Pointwise Convergence for Semigroups in Vector-Valued Lp Spaces

Pointwise Convergence for Semigroups in Vector-valued Lp Spaces

Robert J Taggart 31 March 2008

Abstract

2 Suppose that {Tt : t ≥ 0} is a symmetric diusion semigroup on L (X) and denote by {Tet : t ≥ 0} its tensor product extension to the Bochner space Lp(X, B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the HopfDunfordSchwartz ergodic theorem and show that this extends to a maximal theorem for analytic p continuations of {Tet : t ≥ 0} on L (X, B). As an application, we show that such continuations exhibit pointwise convergence.

1 Introduction

The goal of this paper is to show that two classical results about symmetric diusion semigroups, which go back to E. M. Stein's monograph [24], can be extended to the setting of vector-valued Lp spaces. Suppose throughout that (X, µ) is a positive σ-nite measure space.

Denition 1.1. Suppose that {Tt : t ≥ 0} is a semigroup of operators on L2(X). We say that

(a) the semigroup {Tt : t ≥ 0} satises the contraction property if 2 q (1) kTtfkq ≤ kfkq ∀f ∈ L (X) ∩ L (X) whenever t ≥ 0 and q ∈ [1, ∞]; and

(b) the semigroup {Tt : t ≥ 0} is a symmetric diusion semigroup if it satises 2 the contraction property and if Tt is selfadjoint on L (X) whenever t ≥ 0. It is well known that if 1 ≤ p < ∞ and 1 ≤ q ≤ ∞ then Lq(X) ∩ Lp(X) is p 2 dense in L (X). Hence, if a semigroup {Tt : t ≥ 0} acting on L (X) has the p contraction property then each Tt extends uniquely to a contraction of L (X) whenever p ∈ [1, ∞). By abuse of notation, we shall also denote by {Tt : t ≥ 0} the unique semigroup extension which acts on Lp(X). The class of symmetric diusion semigroups is widely used in applications and includes the Gaussian and Poisson semigroups on L2(Rn). Despite the simplicity of the axioms dening this class, symmetric diusion semigroups have a rich theory. For example, if {Tt : t ≥ 0} is a symmetric diusion semigroup on L2(X) then the semigroup can also be continued analytically to sectors of the

1 complex plane. To be precise, given a positive angle ψ, let Γψ denote the cone and its closure. We shall denote the interval by {z ∈ C : | arg z| < ψ} Γψ [0, ∞) Γ0. Using spectral theory and complex interpolation, Stein proved the following result.

Theorem 1.2 (Stein [24]). Suppose that 1 < p < ∞,

ψ/π = 1/2 − |1/p − 1/2| > 0, and is a symmetric diusion semigroup on 2 . Then {Tt : t ∈ R} L (X) {Tt : p t ≥ 0} extends uniquely to a semigroup {Tz : z ∈ Γψ} of contractions on L (X) such that the operator-valued function z 7→ Tz is holomorphic in Γψ and weak operator topology continuous in Γψ. We now recall two results of M. Cowling [5], developing the fundamental work of Stein [24], which the current paper generalises. The rst is a useful technical tool. For f in Lp(X), dene the maximal function Mψf by

ψ M f = sup{|Tzf| : z ∈ Γψ}. The maximal theorem, stated below, says that the maximal function operator Mψ is bounded on Lp(X). Theorem 1.3 (SteinCowling [5]). Suppose that 1 < p < ∞ and that

0 ≤ ψ/π < 1/2 − |1/p − 1/2|.

p If {Tz : z ∈ Γψ} is the semigroup on L (X) given by Theorem 1.2 then there is a positive constant C such that

ψ p M f p ≤ C kfkp ∀f ∈ L (X). The maximal theorem allows one to deduce a pointwise convergence result for the semigroup {Tz : z ∈ Γψ}. Corollary 1.4 (SteinCowling [5]). Assume the hypotheses of Theorem 1.3. p If f ∈ L (X) then (Tzf)(x) → f(x) for almost every x in X as z tends to 0 in Γψ. The earliest form of the maximal theorem appeared in Stein [24, p. 73] for the case when ψ = 0. From this Stein deduced the pointwise convergence of Ttf to f as t → 0+. Using a simpler approach, Cowling [5] extended Stein's result to semigroups {Tz : z ∈ Γψ}, holomorphic in the sector Γψ, without additional hypotheses. Given z ∈ Γψ, Cowling's strategy was to decompose the operator Tz into two parts: Z t Z t 1 −sL h −zL 1 −sL i Tzf = e f ds + e f − e f ds , (2) t 0 t 0 where t = |z| and −L is the generator of the semigroup. The Lp norm of the rst term on the right-hand side can be controlled by the HopfDunfordSchwartz ergodic theorem. A clever use of the Mellin transform allows the bracketed terms to be controlled by bounds on the imaginary powers of L.

