ( g | F ) ∈ L F ( µ × ν ). Since I ∗ g ∈ ( L p ′ F ( µ ; L q ′ ( ν ))) ∗ , bythe assumption of the theorem we may identify I ∗ g with a function in L p ( µ ; L q ( ν )).We conclude that E ( g | F ) = I ∗ g ∈ L p F ( µ ; L q ( ν )). (cid:3) If we make a stronger assumption, more can be said:
Theorem 2.2.
Suppose that < p, q < ∞ and let X be a non-zero Banach space.Then the following assertions are equivalent: (1) the conditional expectation operator E ( ·| F ) restricts to a bounded projectionon the space L p ( µ ; L q ( ν ; X )) ; (2) the conditional expectation operator E ( ·| F ) restricts to a bounded projectionon the space L p ′ ( µ ; L q ′ ( ν ; X )) ; (3) f φ f induces an isomorphism of Banach spaces L p F ( µ ; L q ( ν )) ≃ ( L p ′ F ( µ ; L q ′ ( ν ))) ∗ . Remark . In [9] it is shown that condition (3) is satisfied if(2.1) I × E ν maps L F ( µ × ν ) into itself.Here E ν denotes the bounded operator on L ( ν ) defined by E ν f := ( E ν f ) , with E ν f = R f d ν .The proof of Theorem 2.2 is based on the following elementary lemma. Lemma 2.4.
Let P be a bounded projection on a Banach space X . Let X = R ( P ) , X = N ( P ) , Y = R ( P ∗ ) and Y = N ( P ∗ ) , so that we have direct sum decompo-sitions X = X ⊕ X and X ∗ = Y ⊕ Y . Then we have natural isomorphisms ofBanach spaces X ∗ = Y and X ∗ = Y .Proof of Theorem 2.2. We have already proved (3) ⇒ (1). For proving (1) ⇒ (2) ⇒ (3)there is no loss of generality in assuming that X is the scalar field, for instanceby observing that the proof of Theorem [6, Theorem 2.1.3] also works for mixed L p ( L q )-spaces.(1) ⇒ (2): The assumption (1) implies that L p F ( µ ; L q ( ν )) is the range of thebounded projection ( E ( ·| F )) in L p ( µ ; L q ( ν )). Moreover, h E ( f | F ) , g i = h f, E ( g | F ) i for all f ∈ L p ( µ ; L q ( ν )) and g ∈ L p ′ ( µ ; L q ′ ( ν )), since this is true for f and g inthe (dense) intersections of these spaces with L ( µ × ν ). It follows that the condi-tional expectation E ( ·| F ) is bounded on L p ′ ( µ ; L q ′ ( ν )) = ( L p ( µ ; L q ( ν ))) ∗ and equals( E ( ·| F )) ∗ . Clearly it is a projection and its range equals L p ′ F ( µ ; L q ′ ( ν )).(2) ⇒ (3): This implication follows Lemma 2.4. (cid:3) Inspection of the proof of Theorem 2.1 shows that if for all f ∈ L p F ( µ ; L q ( ν )) wehave k f k L p F ( µ ; L q ( ν )) = k f k ( L p ′ F ( µ ; L q ′ ( ν ))) ∗ , then E ( ·| F ) is contractive on L p ( µ ; L q ( ν )).The next example, due to Qiu [10], shows that the conditional expectation, whenit is bounded, may fail to be contractive. QI L¨U AND JAN VAN NEERVEN
Example . Let A = B = { , } with A = B = { ∅ , { } , { } , { , }} and µ = ν the measure on { , } that gives each point mass , and let F be the σ -algebragenerated by the three sets { (0 , } , { (1 , } , { (0 , , (1 , } . If we think of B asdescribing discrete ‘time’, then F is the progressive σ -algebra corresponding to thefiltration ( F t ) t ∈{ , } in A given by F = { ∅ , { , }} and F = { ∅ , { } , { } , { , }} .Let f : A × B → R be defined by f (0 ,
0) = 0 , f (1 ,
0) = 1 , f (0 ,
1) = 1 , f (1 ,
1) = 0 . Then E ( f | F )(0 ,
0) = 12 , E ( f | F )(1 ,
0) = 12 , E ( f | F )(0 ,
1) = 1 , E ( f | F )(1 ,
