Vector-valued Littewood-Paley-Stein theory for semigroups II
aa r X i v : . [ m a t h . F A ] S e p VECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY FORSEMIGROUPS II
QUANHUA XU
Abstract.
Inspired by a recent work of Hyt¨onen and Naor, we solve a problem left open in ourprevious work joint with Mart´ınez and Torrea on the vector-valued Littlewood-Paley-Stein theoryfor symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any T which is the square of a symmetric Markovian operator on a measure space (Ω , µ ). Moreover,we show that T ⊗ Id X extends to an analytic contraction on L p (Ω; X ) for any 1 < p < ∞ andany uniformly convex Banach space X . Introduction
Let (Ω , A , µ ) be a σ -finite measure space. By a symmetric diffusion semigroup on (Ω , A , µ ) inStein’s sense [24, section III.1], we mean a family { T t } t> of linear maps satisfying the followingproperties: • T t is a contraction on L p (Ω) for every 1 ≤ p ≤ ∞ ; • T t T s = T t + s ; • lim t → T t f = f in L (Ω) for every f ∈ L (Ω); • T t is positive (i.e. positivity preserving) and T t • T t is selfadjoint on L (Ω).It is a classical fact that the orthogonal projection from L (Ω) onto the fixed point subspace of { T t } t> extends to a contractive projection on L p (Ω) for every 1 ≤ p ≤ ∞ . We will denote thisprojection by F . Then F is also positive and F (cid:0) L p (Ω) (cid:1) is the fixed point subspace of { T t } t> on L p (Ω) (cf. e.g. [4]).Stein proved in [24, chapter IV] the following result which considerably extends the classicalinequality on the Littlewood-Paley g -function in harmonic analysis: For every 1 < p < ∞ (1) k f − F ( f ) k L p (Ω) ≈ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:12)(cid:12)(cid:12) t ∂∂t T t f (cid:12)(cid:12)(cid:12) dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , ∀ f ∈ L p (Ω) , where the equivalence constants depend only on p .The vector-valued Littlewood-Paley-Stein theory was developed in [26, 15]. Given a Banachspace X , we denote by L p (Ω; X ) the usual L p space of strongly measurable functions from Ω to X .It is a well known elementary fact that if T is a positive bounded operator on L p (Ω) with 1 ≤ p ≤ ∞ ,then T ⊗ Id X is bounded on L p (Ω; X ) with the same norm. For notational convenience, throughoutthis paper, we will denote T ⊗ Id X by T too. Thus { T t } t> is also a semigroup of contractions on L p (Ω; X ) for any Banach space X .The one-sided vector-valued extension of (1) was obtained in [15] not for the semigroup { T t } t> itself but for its subordinated Poisson semigroup { P t } t> that is defined by P t f = 1 √ π Z ∞ e − s √ s T t s f ds. { P t } t> is again a symmetric diffusion semigroup. Recall that if A denotes the negative infinitesimalgenerator of { T t } t> , then P t = e −√ A t . Primary: 46B20, 42B25. Secondary: 47B06, 47A35.
Key words:
Analytic semigroups, analytic contractions, Littewood-Paley-Stein inequalities, uniformly convexBanach spaces, martingale type and cotype.
Let 1 < q < ∞ . Recall that a Banach space X is of martingale cotype q if there exists a positiveconstant C such that every finite X -valued L q -martingale ( f n ) defined on some probability spacesatisfies the following inequality X n E (cid:13)(cid:13) f n − f n − (cid:13)(cid:13) qX ≤ C q sup n E (cid:13)(cid:13) f n (cid:13)(cid:13) qX , where E denotes the expectation on the underlying probability space. We then must have q ≥ X is of martingale type q if the reverse inequality holds. It is easy to see that X is of martingalecotype q iff the dual space X ∗ is of martingale type q ′ , where q ′ denotes the conjugate index of q .We refer to [19, 20] for more information.The following is the principal result of [15]. In the sequel, we will use the abbreviation ∂ = ∂/∂t . Theorem 1 (Mart´ınez-Torrea-Xu) . Let < q < ∞ and X be a Banach space. (i) X is of martingale cotype q iff for every < p < ∞ ( or equivalently, for some < p < ∞ ) there exists a constant C such that every subordinated Poisson semigroup { P t } t> as abovesatisfies the following inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t ∂P t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) . (ii) X is of martingale type q iff for for every < p < ∞ ( or equivalently, for some < p < ∞ ) there exists a constant C such that every subordinated Poisson semigroup { P t } t> as abovesatisfies the following inequality (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) ≤ (cid:13)(cid:13) F ( f ) (cid:13)(cid:13) L p (Ω; X ) + C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t ∂P t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , ∀ f ∈ L p (Ω; X ) . Note that the above theorem for the Poisson semigroup of the torus T was first proved in [26].The main problem left open in [15] asks whether the theorem holds for the semigroup { T t } t> itself instead of its subordinated Poisson semigroup { P t } t> (see Problem 2 on page 447 of [15]).Very recently, Hyt¨onen and Naor [8] proved that the answer is affirmative for the heat semigroupof R n and for p = q ; the resulting inequality plays a key role in their work on the approximationof Lipschitz functions by affine maps. Stimulated by their result and using a clever idea of them,we are able to resolve the problem in full generality. Theorem 2.