2 The chief contribution of this paper is to observe that, under certain assump- tions, this argument may be adapted to the setting of Lp spaces of Banach-space- valued functions. Several other results contained in Stein's monograph [24] have already been pushed in this direction (see, for example, [28], [19] and [14]). In a broader context, there has been much recent interest in operators which act on such spaces, particularly when the Banach space has the so-called UMD property (see the ground breaking work of J. Bourgain [1] and D. Burkholder [3]). Developments which are perhaps most pertinent to our results include studies on bounded imaginary powers of operators (of which the article [9] of G. Dore and A. Venni is now a classic), H∞-functional calculi for sectorial operators (see especially the paper [6] of A. McIntosh and his collaborators) and maximal Lp-regularity (see L. Weis [27] and the references therein). The article [18] of P. Kunstmann and L. Weis gives an excellent exposition of the interplay between these motifs in the vector-valued setting as well as an extensive bibliography detailing the key contributions made to the eld over the last two decades. Suppose that B is a (complex) Banach space and let Lp(X, B) denote the Bochner space of B-valued p-integrable functions on X. Given a symmetric 2 diusion semigroup {Tt : t ≥ 0} on L (X), its tensor product extension {Tet : t ≥ p 0} to L (X, B) exists by the contraction property (see Section 2). If {Tet : t ≥ 0} can be continued analytically to some sector Γψ+, where 0 < ψ < π/2 and is a (suciently) small positive number, then denote this continuation by {Tez : z ∈ Γψ+}. If such a continuation does not exist, we take ψ to be 0. Given any function in p , one denes the maximal function ψ by F L (X, B) MBF ψ (3) MBF = sup{|TezF |B : z ∈ Γψ}. The theorem below is the main result of this paper.

Theorem 1.5. Suppose that (X, µ) is a σ-nite measure space and that {Tt : t ≥ 0} is a symmetric diusion semigroup on L2(X). Suppose also that B is a Banach space isomorphic to a closed subquotient of a complex interpolation space (H, U)[θ], where H is a Hilbert space, U is a UMD space and 0 < θ < 1. If 1 < p < ∞, |2/p − 1| < θ and π 0 ≤ ψ < (1 − θ) 2 then

(a) {Tet : t ≥ 0} has a bounded analytic continuation to the sector Γψ in Lp(X, B), (b) there is a positive constant C such that

ψ p MBF ≤ C kF kLp(X,B) ∀F ∈ L (X, B), Lp(X) and

p (c) if F ∈ L (X, B) then TezF (x) converges to F (x) for almost every x in X as z tends to 0 in the sector Γψ.

3 It is noteworthy that the class of Banach spaces B satisfying the interpolation hypothesis of Theorem 1.5 is a subset of those Banach spaces possessing the UMD property. It includes those classical Lebesgue spaces, Sobolev spaces and Schattenvon Neumann ideals that are reexive. The reader is directed to Section 6 for further remarks on these spaces. The structure and content of the rest of this paper is as follows. Section 2 presents some standard results on tensor product extensions of operators to vector-valued Lp spaces. For example, it is well-known that such extensions exist for semigroups with the contraction property and consequently these semigroups are subpositive. In Section 3 we prove a stronger result; namely that, whenever 2 1 ≤ p < ∞, every measurable semigroup {Tt : t ≥ 0} on L (X) satisfying the contraction property is dominated on Lp(X) by a measurable positive semigroup with the contraction property. This result allows us to easily deduce, in Section 4, a vector-valued version of the HopfDunfordSchwartz ergodic theorem. Parts (a) and (b) of Theorem 1.5 are proved in Sections 5 and 6. Following techniques used in [5], we begin by proving a maximal theorem for the tensor p product extension {Tet : t ≥ 0} to L (X, B) of a symmetric diusion semigroup {Tt : t ≥ 0} satisfying the contraction property. Here we assume that 1 < p < ∞ and B is any reexive Banach space, provided that the generator −Le of the B- valued extension has bounded imaginary powers on Lp(X, B) with a power angle less than π/2 − ψ. Section 6 discusses circumstances under which this condition holds. In general, it is necessary that B has the UMD property. Moreover, by exploiting the subpositivity of {Tt : t ≥ 0} and adapting arguments of M. Hieber and J. Prüss [13], we show that if B has the UMD property then Le has an H∞-functional calculus. This, along with spectral theory and interpolation, allows us to remove the bounded imaginary powers hypothesis at the cost of restricting the class of Banach spaces B for which the maximal theorem is valid. In Section 7 we show that that the pointwise convergence of {Tez : z ∈ Γψ} is easily deduced from the pointwise convergence of {Tz : z ∈ Γψ} and the maximal theorem. This completes the proof of Theorem 1.5.

2 Vector-valued extensions of contraction semigroups

Suppose that B is a (complex) Banach space with norm |·|B and that (X, µ) is a σ-nite measure space. We also assume throughout this section that p ∈ [1, ∞). Denote by Lp(X, B) the Bochner space of all B-valued measurable functions F on X satisfying

Z 1/p p kF kLp(X,B) := |F (x)|B dµ(x) < ∞. X (As is customary, we will not distinguish between equivalence classes of functions and members of each equivalence class.) Let Lp(X) ⊗ B denote the set of all nite linear combinations of B-valued functions of the form uf, where u ∈ B and f ∈ Lp(X). It is known that this set is dense in Lp(X, B). Many operators acting on scalar-valued function spaces can be extended to act on B-valued function spaces in the following canonical way.

4 p Denition 2.1. Suppose that T is a bounded operator on L (X). If IB denotes p the identity operator on B then dene the tensor product T ⊗ IB on L (X) ⊗ B by n ! n X X T ⊗ IB ukfk = ukT fk k=1 k=1 whenever +, , p and . We say that a bounded n ∈ Z uk ∈ B fk ∈ L (X) k = 1, . . . , n p p operator Te : L (X, B) → L (X, B) is a B-valued extension of T if Te = T ⊗ IB on Lp(X) ⊗ B. In this case Te is also called a tensor product extension of T to Lp(X, B).