1) = 0 . Hence in this example we have k f k L p ( µ ; L ( ν )) = h(cid:16) (cid:17) p/ + (cid:16) (cid:17) p/ i /p , k E ( f | F ) k L p ( µ ; L ( ν )) = h(cid:16) (cid:17) p/ + (cid:16) (cid:17) p/ i /p . Consequently, for large enough p the conditional expectation fails to be contractivein L p ( µ ; L ( ν )).We continue with two examples showing that the condition expectation operatoron L ( µ × ν ) may fail to restrict to a bounded operator on L p ( µ ; L q ( ν )). The firstwas communicated to us by Gilles Pisier. Example . Let ( A, A , µ ) and ( B, B , ν ) be probability spaces and let ( C, C , P ) =( A, A , µ ) × ( B, B , ν ) be their product. Consider the infinite product ( C, C , P ) N =( C N , C N , P N ); with an obvious identification it may be identified with ( A N , A N , µ N ) × ( B N , B N , ν N ).Consider the sub- σ -algebra F N of A N × B N = C N , where F ⊆ A × B = C is a given sub- σ -algebra. Let T := E ( ·| F ) and T N := E ( ·| F N ) be the conditionalexpectation operators on L ( µ × ν ) and L ( µ N × ν N ), respectively. For a function f ∈ L ∞ ( µ N × ν N ) of the form f = f ⊗ · · · ⊗ f N ⊗ ⊗ ⊗ . . . with f n ∈ L ( µ × ν )for all n = 1 , . . . , N , we have T N f = T f ⊗ · · · ⊗ T f N ⊗ ⊗ ⊗ . . . By an elementary computation, k f k L p ( µ N ; L q ( ν N )) = N Y n =1 k f n k L p ( µ ; L q ( ν )) and k T N f k L p ( µ N ; L q ( ν N )) = N Y n =1 k T f n k L p ( µ ; L q ( ν )) . This being true for very N > T N is bounded if and only if T iscontractive. Example 2.5, however, shows that the latter need not always be thecase.The second example is due to Tuomas Hyt¨onen: N CONDITIONAL EXPECTATIONS IN L p ( µ ; L q ( ν ; X )) 5 Example . Let B the Borel σ -algebra of [0 , A ∈ B × B , let e A := { ( y, x ) : ( x, y ) ∈ A } and let F := { A ∈ B × B : e A = A } be the symmetric sub- σ -algebra of the product σ -algebra. Then E ( ·| F ) does notrestrict to a bounded operator on L p ( L q ) := L p (0 , L q (0 , p = q . To seethis let e f ( x, y ) := f ( y, x ). One checks that E ( f | F ) = 12 ( f + e f ) > e f if f >
0. In particular, E ( φ ⊗ ψ | F ) > ψ ⊗ φ if φ, ψ >
0. Let then φ ∈ L p (0 , ψ ∈ L q (0 ,
1) be positive functions such that only one of them is in L p ∨ q (0 , f = φ ⊗ ψ , then k f k L p ( L q ) = k φ k L p k ψ k L q < ∞ but k E ( f | F ) k L p ( L q ) > k ψ ⊗ φ k L p ( L q ) = 12 k ψ k L p k φ k L q . If p > q , then k ψ k L p = ∞ , and if p < q , then k φ k L q = ∞ , so that in either case k E ( f | F ) k L p ( L q ) = ∞ .Let us check that (2.1) fails in the above examples. As in Example 2.5 let A = B = { , } with A = B = { ∅ , { } , { } , { , }} , µ = ν the measure on { , } that gives each point mass , and F the σ -algebra generated by the three sets { (0 , } , { (1 , } , { (0 , , (1 , } . Let f : A × B → R be defined by f (0 ,
0) = 1 , f (1 ,
0) = 1 , f (0 ,
1) = 0 , f (1 ,
1) = 1 . This function is F -measurable, but ( I ⊗ E ν ) f is not:( I ⊗ E ν ) f (0 ,
0) = 12 , ( I ⊗ E ν ) f (1 ,
0) = 1 , ( I ⊗ E ν ) f (0 ,
1) = 12 , ( I ⊗ E ν ) f (1 ,
1) = 1 . Thus (2.1) fails in Example 2.5. It is clear that if we start from this example, (2.1)also fails in Example 2.6. In Example 2.7 (2.1) also fails, for obvious reasons.An interesting example where condition (2.1) is satisfied is the case when A =[0 ,
1] is the unit interval, B = Ω a probability space, and F = P the progressive σ -algebra in [0 , × Ω. From Theorem 2.1 we therefore obtain the following result:
Corollary 2.8.