Let X be a Banach space and k a positive integer. (i) If X is of martingale cotype q with ≤ q < ∞ , then for every symmetric diffusion semigroup { T t } t> and for every < p < ∞ we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) , where C is a constant depending only on p, q, k and the martingale cotype q constant of X . (ii) If X is of martingale type q with < q ≤ , then for every symmetric diffusion semigroup { T t } t> and for every < p < ∞ we have (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) ≤ (cid:13)(cid:13) F ( f ) (cid:13)(cid:13) L p (Ω; X ) + C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , ∀ f ∈ L p (Ω; X ) , where C is a constant depending only on p, q, k and the martingale type q constant of X . Remark 3.
Applied to the heat semigroup { H t } t> of R n , the above theorem implies a dimensionfree estimate for the g -function associated to { H t } t> : (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t∂H t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) L p ( R n ; X ) , ∀ f ∈ L p ( R n ; X )when X is of martingale cotype q . Compare this with [8, Theorem 17] (and the paragraph there-after). ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 3
Remark 4.
Theorem 2 allows one to improve some recent results of Hong and Ma on vector-valuedvariational inequalities associated to symmetric diffusion semigroups. For instance, using it, onecan extend [6, Theorem 5.2] to any Banach space X of martingale cotype q . See also [5] for relatedresults in the Banach lattice case.Theorem 2 admits a discrete analogue. First recall that a power bounded operator R on aBanach space Y is said to be analytic ifsup n ≥ n (cid:13)(cid:13) R n ( R − (cid:13)(cid:13) < ∞ , where the norm is the operator norm on Y . It is known that the analyticity of R is equivalent tosup z ∈ C , | z | > | − z | (cid:13)(cid:13) ( z − R ) − (cid:13)(cid:13) < ∞ . Moreover, if R is analytic, its spectrum σ ( R ) is contained in B γ for some 0 < γ < π/
2, where B γ denotes the Stolz domain which is the interior of the convex hull of 1 and the disc D (0 , sin γ ) (seefigure 1). We refer to [2, 17] for more information. γ B γ Figure 1.
Now consider a symmetric Markovian operator T on (Ω , A , µ ), that is, T satisfies the followingconditions: • T is a linear contraction on L p (Ω) for every 1 ≤ p ≤ ∞ ; • T is positivity preserving and T • T is a selfadjoint operator on L (Ω).With a slight abuse of notation, we use again F to denote the projection on the fixed point subspaceof T . Both T and F extend to contractions on L p (Ω; X ) for any Banach space X . In the followingtwo theorems, T = S with S a symmetric Markovian operator, so T is a symmetric Markovianoperator too, The following is the discrete analogue of a theorem of Pisier [21] for semigroups. Theorem 5.
Let T = S with S a symmetric Markovian operator, < p < ∞ and X be auniformly convex Banach space. Then the extension of T to L p (Ω; X ) is analytic. More precisely,there exist constants C and γ ∈ (0 , π/ , depending only on p and the modulus of uniform convexityof X , such that (2) σ ( T ) ⊂ B γ and (cid:13)(cid:13) ( z − T ) − (cid:13)(cid:13) ≤ C | − z | , ∀ z ∈ C \ B γ . The discrete analogue of Theorem 2 is the following
Theorem 6.
Let T = S be as above and < p < ∞ . (i) If X is of martingale cotype q with ≤ q < ∞ , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q − (cid:13)(cid:13) ( T n − T n − ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) , QUANHUA XU where the constant C depends only on p, q and the martingale cotpye q constant of X . (ii) If X is of martingale type q with < q ≤ , then (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13) F f (cid:13)(cid:13) qX + ∞ X n =1 n q − (cid:13)(cid:13) ( T n − T n − ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , ∀ f ∈ L p (Ω; X ) , where the constant C depends only on p, q and the martingale tpye q constant of X . Remark 7.
If the inequality in Theorem 6 (i) holds for every positive symmetric Markovianoperator T , then the corresponding inequality of Theorem 1 holds for every subordinated Poissonsemigroup { P t } t> . Thus X is of martingale cotype q . Therefore, the validity of the inequality inTheorem 6 (i) characterizes the martingale cotype q of X . A similar remark applies to part (ii). Remark 8.