If it exists, a B-valued extension Te of T is necessarily unique by the density of Lp(X) ⊗ B in Lp(X, B). It is well-known that if an operator T on L2(X) extends to a contraction on q p L (X) for all q in [1, ∞] then T ⊗ IB extends to a contraction Te on L (X, B). (This is not hard to show if p = 1; for other values of p the result can be deduced by duality and interpolation.) Consequently, any semigroup {Tt : t ≥ 0} on 2 L (X) with the contraction property extends to a semigroup {Tet : t ≥ 0} of p contractions on L (X, B). Moreover, if {Tt : t ≥ 0} is strongly continuous p p on L (X) then {Tet : t ≥ 0} is strongly continuous on L (X, B). This is a consequence of the following lemma.

Lemma 2.2. Suppose that {St : t ≥ 0} is a family of bounded operators on p p L (X) with a B-valued extension {Set : t ≥ 0} to L (X, B). If the mapping t 7→ St is strongly continuous and {Set : t ≥ 0} is locally (with respect to t) uniformly bounded in norm then the mapping t 7→ Set is strongly continuous. Proof. Suppose that F ∈ Lp(X, B). Then we may approximate F by a function of the form Pn , where , and p . Since the G k=1 ukgk n ∈ N uk ∈ B gk ∈ L (X) map t 7→ St is strongly continuous, the map t 7→ SetG is continuous, and by a standard 3 argument, so is the map t 7→ SetF .

2 Suppose that {Tt : t ≥ 0} is a strongly continuous semigroup on L (X) with the contraction property. The generator B of tensor extension {Tet : t ≥ 0} to Lp(X, B) is given, as usual, by

TetF − F BF = lim t→0+ t for all F in Lp(X, B) for which the limit exists. The collection of such F is called the domain of B. Let −L denote the generator of {Tt : t ≥ 0}. It is easy to show that Dom(L) ⊗ B ⊆ Dom(B) and that B = −L ⊗ IB on Dom(L) ⊗ B. Therefore we shall denote B by −Le. It is of later interest to know when a family {Tez : z ∈ Γψ} of B-valued extensions inherits analyticity from {Tz : z ∈ Γψ}.

Lemma 2.3. Suppose that B is a reexive, 1 < p < ∞ and {Sz : z ∈ Γψ} is a p family of operators on L (X) with a uniformly bounded extension {Sez : z ∈ Γψ} p to L (X, B). If the mapping z 7→ Sz is analytic for z in Γψ then so is the mapping z 7→ Sez.

5 Proof. Assume the hypotheses of the lemma and dene q by 1/p + 1/q = 1. q ∗ p Whenever G ∈ L (X, B ), let LG denote the linear functional acting on L (X, B) associated by duality to G. Suppose now that > 0, F ∈ Lp(X, B) and G ∈ Lq(X, B∗). By appropriately approximating F and G with functions F 0 in Lp(X) ⊗ B and G0 in Lq(X) ⊗ B∗, it is not hard to show that

0 |LG(SezF ) − LG0 (SezF )| < ∀z ∈ Γψ.

Moreover, since z 7→ Sz is analytic in Γψ, it is easily shown that that z 7→ 0 LG0 (SezF ) is also analytic. Consequently z 7→ LG(SezF ) is analytic, whence the lemma follows. Bounded operators with vector-valued extensions can be characterised in terms of subpositivity, a property that proves useful in subsequent sections.

Denition 2.4. Suppose that T is a linear operator on Lp(X). We say that (a) T is positive if T f ≥ 0 whenever f ≥ 0 for f in Lp(X); (b) T is subpositive if there exists a bounded positive operator S on Lp(X) such that |T f| ≤ S|f| whenever f ∈ Lp(X), in which case we also say that T is dominated by S.

If R is an operator on Lp(X) then dene R by the formula Rf = Rf¯ whenever f ∈ Lp(X), and dene Re(R) by (R + R)/2. Lemma 2.5. Suppose that T is a bounded operator on Lp(X). Then the following are equivalent.

(a) The operator T is a subpositive contraction on Lp(X). (b) For any nite subset n of p , {fk}k=1 L (X)

sup |T fk| ≤ sup |fk| . k Lp(X) k Lp(X)

p (c) The tensor product T ⊗ IB extends to a contraction Te on L (X, B). (d) There exists a positive contraction S such that S + Re(eiθT ) is positive whenever θ ∈ R. The equivalence of (a), (b) and (c) are well-known (see, for example, [21] and [20]) and will be used in Section 5. Statement (d) is the denition of subpositive contractivity given by R. Coifman, R. Rochberg and G. Weiss [4, p. 54]. The fact that (a) implies (d) is easy to establish and is used to prove Theorem 6.1. The converse is harder to prove (see, for example, [25, Section 2.1]) but will not be needed for the results of this paper.

6 3 Subpositivity for contraction semigroups

One of the consequences of Lemma 2.5 is that any semigroup on L2(X) that enjoys the contraction property extends to a semigroup of subpositive contractions on Lp(X) whenever 1 ≤ p < ∞. The goal of this section is to prove a stronger result needed in Section 4, namely that every semigroup on L2(X) with the contraction property is, when extended to a semigroup on Lp(X) for p in [1, ∞), dominated by a positive contraction semigroup on Lp(X). We begin with a few preliminaries. Suppose that 1 ≤ p < ∞ and T is a bounded linear operator on p . If and for all L (X) 1 ≤ q < ∞ kT fkq ≤ C kfkq f in Lq(X) ∩ Lp(X) then T has a unique bounded linear extension acting on Lq(X). By abuse of notation we will also denote this extension by T . We say that a family of operators {Tt : t ≥ 0} is (strongly) measurable on p p p L (X) if, for every f in L (X), the L (X)-valued map t 7→ Ttf is measurable with respect to Lebesgue measure on [0, ∞). The family is said to be weakly measurable if the complex-valued map t 7→ hTtf, gi is measurable with respect 0 to Lebesgue measure on [0, ∞) whenever f ∈ Lp(X) and g ∈ Lp (X). Here, p0 denotes the conjugate exponent of p, given by 1/p+1/p0 = 1. If 1 ≤ p < ∞ then Lp(X) is a separable Banach space and hence strong measurability and weak measurability coincide by the Pettis measurability theorem (see [10, Theorem III.6.11]). We now state the main result of this section.