For all < p, q < ∞ and all Banach spaces X , the conditionalexpectation with respect to the progressive σ -algebra on [0 , × Ω is bounded on L p (0 , L q (Ω; X )) . This quoted result of [9] plays an important role in the study of well-posednessand control problems for stochastic partial differential equations. For example, in[8], it is used to show the well-posedness of stochastic Schr¨odinger equations withnon-homogeneous boundary conditions in the sense of transposition solutions, in[7] it is applied to obtain a relationship between null controllability of stochasticheat equations, and in [9] it is used to establish a Pontryagin type maximum forcontrolled stochastic evolution equations with non-convex control domain.As a consequence of (a special case of) [3, Theorem A.3] we obtain that theassumptions of Theorem 2.1 are also satisfied for progressive σ -algebra F = P if we replace L p (0 , L q (Ω; X )) by L p (Ω; L q (0 , X )). The quoted theorem is QI L¨U AND JAN VAN NEERVEN stated in terms of the predictable σ -algebra G . However, since every progres-sively measurable set P ∈ P is of the form P = G ∆ N with G ∈ G and N anull set in the product σ -algebra F × B ([0 , L p G (Ω; L q (0 , X )) = L p P (Ω; L q (0 , X )). Therefore, [3, Theorem A.3] remains trueif we replace the predictable σ -algebra by the progressive σ -algebra and we obtainthe following result: Corollary 2.9.
For all < p, q < ∞ and all Banach spaces X , the conditionalexpectation with respect to the progressive σ -algebra on Ω × [0 , is bounded on L p (Ω; L q (0 , X )) .Proof. In the scalar-valued case we apply [3, Theorem A.3] (with J a singleton).The vector-valued case then follows from the observation, already made in the proofof Theorem 2.2, that Theorem [6, Theorem 2.1.3] also holds for mixed L p ( L q )-spaces. (cid:3) Our final example shows that condition (2) in Theorem 2.2 fails for the pair p = 1, q = 2 even when X is the scalar field. Example . Let { F t } t ∈ [0 , be the filtration generated by a one-dimensionalstandard Brownian motion { W ( t ) } t ∈ [0 , defined on a probability space (Ω , F , P ).Let P be the associated progressive σ -algebra on Ω × [0 , L ∞ P (Ω; L (0 , ( ( L P (Ω; L (0 , ∗ in the sense that the former is contained isometrically as a proper closed subspaceof the latter.For v ∈ L P (Ω; L (0 , x to the following problem:(2.2) ( d x ( t ) = v ( t ) d W ( t ) , t ∈ [0 , ,x (0) = 0 . By the classical well-posedness theory of SDEs (e.g. [11, Chapter V, Section 3]), x ∈ L P (Ω; C ([0 , k x k L P (Ω; C ([0 , C k v k L P (Ω; L (0 , for some constant C independent of v . Let ξ ∈ L ∞ F (Ω). Define a linear functional L on L P (Ω; L (0 , L ( v ) := E ( ξx (1)) . By (2.3), L is bounded. Suppose now, for a contradiction, that ( L P (Ω; L (0 , ∗ = L ∞ P (Ω; L (0 , f ∈ L ∞ P (Ω; L (0 , L ( v ) = E Z f ( t ) v ( t ) d t for all v ∈ L P (Ω; L (0 , g ∈ L P (Ω; L (0 , ξ = E ( ξ ) + Z g ( t ) d W ( t ) . N CONDITIONAL EXPECTATIONS IN L p ( µ ; L q ( ν ; X )) 7 Take now v ∈ L P (Ω; L (0 , E ( ξx (1)) = E Z g ( t ) v ( t ) d t. Since (2.4) and (2.6) hold for all v ∈ L P (Ω; L (0 , f = g foralmost all ( t, ω ) ∈ (0 , × Ω. Hence, g ∈ L ∞ P (Ω; L (0 , { ξ ∈ L F (Ω) : E ξ = 0 } into L P (Ω; L (0 , { ξ ∈ L ∞ F (Ω) : E ξ = 0 } into L ∞ P (Ω; L (0 , L P (Ω; L (0 , Remark . In [9], the authors proved that ( L P (0 , L (Ω))) ∗ = L ∞ P (0 , L (Ω)) . It seems that this result cannot be obtained by the method in this paper.
Acknowledgment – The authors thank Gilles Pisier for pointing out an error inan earlier version of the paper and communicating to us Example 2.6 and TuomasHyt¨onen for showing us Example 2.7.
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