It is worth to note that all constants involved in the preceding theorems are indepen-dent of the semigroup { T t } t> or contraction T in consideration. They depend only on the indices p, q and the relevant geometric constants of the space X .The preceding three theorems will be proved in the next three sections. The proofs of Theorem 2and Theorem 6 follow the same pattern although the latter one is more involved. The last sectioncontains some open problems.We will use the symbol . to denote an inequality up to a constant factor; all constants willdepend only on X , p, q , etc. but never on the function f in consideration.2. A spectral estimate
This section contains a spectral estimate for positive symmetric Markovian operators. Let X bea uniformly convex Banach space and 1 < p < ∞ . Then Y = L p (Ω; X ) is uniformly convex too.By Pisier’s renorming theorem [19], we can assume that Y is uniformly convex of power type q forsome 2 ≤ q < ∞ , namely,(3) (cid:13)(cid:13)(cid:13)(cid:13) x + y (cid:13)(cid:13)(cid:13)(cid:13) q + δ (cid:13)(cid:13)(cid:13)(cid:13) x − y (cid:13)(cid:13)(cid:13)(cid:13) q ≤ (cid:0) k x k q + k y k q (cid:1) , ∀ x, y ∈ Y for some positive constant δ . Note that the above inequality implies the martingale cotype q of Y .Conversely, if Y is of martingale cotype q , then it admits an equivalent norm which satisfies (3).Let T = S with S a symmetric Markovian operator on (Ω , A , µ ). We extend T to a contractionon Y , still denoted by T . In the following the norm and spectrum of T is taken for T viewed as anoperator on Y . Lemma 9.
Under the above assumptions we have (i) k − T k ≤ min (cid:0) , − δ q ) /q (cid:1) < ; (ii) the spectrum of T is contained in a Stolz domain B γ for some γ ∈ (0 , π/ depending onlyon δ and q in (3) . Part (i) above is already contained in [21] (see, in particular, Remark 1.8 there). In fact, ourproof below is modeled on that of [21, Lemma 1.5]. As in [21], We will need the following one stepversion of Rota’s dilation theorem for positive symmetric Markovian operators. We refer to [24,Chapter IV] for its proof as well as its full version.
Lemma 10 (Rota) . Let T = S with S a symmetric Markovian operator on (Ω , A , µ ) . Then thereexist a larger measure space ( e Ω , e A , e µ ) containing (Ω , A , µ ) , and a σ -subalgebra B of e A such that T f = E A E B f, ∀ f ∈ L p (Ω , A , µ ) , where E A denotes the conditional expectation relative to A (and similarly for E B ).Proof of Lemma 9. Rota’s dilation extends to X -valued functions: T = E A E B (cid:12)(cid:12) Y . Here we have used our usual convention that E A ⊗ Id X and E B ⊗ Id X are abbreviated to E A and E B , respectively. Thus for any λ ∈ C (with P = E B ) λ + T = E A ( λ + P ) (cid:12)(cid:12) Y . ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 5
Let y be a unit vector in Y . Using (3), we get (cid:13)(cid:13)(cid:13)(cid:13) λy + P y (cid:13)(cid:13)(cid:13)(cid:13) q + δ (cid:13)(cid:13)(cid:13)(cid:13) λy − P y (cid:13)(cid:13)(cid:13)(cid:13) q ≤
12 ( | λ | q + 1) . However (noting that P is a contractive projection), (cid:13)(cid:13) λy − P y (cid:13)(cid:13) ≥ | − λ | (cid:13)(cid:13) P y (cid:13)(cid:13) ≥ | − λ | (cid:0)(cid:13)(cid:13) λy + P y (cid:13)(cid:13) − | λ | (cid:1) ≥ | − λ | (cid:0)(cid:13)(cid:13) λy + T y (cid:13)(cid:13) − | λ | (cid:1) . When k λy + T y k approaches k λ + T k , we then deduce(4) (cid:13)(cid:13)(cid:13)(cid:13) λ + T (cid:13)(cid:13)(cid:13)(cid:13) q + δ | − λ | q (cid:13)(cid:13) λ + T (cid:13)(cid:13) − | λ | ! q ≤
12 ( | λ | q + 1) . In particular, for λ = − k − T k q + δ q ( k − T k − q ≤ q , which implies k − T k ≤ min (cid:0) , − δ q ) /q (cid:1) . This is part (i). On the other hand, if λ ∈ σ ( T ), then (4) yields | λ | q + δ | − λ | q | λ | q ≤ (cid:0) | λ | q + 1 (cid:1) , whence | − λ | | λ | ≤ (cid:16) q δ (cid:17) /q (1 − | λ | ) . The last inequality implies (in fact, is equivalent to) that λ ∈ B γ for some γ ∈ (0 , , π/
2) dependingonly on the constant (cid:0) q/ (2 δ ) (cid:1) /q . The proof of the lemma is thus complete. (cid:3) Lemma 9 (i) implies the following result which is [21, Remark 1.8].