2 Theorem 3.1. Suppose that {Tt : t ≥ 0} is a semigroup on L (X) satisfying the contraction property. Then there exists a positive semigroup {St : t ≥ 0} on L2(X), satisfying the contraction property, such that p |Ttf| ≤ St|f| ∀f ∈ L (X) whenever 1 ≤ p < ∞ and t ≥ 0. If {Tt : t ≥ 0} is a measurable semigroup on 2 p L (X) then {St : t ≥ 0} extends to a measurable semigroup on L (X) whenever 1 ≤ p < ∞. We shall prove the theorem via a sequence of lemmata. The nal stage of the proof draws heavily on the work of Y. Kubokawa [17] and C. Kipnis [15], who independently proved a similar result for L1 contraction semigroups. Denote by the p the set of nonnegative functions in p . L+(X) L (X) Lemma 3.2. Suppose that 1 < p < ∞. Assume also that T and S are bounded operators on 1 such that and whenever L (X) kT fkp ≤ kfkp kSfkp ≤ kfkp f ∈ L1(X) ∩ Lp(X). If S is positive and dominates T on L1(X) then |T f| ≤ S|f| ∀f ∈ Lp(X). Proof. Assume the hypotheses of the lemma and suppose that f ∈ Lp(X). Since 1 p is dense in p there is a sequence in 1 p L (X)∩L (X) L (X) {fn}n∈N L (X)∩L (X) p p such that fn → f in L (X). By continuity, |T fn| → |T f| in L (X) and similarly p S|fn| → S|f| in L (X). Moreover, |T fn| ≤ S|fn| for all n. If gn = S|fn| − |T fn| p and g = S|f| − |T f| then each gn is nonnegative and gn → g in L (X). Now p is a closed subset of p , so and the proof is complete. L+(X) L (X) g ≥ 0 Lemma 3.3. Suppose that p and q both lie in the interval [1, ∞) and that 2 {Tt : t ≥ 0} is a family of operators on L (X) satisfying the contraction property. p q If {Tt : t ≥ 0} is measurable on L (X) then {Tt : t ≥ 0} is measurable on L (X).

7 0 Proof. Assume the hypotheses and suppose that f ∈ Lq(X) and g ∈ Lq (X). It suces to show that the map φ : [0, ∞) → C, dened by

φ(t) = hTtf, gi , is measurable. Choose any sequence of measurable sets satisfying {Xn}n∈N Xn ⊂ Xn+1 and ∪n∈ Xn = X. By carefully choosing a sequence {gn}n∈ 0 N 0 0 N contained in Lq (X) ∩ Lp (X) which converges in Lq (X) to g, and a sequence contained in q p which converges in q to , one can {fn}n∈N L (X) ∩ L (X) L (X) f show that the sequence , given by {φn}n∈N

φn(t) = hTtfn, 1Xn gni , converges pointwise to φ. Since each φn is measurable, φ is also. Lemma 3.4. Suppose that T is a linear contraction on L1(X) with the property that whenever q 1 and . Then there kT fkq ≤ kfkq f ∈ L (X) ∩ L (X) 1 ≤ q ≤ ∞ is a unique bounded linear positive operator T on L1(X) such that (a) the operator norms of T and T on L1(X) are equal, (b) whenever q 1 and , kTfkq ≤ kfkq f ∈ L (X) ∩ L (X) 1 ≤ q ≤ ∞ (c) |T f| ≤ T|f| whenever f ∈ Lp(X) and 1 ≤ p ≤ ∞, and (d) 1 whenever 1 . Tf = sup{|T g| : g ∈ L (X), |g| ≤ f} f ∈ L+(X) Proof. For the existence of a unique operator T satisfying properties (a), (c) (for the case when p = 1) and (d), see, for example, [16, Theorem 4.1.1]. Property (b) holds by [10, Lemma VIII.6.4]. We can now deduce property (c), for the case when 1 < p < ∞, from Lemma 3.2. The operator T introduced in the lemma is called the linear modulus of 1 T . If {Tt : t ≥ 0} is a bounded semigroup on L (X) then Ts+t ≤ TsTt for all nonnegative s and t. However, equality may not hold and thus the family {Tt : t ≥ 0} of bounded positive operators will not, in general, be a semigroup. Nevertheless, Kubokawa [17] and Kipnis [15] (see [16, Theorems 4.1.1 and 7.2.7] for a more recent exposition) showed that the linear modulus Tt could be used to construct a positive semigroup {St : t ≥ 0}, known as the modulus semigroup, which dominates {Tt : t ≥ 0}. The following proof uses this construction. Proof of Theorem 3.1. Assume the hypothesis of Theorem 3.1 and suppose that t > 0. Let Dt denote the family of all nite subdivisions (si) of [0, t] satisfying

0 = s0 < s1 < s2 < . . . < sn = t. If and 0 0 are two elements of then we write 0 whenever s = (si) s = (sj) Dt s < s 0 s is a renement of s. With this partial order Dt is an increasingly ltered set. For in 1 , put f L+(X)

Φ(s, f) = Ts1 Ts2−s1 ... Tsn−sn−1 f, where Tα is the linear modulus of Tα whenever α ≥ 0. It follows from Tα+β ≤ 0 0 TαTβ that s < s implies Φ(s, f) ≤ Φ(s , f). Since the operator Tα is contraction whenever , we have . We now dene on 1 α ≥ 0 kΦ(s, f)k1 ≤ kfk1 St L+(X) by

Stf = sup{Φ(s, f): s ∈ Dt}.

8 Note that

sup{Φ(s, f): s ∈ Dt} = lim Φ(s, f) s∈Dt so St is well-dened by the monotone convergence theorem for increasingly l- tered families.

It is easy to check that St(f + g) = Stf + Stg and St(λf) = λStf whenever and belong to 1 and . Moreover, if 1 . f g L+(X) λ ≥ 0 kStfk1 ≤ kfk1 f ∈ L+(X) 1 Therefore St can now be dened for all f in L (X) as a linear contraction of 1 1 L (X). We dene S0 as the identity operator on L (X). 0 We now show that {St : t ≥ 0} is a semigroup. Suppose that t and t are both positive. If

0 = s0 < s1 < s2 < . . . < sn = t and 0 0 0 0 0 0 = s0 < s1 < s2 < . . . < sn = t form subdivisions of [0, t] and [0, t0] then

0 0 0 0 0 = s0 < s1 < s2 < . . . < sn = sn + s0 < sn + s1 < sn + s2 < . . . < sn + sn forms a subdivision of [0, t + t0]. Conversely every subdivision of [0, t + t0] which is ne enough to contain t is of this form. This yields St+t0 = StSt0 . By Lemma 3.4 (b), it is easy to check that {St : t ≥ 0} extends to a contraction semigroup on L2(X) which satises the contraction property. Moreover, 1 the construction shows that |Ttf| ≤ St|f| whenever f ∈ L (X) and t ≥ 0. By an application of Lemma 3.2, we deduce that each Tt is dominated by St on Lp(X) whenever 1 ≤ p < ∞. 2 It remains to show that if {Tt : t ≥ 0} is measurable on L (X) then {St : t ≥ 0} is measurable on Lp(X) whenever 1 ≤ p < ∞. In view of Lemma 3.3, 1 we can assume that {Tt : t ≥ 0} is measurable on L (X) and it suces to 1 1 show that {St : t ≥ 0} is measurable on L (X). Fix f in L (X) and dene 1 by . We will construct a sequence of φ : [0, ∞) → L (X) φ(t) = Stf {φn}n∈N measurable functions converging pointwise to φ, completing the proof. Since f can be decomposed as a linear combination of four nonnegative functions (the positive and negative parts of Re(f) and Im(f)) and each St is linear, we can assume, without loss of generality, that f ≥ 0. When t > 0, n ∈ N and m is the smallest integer such that m ≥ 2nt, let denote the subdivision m of given by s(n, t) (sk(n, t))k=0 [0, t] ( k2−n if k = 0, 1, . . . , m − 1 sk(n, t) = t if k = m.

1 Now dene φn : [0, ∞) → L (X) by

φn(t) = Φ(s(n, t), f) when t > 0 and φn(0) = f. By the denition of St, |φ(t)−φn(t)| → 0 as n → ∞ for each t ≥ 0. Our task is to demonstrate that is measurable for each in . Note that φn n N −n −n φn(t), when t is restricted to the interval [k2 , (k + 1)2 ), is of the form

Bk,nTt−k2−n f

9 1 where Bk,n is a contraction on L (X). It follows that if E is an open set of L1(X) then

∞ −1 [ −n −n φn (E) = t ∈ [k2 , (k + 1)2 ): Bk,nTt−k2−n f ∈ E . k=1

Hence if the map ϕ : [0, 2−n) → L1(X), dened by

ϕ(t) = Ttf, is measurable then −1 can be written as a countable union of measurable φn (E) sets and consequently φn is measurable. But by Lemma 3.4 (d) there is a sequence in 1 such that as . In other {fj}j∈N L+(X) |Ttf − Ttfj| → 0 j → ∞ words, is the pointwise limit of a sequence of measurable functions, ϕ {ϕj}j∈N dened by ϕj(t) = Ttfj, and hence ϕ is measurable. This completes the proof of Theorem 3.1. In preparation for the next section, we present the following lemma. Its proof will be omitted.

2 Lemma 3.5. Suppose that {Tt : t ≥ 0} is a semigroup on L (X) satisfying the p contraction property. If {Tt : t ≥ 0} is strongly continuous on L (X) for some q p in [1, ∞) then {Tt : t ≥ 0} is strongly continuous on L (X) for all q in (1, ∞).

4 The HopfDunfordSchwartz ergodic theorem

We now obtain a vector-valued version of the HopfDunfordSchwartz ergodic theorem for use in Section 5. If T is a bounded strongly measurable semigroup p {Ts : s ≥ 0} on L (X) then dene the operator A(T , t), for positive t, by the formula Z t 1 p A(T , t)f = Tsf ds ∀f ∈ L (X). t 0 For f in Lp(X), we now dene a maximal ergodic function AT f by

AT f = sup |A(T , t)f|. (4) t>0 A simplied version of the classical HopfDunfordSchwartz ergodic theorem may be stated as follows.