Lemma 11.
Let X and p be as above and { T t } t> be a symmetric diffusion semigroup on (Ω , A , µ ) .Then the extension of { T t } t> to Y = L p (Ω; X ) is analytic. Consequently, { t∂T t } t> is a uniformlybounded family of operators on Y , namely, (5) sup t> (cid:13)(cid:13) t∂T t (cid:13)(cid:13) ≤ C, where C is a constant depending only on δ and q in (3) .Proof. Applying Lemma 9 to T = T t , we getsup t> (cid:13)(cid:13) − T t (cid:13)(cid:13) ≤ min (cid:0) , − δ q ) /q (cid:1) < . Then using Kato’s characterization of analytic semigroups in [10], we deduce (5). (cid:3) Proof of Theorem 2
This section is devoted to the proof of Theorem 2. Let us first note that assertion (ii) followseasily from (i) by duality. Indeed, let { e λ } be the resolution of the identity of { T t } t> on L (Ω): T t f = Z ∞ e − λt de λ f, f ∈ L (Ω) . Then ∂ k T t f = ( − k Z ∞ λ k e − λt de λ f. QUANHUA XU
It thus follows that Z Ω Z ∞ (cid:12)(cid:12) t k ∂ k T t f (cid:12)(cid:12) dtt dµ = Z ∞ Z ∞ t k λ k e − λt d h e λ f, f i dtt = 4 − k Z ∞ Z ∞ t k e − t dtt d h e λ f, f i = 4 − k (2 k − Z ∞ d h e λ f, f i = 4 − k (2 k − Z Ω | f − F ( f ) | dµ. By polarization, for f, g ∈ L (Ω) we have Z Ω ( f − F ( f ))( g − F ( g )) dµ = 4 k (2 k − Z Ω Z ∞ (cid:0) t k ∂ k T t f (cid:1)(cid:0) t k ∂ k T t g (cid:1) dtt dµ. We then deduce that for any f ∈ L (Ω) ∩ L ∞ (Ω) ⊗ X and g ∈ L (Ω) ∩ L ∞ (Ω) ⊗ X ∗ Z Ω h g − F ( g ) , f − F ( f ) i dµ = 4 k (2 k − Z Ω Z ∞ h t k ∂ k T t g, t k ∂ k T t f i dtt dµ, where h , i denotes the duality bracket between X and X ∗ . Hence (cid:12)(cid:12)(cid:12)(cid:12)Z Ω h g − F ( g ) , f − F ( f ) i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k (2 k − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t g (cid:13)(cid:13) q ′ X ∗ dtt (cid:19) /q ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ (Ω) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , where r ′ is the conjugate index of r . Under the assumption of (ii) and by duality, we have that X ∗ is of martingale cotype q ′ . Therefore, (i) implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t g (cid:13)(cid:13) q ′ X ∗ dtt (cid:19) /q ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ (Ω) ≤ k C (2 k − (cid:13)(cid:13) g (cid:13)(cid:13) L p ′ (Ω; X ∗ ) . Combining the previous inequalities and taking the supremum over all g in the unit ball of L p ′ (Ω; X ∗ ), we derive assertion (ii).Thus we are left to showing assertion (i). In the rest of this section, we will assume that X is aBanach space of martingale cotype q with 2 ≤ q < ∞ . The following lemma, due to Hyt¨onen andNaor [8, Lemma 24], will play an important role in our argument. Lemma 12 (Hyt¨onen-Naor) . For any f ∈ L q (Ω; X ) we have (cid:18)Z ∞ (cid:13)(cid:13) ( T t − T t ) f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q . (cid:13)(cid:13) f (cid:13)(cid:13) L q (Ω; X ) , ∀ f ∈ L q (Ω; X ) . Based on Rota’s dilation theorem quoted in the previous section, the proof is simple. Below isthe main idea. First write Z ∞ (cid:13)(cid:13) ( T t − T t ) f (cid:13)(cid:13) qL q (Ω; X ) dtt = X k ∈ Z Z k +1 k (cid:13)(cid:13) ( T t − T t ) f (cid:13)(cid:13) qL q (Ω; X ) dtt = Z X k ∈ Z (cid:13)(cid:13) ( T k t − T k +1 t ) f (cid:13)(cid:13) qL q (Ω; X ) dtt . Then Rota’s dilation theorem allows us to turn { T k t − T k +1 t } k for each fixed t into a martingaledifference sequence.The following lemma shows Theorem 2 in the case of p = q . Lemma 13.