Theorem 4.1. [10, Theorem VIII.7.7] Suppose that {Tt : t ≥ 0} is a measurable semigroup on L2(X) satisfying the contraction property. Assume that p ∈ (1, ∞) T and denote {Tt : t ≥ 0} by T . Then the maximal ergodic function operator A satises the inequality

1/p T p p A f ≤ 2 kfk ∀f ∈ L (X). p p − 1 p We will now develop a vector-valued version of this theorem. Fix p in the interval (1, ∞). Suppose that T is a strongly continuous semigroup {Tt : t ≥ 0} on L2(X) satisfying the contraction property. By Lemma 3.5, the semigroup T is a strongly continuous semigroup of contractions when viewed as acting on

10 Lp(X). We rst show that the bounded linear operator A(T , t) on Lp(X) has an extension to Lp(X, B) for all positive t. By Theorem 3.1 there is a measurable p semigroup {St : t ≥ 0} of positive contractions on L (X), which we denote by S, dominating T on Lp(X). Hence A(S, t) is also a positive contraction on Lp(X) for each positive t. Moreover,

Moreover, if p then T is measurable. To see this, observe F ∈ L (X, B) AB F that the mapping t 7→ A(T , t)f is continuous from (0, ∞) to Lp(X) and Ae(T , t) is a contraction on Lp(X) for every t by (5). Hence the vector-valued mapping t 7→ Ae(T , t)F is continuous from (0, ∞) to Lp(X, B) by a simple modication of Lemma 2.2. This implies that the mapping t 7→ |Ae(T , t)F |B is continuous from p (0, ∞) to L (X). Therefore the measurable function sup + |A(T , t)F | , where t∈Q e B + denotes the set of positive rationals, coincides with . Q supt>0 |Ae(T , t)F |B Corollary 4.2. Suppose that B is a Banach space and that T is a strongly 2 continuous semigroup {Tt : t ≥ 0} on L (X) with the contraction property. If then the maximal ergodic function operator T , dened by (6), 1 < p < ∞ AB satises the inequality

1/p T p p A F p ≤ 2 kF k p ∀f ∈ L (X, B). B L (X) p − 1 L (X,B)

Proof. Fix p in (1, ∞) and let S denote the semigroup dominating T that was introduced above. For F in Lp(X, B),

T AB F = sup |Ae(T , t)F |B t>0

≤ sup A(S, t)|F |B t>0 S = A |F |B.

The result follows upon taking the Lp(X) norm of both sides and applying Theorem 4.1.

5 The vector-valued maximal theorem

The main result of this section is a vector-valued version of Theorem 1.3. It gives an p estimate for the maximum function ψ (dened by (3)) under L MBF the assumption that the generator −Le of {Tet : t ≥ 0} has bounded imaginary powers.

11 Theorem 5.1. Suppose that (X, µ) is a σ-nite measure space, B is a reexive Banach space, 1 < p < ∞, 0 ≤ ψ < π/2 and {Tt : t ≥ 0} is a symmetric diusion semigroup on L2(X). If there exists ω less than π/2 − ψ and a positive constant K such that Le has bounded imaginary powers satisfying the norm estimate

iu ω|u| p (7) kLe F kLp(X,B) ≤ Ke kF kLp(X,B) ∀F ∈ L (X, B) ∀u ∈ R,

p then {Tet : t ≥ 0} has a bounded analytic continuation in L (X, B) to the sector and there is a constant such that the maximal function operator ψ Γψ C MB satises the inequality

ψ p (8) MBF ≤ C kF kLp(X,B) ∀F ∈ L (X, B). Lp(X)

Proof. Assume the hypotheses of the theorem. Suppose that f ∈ Lp(X) and iθ ψ0 < π/2 − ω. Write z as e t, where |θ| < ψ0 and t > 0. First we review the representation of Tz given implicitly in [5]. The key idea is to decompose Tzf into two parts:

Z t Z t 1 −sL h −zL 1 −sL i Tzf = e f ds + e f − e f, ds t 0 t 0 Z t Z 1 1 h iθ −stL i = Tsf ds + exp(−e tL) − e ds . (9) t 0 0

Dene the function mθ on (0, ∞) by

Z 1 iθ −sλ mθ(λ) = exp(−e λ) − e ds ∀λ > 0. 0

Write nθ for mθ ◦ exp and observe that Z ∞ 1 iu mθ(λ) = nˆθ(u)λ du, 2π −∞ where nˆθ denotes the Fourier transform of nθ. Calculation using complex analysis shows that

−θu −1 nˆθ(u) = e − (1 + iu) Γ(iu) ∀u ∈ R, and the theory of the Γ-function (see, for example, [26, p. 151]) gives the estimate

(10) |nˆθ(u)| ≤ C0 exp (|θ| − π/2)|u| ∀u ∈ R, where C0 is a constant independent of u and θ. Hence (9) becomes

Z t Z ∞ 1 1 iu iu Tzf = Tsf ds + nˆθ(u)t L f du. (11) t 0 2π −∞ (This representation may be justied when p = 2 by spectral theory and then for other values of p by (10) and the bounded imaginary power estimates for L given in [5].)