Let k be a positive integer. Then (6) (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k T t f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q . (cid:13)(cid:13) f (cid:13)(cid:13) L q (Ω; X ) , ∀ f ∈ L q (Ω; X ) , ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 7 where the relevant constant depends on k and the martingale cotype q constant of X .Proof. We will use the idea of the proof of Theorem 17 of [8]. By virtue of the identity ∂T t + s = ∂T t T s , we write ∂T t f = ∞ X k = − (cid:0) ∂T k +1 t − ∂T k +2 t (cid:1) f = ∞ X k = − ∂T k t (cid:0) T k t − T · k t (cid:1) f. Then by the triangle inequality we get (cid:18)Z ∞ (cid:13)(cid:13) t∂T t f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q ≤ ∞ X k = − (cid:18)Z ∞ (cid:13)(cid:13) t∂T k t (cid:0) T k t − T · k t (cid:1) f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q = ∞ X k = − − k (cid:18)Z ∞ (cid:13)(cid:13) t∂T t (cid:0) T t − T t (cid:1) f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q = 4 (cid:18)Z ∞ (cid:13)(cid:13) t∂T t (cid:0) T t − T t (cid:1) f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q . We are now in a position of using Lemma 11 with p = q . Indeed, since X is of martingalecotype q , so is Y = L q (Ω; X ). Then by [19], Y can be renormalized into a uniformly convex spaceof power type q , that is, Y admits an equivalent norm satisfying (3). Thus we have (5); moreover,the constant C there depends only on q and the martingale cotype q constant of X .Therefore, (cid:13)(cid:13) t∂T t (cid:0) T t − T t (cid:1) f (cid:13)(cid:13) L q (Ω; X ) . (cid:13)(cid:13)(cid:0) T t − T t (cid:1) f (cid:13)(cid:13) L q (Ω; X ) , ∀ t > . Combining the above inequalities together with Lemma 12, we deduce (cid:18)Z ∞ (cid:13)(cid:13) t∂T t f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q . (cid:18)Z ∞ (cid:13)(cid:13)(cid:0) T t − T t (cid:1) f (cid:13)(cid:13) qL q (Ω; X ) dtt (cid:19) /q . (cid:13)(cid:13) f (cid:13)(cid:13) L q (Ω; X ) . This is (6) for k = 1. To handle a general k , by the semigroup identity T t + s = T t T s once more, wehave t k ∂ k T t = k k (cid:18) tk ∂T tk (cid:19) k . Thus, by (5) and the already proved inequality, we obtain Z ∞ (cid:13)(cid:13) t k ∂ k T t f (cid:13)(cid:13) qL q (Ω; X ) dtt = k k Z ∞ (cid:13)(cid:13) ( t ∂T t ) k f (cid:13)(cid:13) qL q (Ω; X ) dtt . Z ∞ (cid:13)(cid:13) t∂T t f (cid:13)(cid:13) qL q (Ω; X ) dtt . (cid:13)(cid:13) f (cid:13)(cid:13) qL q (Ω; X ) . The lemma is thus proved. (cid:3)
To show Theorem 2 for any 1 < p < ∞ , we will use Stein’s complex interpolation machinery.To that end, we will need the fractional integrals. For a (nice) function ϕ on (0 , ∞ ) defineI α ϕ ( t ) = 1Γ( α ) Z t ( t − s ) α − ϕ ( s ) ds, t > . The integral in the right hand side is well defined for any α ∈ C with Re α >
0; moreover, I α ϕ isanalytic in the right half complex plane Re α >
0. Using integration by parts, Stein showed in [24,section III.3] that I α ϕ has an analytic continuation to the whole complex plane, which satisfies thefollowing properties • I α I β ϕ = I α + β ϕ for any α, β ∈ C ; • I ϕ = ϕ ; • I − k = ∂ k ϕ for any positive integer k .We will apply I α to ϕ defined by ϕ ( s ) = T s f for a given function f in L p (Ω; X ) and setM αt f = t − α I α ϕ ( t ) with ϕ ( s ) = T s f. Note that M t f = 1 t Z t T s f ds, M t f = T t f and M − kt f = t k ∂ k T t f for k ∈ N . QUANHUA XU
The following lemma is [15, Theorem 2.3].
Lemma 14.
Let q and X be as in Theorem 2. Then for any < p < ∞ we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t∂ M t f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) . Lemma 15.
Let α and β be complex numbers such that Re α > Re β > − . Then for any positiveinteger k (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M αt f (cid:13)(cid:13) qX dtt (cid:19) /q ≤ Ce π | Im( α − β ) | (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M βt f (cid:13)(cid:13) qX dtt (cid:19) /q on Ω , where C is a constant depending only on Re α and Re β .Proof. Using I α = I α − β I β , we writeM αt f = t − α Γ( α − β ) Z t ( t − s ) α − β − s β M βs f ds = 1Γ( α − β ) Z (1 − s ) α − β − s β M βts f ds. Thus t k ∂ k M αt f = 1Γ( α − β ) Z (1 − s ) α − β − s β ( ts ) k ∂ k M βts f ds, which implies (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M αt f (cid:13)(cid:13) qX dtt (cid:19) /q ≤ | Γ( α − β ) | Z (1 − s ) Re( α − β ) − s Re β ds (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M βt f (cid:13)(cid:13) qX dtt (cid:19) /q . | Γ( α − β ) | (cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M βt f (cid:13)(cid:13) qX dtt (cid:19) /q . Then the desired inequality follows from the following well known estimate on the Γ-function: ∀ x, y ∈ R , | Γ( x + i y ) | ∼ e − π | y | | y | x − as y → ±∞ (see [25, p. 151]). (cid:3) Combining Lemma 14 and Lemma 15 with k = β = 1, we get Lemma 16.
For any < p < ∞ and α ∈ C with Re α > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t∂ M αt f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ Ce π | Im α | (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) , where C depends on Re α , p and the martingale cotype q constant of X . Lemma 17.
For any α ∈ C (7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t∂ M αt f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ Ce π | Im α | (cid:13)(cid:13) f (cid:13)(cid:13) L q (Ω; X ) , ∀ f ∈ L p (Ω; X ) , where C depends on Re α and the martingale cotype q constant of X .Proof. Combining Lemma 13 and Lemma 15 with β = 0, we deduce that for a positive integer k and α ∈ C with Re α > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M αt f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ Ce π | Im α | (cid:13)(cid:13) f (cid:13)(cid:13) L q (Ω; X ) , ∀ f ∈ L q (Ω; X ) , where C depends on k , Re α and the martingale cotype q constant of X . In particular, when k = 1,we get (7) for any α such that Re α > α ∈ C ∂ M αt = − αt − M αt + t − M α − t , we have(9) t k ∂ k M α − t = ( k + α ) t k ∂ k M αt + t k +1 ∂ k +1 M αt . ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 9
This shows that if (8) holds for M α , so does it for M α − instead of M α (with a different constant).Therefore, by what already proved, we deduce that (8) holds for any α ∈ C with Re α > − α ∈ C . In particular for k = 1, we have (7). (cid:3) Now we are ready to show Theorem 2 (i).
Proof of Theorem 2 (i).
We will prove the following more general statement: Under the assumptionof assertion (i), we have for any α ∈ C (10) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) t k ∂ k M αt f (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X ) . Assertion (i) corresponds to (10) for α = 0.Fix α ∈ C . Choose θ ∈ (0 , , r ∈ (1 , ∞ ), α , α ∈ C such that1 p = 1 − θq + θr , α = (1 − θ ) α + θ α , Re α > α = Im α = Im α. Then by the classical complex interpolation on vector-valued L p -spaces (cf. [1]), we have L p (Ω; X ) = (cid:0) L q (Ω; X ) , L r (Ω; X ) (cid:1) θ . Thus for any f ∈ L p (Ω; X ) with norm less than 1 there exists a continuous function F from theclosed strip { z ∈ C : 0 ≤ Re z ≤ } to L q (Ω; X ) + L r (Ω; X ), which is analytic in the interior andsatisfies F ( θ ) = f, sup y ∈ R (cid:13)(cid:13) F (i y ) (cid:13)(cid:13) L q (Ω; X ) < y ∈ R (cid:13)(cid:13) F (1 + i y ) (cid:13)(cid:13) L r (Ω; X ) < . Define F t ( z ) = e z − θ t∂ M (1 − z ) α + zα t F ( z ) . Viewed as a function of z on the strip { z ∈ C : 0 ≤ Re z ≤ } , F takes values in L q (Ω; L q ( R + ; X ))+ L r (Ω; L q ( R + ; X )), where R + is equipped with the measure dtt . By the analyticity of M (1 − z ) α + zα in z , we see that F is analytic in the interior of the strip. Moreover, by Lemma 17 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:13)(cid:13) F t (i y ) (cid:13)(cid:13) qX dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ C ′ e − y − θ e π ( | Im α | + | Re( α − α ) y | ) , ∀ y ∈ R , where C ′ is a constant depending on α, α , α and X . Hencesup y ∈ R kF (i y ) k L q (Ω; L q (( R + , dtt ); X )) ≤ C . Similarly, Lemma 16 implies sup y ∈ R kF (1 + i y ) k L r (Ω; L q (( R + , dtt ); X )) ≤ C . We then deduce that F ( θ ) belongs to the complex interpolation space (cid:0) L q (Ω; L q ( R + ; X )) , L r (Ω; L q ( R + ; X )) (cid:1) θ with norm majorized by C − θ C θ . However, the latter space coincides with L p (Ω; L q ( R + ; X ))isometrically. Since F t ( θ ) = t∂ M αt F ( θ ) = t∂ M αt f, we get (10) for k = 1. Then using (9) and an induction argument, we derive (10) for any k . Thusthe theorem is completely proved. (cid:3) Proofs of Theorem 5 and Theorem 6
The main part of Theorem 5 is already contained in Lemma 9. Armed with that lemma, wecan easily show Theorem 5. Let us first recall the following well known characterization of theanalyticity of power bounded operators (cf. [2, Theorem 2.3] and [17, Theorem 4.5.4]). Let D denote the open unit disc of the complex plane and T the boundary of D . Lemma 18.