12 iu p Since Ts⊗IB and L ⊗IB extend to bounded operators on L (X, B) whenever and , one obtains from (11) the -valued extension on p s ≥ 0 u ∈ R B Tez L (X, B) given by

Z t Z ∞ 1 1 iu iu p TezF = TesF ds + nˆθ(u)t Le F du ∀F ∈ L (X, B) (12) t 0 2π −∞ whenever . Here we have used estimates (7) and (10) to ensure the z ∈ Γψ0 convergence of the second integral. In fact, these estimates give

C K Z ∞ 0 (ψ0−π/2+ω)|u| kTezF kLp(X,B) ≤ kF kLp(X,B) + e du kF kLp(X,B) 2π −∞ C K = 1 + 0 C kF k , (13) 2π ψ0,ω Lp(X,B) from which we conclude that the family is uniformly bounded in {Tez : z ∈ Γψ0 } norm. It follows from Lemma 2.3 that the map is analytic in . z 7→ Tez Γψ0 It remains to prove the bound (8) on the maximal function. Formally, the representation (12) of Tez implies that

ψ T M F ≤ A F p + sup sup |mθ(tLe)F | , B p B L (X) L (X) t>0 |θ|≤ψ Lp(X) where denotes the semigroup and T is the operator dened T {Tt : t ≥ 0} AB by (6). By Corollary 4.2, the rst term on the right-hand side is majorised by 1/p . The second term is controlled by 2[p/(1 − p)] kF kLp(X,B) C1 kF kLp(X,B) for some constant C1 independent of F , by a calculation similar to (13). This formal calculation is justied provided that the function

sup sup |mθ(tLe)F |B (14) t>0 |θ|≤ψ

p is measurable. Since the map z 7→ TezF is continuous from Γψ to L (X, B), the p map (t, θ) 7→ |mθ(tLe)F |B is continuous from (0, ∞) × [−ψ, ψ] to L (X). Hence

6 Bounded imaginary powers of the generator

In this section we examine circumstances under which the bounded imaginary power estimate (7), one of the hypotheses of the preceding theorem and corollary, is satised. A fruitful (and in our context, necessary) setting is when the Banach space B has the UMD property. A Banach space B is said to be a UMD space if one of the following equivalent statements hold:

13 (a) The Hilbert transform is bounded on Lp(X, B) for one (and hence all) p in (1, ∞). (b) If 1 < p < ∞ then B-valued martingale dierence sequences on Lp(X, B) converge unconditionally.

(c) If then iu extends to a bounded operator on p 1 < p < ∞ (−∆) ⊗IB L (R, B) for every u in R (a result due to S. Guerre-Delabrière [12]). Several other characterisations of UMD spaces exist (see, for example, [2] and the survey in [23]) but those cited here are, for dierent reasons, the most relevant to our discussion. If the Hilbert transform, which corresponds to the multiplier function u 7→ i sgn(u), is bounded on Lp(X, B) then one can establish vector- valued versions of some Fourier multiplier theorems (such as Mikhlin's multiplier theorem [29]). This fact is used below to establish Theorem 6.1. The second characterisation gave rise to the name UMD. The third characterisation shows that, in general, B must be a UMD space if Le is to have bounded imaginary powers, since −∆ generates the Gaussian semigroup. Examples of UMD spaces include, when 1 < p < ∞, the classical Lp(X) spaces and the Schattenvon Neumann ideals Cp. Moreover, if B is a UMD space then its dual B∗, closed subspaces of B, quotient spaces of B and Lp(X, B) when 1 < p < ∞ also inherit the UMD property. It was shown by Hieber and Prüss [13] that when 1 < q < ∞ the generator of a UMD-valued extension of a bounded strongly continuous positive semigroup on Lq(X) has a bounded H∞-functional calculus. The next result says that the same is true if the positivity condition is relaxed to subpositivity (assuming that the UMD-valued extension of the semigroup is bounded), though it is convenient in the present context to state it for semigroups possessing the contraction ∞ property. First we introduce some notation. If σ ∈ (0, π] then let H (Γσ) denote the Banach space of all bounded analytic functions dened on Γσ with norm

kfk ∞ = sup |f(z)|. H (Γσ ) z∈Γσ

Theorem 6.1. Suppose that 1 < q < ∞ and B is a UMD space. If {Tt : t ≥ 0} is a strongly continuous semigroup on L2(X) satisfying the contraction property q and −Le is the generator of its tensor extension {Tet : t ≥ 0} to L (X, B), then ∞ Le has a bounded H (Γσ)-calculus for all σ in (π/2, π]. Consequently, for every σ ∈ (π/2, π] there exists a positive constant Cq,σ such that iu σ|u| q (15) kLe F kLq (X,B) ≤ Cq,σe kF kLq (X,B) ∀F ∈ L (X, B) ∀u ∈ R.

Proof. Since the semigroup {Tt : t ≥ 0} can be extended to a subpositive strongly continuous semigroup of contractions on Lq(X), it has a dilation to a q 0 0 0 bounded c0-group on L (X ) for some measure space (X , µ ). In other words, 0 0 there exists a measure space (X , µ ), a strongly continuous group {Ut : t ∈ R} of subpositive contractions on Lq(X0), a positive isometric embedding D : Lq(X) → Lq(X0) and a subpositive contractive projection P : Lq(X0) → Lq(X0) such that

DTt = PUtD ∀t ≥ 0 (see the result of G. Fendler [11, pp. 737738] which extends the work of Coif- man, Rochberg, and Weiss [4]). Lifting this identity to its B-valued extension,

14 q we see that the semigroup {Tet : t ≥ 0} on L (X, B) has a dilation to a bounded -group on q 0 . c0 {Uet : t ∈ R} L (X , B) Let −Le denote the generator of {Tet : t ≥ 0}. Then the dilation implies that ∞ Le has a bounded H (Γσ)-calculus for all σ in (π/2, π] (see [13] or the exposition in [18, pp. 212214], where the H∞-calculus is rst constructed for the generator of the group using the vector-valued Mikhlin multiplier theorem {Uet : t ∈ R} in conjunction with the transference principle, and then projected back to the generator −Le of {Tet : t ≥ 0} via the dilation). ∞ The bounded H (Γσ)-calculus gives a positive constant Cq,σ such that

∞ kf(L)k q ≤ C kfk ∞ ∀f ∈ H (Γ ). e L (X,B) q,σ H (Γσ ) σ

If f(z) = ziu for u in R then (15) follows. The theorem above suggests that the problem of nding bounded imaginary powers of Le is critical to L2(X, B). That is, if

iu ω|u| 2 kLe F kL2(X,B) ≤ Ce kF kL2(X,B) ∀F ∈ L (X, B) ∀u ∈ R for some ω less than π/2−ψ then one could interpolate between the L2 estimate and (15) to obtain (7). Unfortunately, suitable L2(X, B) bounded imaginary power estimates, where B is a nontrivial UMD space, appear to be absent in the literature, even when Le is the Laplacian. However, if B is a Hilbert space, such estimates are available via spectral theory.