Let T be a power bounded linear operator on a Banach space Y . Then T is analyticiff the semigroup { e t ( T − } t> is analytic and σ ( T ) ⊂ D ∪ { } .Proof of Theorem 5. Note that { e t ( T − } t> is a symmetric diffusion semigroup on (Ω , A , µ ). Thus,by Lemma 11, its extension to Y = L p (Ω; X ) is analytic. Then Theorem 5 immediately followsfrom Lemmas 9 and 18. (cid:3) The difficult part ( Lemma 9) of the above proof concerns the quantitative dependence on thegeometry of X of the angle γ of the Stolz domain which contains the spectrum of the operator T .If we only need to show the analyticity of T on Y , the proof can be largely shortened by virtue ofthe following simple fact which, together with Lemma 10, ensures that σ ( T ) ⊂ D ∪ { } . Remark 19.
Let P be a contractive linear projection on a uniformly convex Banach space Y .Then k λ − P k < λ ∈ T \ {− } .This remark is a weaker form of Lemma 9. Let λ ∈ T such that k λ − P k = 2. Choose a sequence { y k } of unit vectors in Y such that k y k − P y k k → k → ∞ . Then the uniform convexity of Y implies k λy k + P y k k →
0. However, | λ + 1 | k P y k k = k P ( λ + P ) y k k ≤ k ( λ + P ) y k k and k P y k k ≥ k λy k − P y k k − → . It thus follows that | λ + 1 | = 0, that is, λ = − X is uniformly convex. Proof of Theorem 6 (ii) . Using the spectral resolution of the identity of T on L (Ω), we obtain (cid:13)(cid:13) f − F ( f ) (cid:13)(cid:13) L (Ω) = ∞ X n =1 n (cid:13)(cid:13) T n − (1 − T ) f (cid:13)(cid:13) L (Ω) , f ∈ L (Ω) . Polarizing this identity, we deduce, for f ∈ L (Ω) ∩ L ∞ (Ω) ⊗ X and g ∈ L (Ω) ∩ L ∞ (Ω) ⊗ X ∗ , that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω h f − F ( f ) , g − F ( g ) i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q ′ − (cid:13)(cid:13) T n − (1 − T ) g (cid:13)(cid:13) q ′ X ∗ ! /q ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ (Ω) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q − (cid:13)(cid:13) T n − (1 − T ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q ′ − (cid:13)(cid:13) T n − (1 − T ) g (cid:13)(cid:13) q ′ X ∗ ! /q ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ (Ω) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q − (cid:13)(cid:13) T n − (1 − T ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . Thus under the assumption of (ii) and admitting (i), we obtain (cid:13)(cid:13) f − F ( f ) (cid:13)(cid:13) L p (Ω; X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q − (cid:13)(cid:13) T n − (1 − T ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . Thus assertion (ii) is proved. (cid:3)
ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 11
We will need some preparations on the H ∞ functional calculus for the proof of Theorem 6 (i).Our reference for the latter subject is [3]. Let A be a sectorial operator on a Banach space Y withangle γ and ω > γ . Define H ∞ (Σ ω ) to be the space of all bounded analytic functions ϕ on thesector Σ ω for which there exist two positive constants s and C such that | ϕ ( z ) | ≤ C min {| z | s , | z | − s } , ∀ z ∈ Σ ω . For any ϕ ∈ H ∞ (Σ ω ), we define ϕ ( A ) = 12 π i Z Γ θ ϕ ( z )( z − A ) − dz, where θ ∈ ( γ, ω ) and Γ θ is the boundary ∂ Σ θ oriented counterclockwise. Then ϕ ( A ) is a boundedoperator on Y .The following result is a variant of [16, Theorem 5]. The proof there works equally for thepresent setting without change. This was pointed to us by Christian Le Merdy (see [13, page 719]). Lemma 20.