Lemma 6.2. Suppose that H is a Hilbert space. If {Tt : t ≥ 0} is a symmetric diusion semigroup on L2(X) then the generator −Le of the H-valued extension 2 {Tet : t ≥ 0} to L (X, H) satises

iu 2 (16) kLe F kL2(X,H) ≤ kF kL2(X,H) ∀F ∈ L (X, H) ∀u ∈ R.

Proof. It is not hard to check that the tensor product extension to L2(X, H) of the semigroup {Tt : t ≥ 0} is a semigroup of selfadjoint contractions on L2(X, H). Its generator −Le is therefore selfadjoint on L2(X, H) and hence Le has nonnegative spectrum. Spectral theory now gives estimate (16). To obtain (7) we shall interpolate between (15) and (16). Hence we consider the class of UMD spaces whose members B are isomorphic to closed subquo- tients of a complex interpolation space (H, U)[θ], where H is a Hilbert space, U is a UMD space and 0 < θ < 1. Members of this class include the UMD function lattices on a σ-nite measure space (such as the reexive Lp(X) spaces) by a result of Rubio de Francia (see [23, Corollary, p. 216]), the reexive Sobolev spaces (which are subspaces of products of Lp spaces) and the reexive Schattenvon Neumann ideals. This class can be further extended to include many operator ideals by combining Rubio de Francia's theorem with results due to P. Dodds, T. Dodds and B. de Pagter [8] which show that the interpolation properties of noncommutative spaces coincide with those of their commutative counterparts under fairly general conditions. It was asked in [23] whether the described class of UMD spaces includes all UMD spaces. It appears that this is still an open question.

15 Corollary 6.3. Suppose that B is a UMD space isomorphic to a closed subquotient of a complex interpolation space (H, U)[θ], where H is a Hilbert space, U is a UMD space and 0 < θ < 1. Suppose also that {Tt : t ≥ 0} is a symmetric diusion semigroup on L2(X) and denote by −Le the generator of its tensor extension to Lp(X, B), where 1 < p < ∞. If |2/p − 1| < θ (17) and π 0 ≤ ψ < (1 − θ) 2 then there exists ω less than π/2 − ψ such that Le has bounded imaginary powers on Lp(X, B) satisfying estimate (7). Proof. Assume the hypotheses of the corollary. Note that π 1 π < − ψ 2 θ 2 so that if σ is the arithmetic mean of π/2 and (π/2 − ψ)/θ then σ > π/2 and σθ < π/2 − ψ. Now choose q such that 1 1 − θ θ = + . p 2 q Inequality (17) guarantees that 1 < q < ∞. Interpolating between (16) and (15) (for the space Lq(X, U)) gives

iu θ σθ|u| p kLe kLp(X,B) ≤ Cq,σe kF kLp(X,B) ∀F ∈ L (X, B) ∀u ∈ R. If ω = σθ then (7) follows, completing the proof.

7 Proof of Theorem 1.5

In this nal section we complete the proof of Theorem 1.5. Suppose the hypotheses of the theorem. Parts (a) and (b) follow immediately from Theorem 5.1 and Corollary 6.3. Part (c) will be deduced from the vector-valued maximal theorem and the pointwise convergence of {Tt : t ≥ 0} (see Corollary 1.4). For ease of notation, write z → 0 as shorthand for z → 0 with z in Γψ. Suppose that F ∈ Lp(X, B) and > 0. There exists a function G in Lp(X)⊗ such that . Write as Pn , where is a positive B kG − F kLp(X,B) < G k=1 ukfk n integer, n is contained in and n is contained in p . Hence, {uk}k=1 B {fk}k=1 L (X) for almost every x in X,

+ lim sup |TezG(x) − G(x)|B z→0

≤ sup |Tez(F − G)(x)|B + |G(x) − F (x)|B z∈Γψ n X + uk B lim sup Tzfk(x) − fk(x) z→0 k=1 ψ ≤ 2MB(G − F )(x),

16 since Corollary 1.4 implies that

lim |Tzfk(x) − fk(x)| = 0 z→0 for each k and for almost every x in X. By taking the Lp(X) norm and applying Theorem 5.1 we obtain

ψ lim sup |TezF − F |B ≤ 2 MB(G − F ) < 2C, z→0 p p where the positive constant C is independent of F and G. Since is an arbitrary positive number,

lim sup |TezF (x) − F (x)|B = 0 z→0 for almost every x in X, proving the theorem. Acknowledgments It is a pleasure to acknowledge the contribution of Michael Cowling, who introduced me to this problem and gave many helpful comments and suggestions. Thanks also go to Ian Doust, who helped shed light on a cou- ple of technical obstacles, and to Pierre Portal and Tuomas Hytönen for the interest they showed in this work. The referee made several valuable comments, one which helped strengthened the main result of this paper. This research was funded by an Australian Postgraduate Award and by the Australian Research Council's Centre of Excellence for Mathematics and Statistics of Complex Sys- tems.

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