Let < q < ∞ and ϕ, ψ ∈ H ∞ (Σ ω ) with Z ∞ ψ ( t ) dtt = 0 . Then there exists a positive constant C , depending only on ϕ, ψ and q , such that (cid:18)Z ∞ (cid:13)(cid:13) ϕ ( tA ) y (cid:13)(cid:13) q dtt (cid:19) /q ≤ C (cid:18)Z ∞ (cid:13)(cid:13) ψ ( tA ) y (cid:13)(cid:13) q dtt (cid:19) /q , ∀ y ∈ Y. Proof of Theorem 6 (i) . We will follow the pattern set up in the proof of Theorem 2. The maindifficulty is to prove the following discrete analogue of Lemma 13:(11) ∞ X n =1 n q − (cid:13)(cid:13) T n ( T − f (cid:13)(cid:13) qL q (Ω; X ) . (cid:13)(cid:13) f (cid:13)(cid:13) qL q (Ω; X ) , ∀ f ∈ L q (Ω; X ) . Contrary to Lemma 13, the proof of the above inequality is much more involved. We will adaptthe proof of [14, Proposition 3.2] which is based on the H ∞ functional calculus.By Theorem 5, T is analytic as an operator on Y = L q (Ω; X ) and we have (2). Let A = 1 − T .Then A is a sectorial operator on Y with angle γ . Fix θ ∈ ( γ, π/ L θ be the boundary of1 − B θ oriented counterclockwise (see figure 2). θ L θ σ ( A )cos( θ ) e i θ Figure 2.
Let ϕ n ( z ) = n /q ′ z (1 − z ) n . Then by the Dunford functional calculus12 π i Z L θ ϕ n ( z )( z − A ) − dz = ϕ n ( A ) and 12 π i Z L θ ϕ n ( z )( z + A ) − dz = 0 . Thus n /q ′ T n (1 − T ) = ϕ n ( A ) = 1 π i Z L θ ϕ n ( z ) A ( z − A ) − ( z + A ) − dz . Fix f in the unit ball of Y . Then ∞ X n =1 n q − (cid:13)(cid:13) T n ( T − f (cid:13)(cid:13) qY . Z L θ ∞ X n =1 | ϕ n ( z ) | q (cid:13)(cid:13) A ( z − A ) − ( z + A ) − f (cid:13)(cid:13) qY | dz | . Note that for any z ∈ L θ , an elementary calculation shows that ∞ X n =1 | ϕ n ( z ) | q ≤ sup λ ∈ B θ ∞ X n =1 n q − | λ | nq | − λ | q . sup λ ∈ B θ | − λ | q (1 − | λ | ) q . , where the relevant constants depend only on q and θ . On the other hand, by the H ∞ functionalcalculus, A /q ( z + A ) − is a bounded operator on Y . Then we deduce ∞ X n =1 n q − (cid:13)(cid:13) T n ( T − f (cid:13)(cid:13) qY . Z L θ (cid:13)(cid:13) A /q ′ ( z − A ) − f (cid:13)(cid:13) qY | dz | . The contour L θ is the juxtaposition of a part L θ, of Γ θ (recalling that Γ θ is the boundary of thesector Σ θ ) and the curve L θ, going from cos( θ ) e − i θ to cos( θ ) e i θ counterclockwise along the circleof center 1 and radius sin θ . Accordingly, Z L θ (cid:13)(cid:13) A /q ′ ( z − A ) − f (cid:13)(cid:13) qY | dz | = Z L θ, (cid:13)(cid:13) A /q ′ ( z − A ) − f (cid:13)(cid:13) qY | dz | + Z L θ, (cid:13)(cid:13) A /q ′ ( z − A ) − f (cid:13)(cid:13) qY | dz | . Since L θ, ∩ σ ( A ) = ∅ , the function z
7→ k A /q ′ ( z − A ) − k is bounded on L θ, . Thus the secondintegral in the right hand side above is majorized by a constant independent of f (recalling that k f k Y ≤ Z L θ, (cid:13)(cid:13) A /q ′ ( z − A ) − f (cid:13)(cid:13) qY | dz | ≤ X ε = ± Z ∞ (cid:13)(cid:13) A /q ′ ( te ε i θ − A ) − f (cid:13)(cid:13) qY dt = X ε = ± Z ∞ (cid:13)(cid:13) ( tA ) /q ′ ( e ε i θ − tA ) − f (cid:13)(cid:13) qY dtt = X ε = ± Z ∞ (cid:13)(cid:13) ϕ ε ( tA ) f (cid:13)(cid:13) qY dtt , where ϕ ε ( z ) = z /q ′ e ε i θ − z , ε = ± . Note that ϕ ε ∈ H ∞ (Σ ω ) for ω ∈ ( θ, π/ ψ defined by ψ ( z ) = ze − z belongs to H ∞ (Σ ω ) too. Thus applying Lemma 20, we get Z ∞ (cid:13)(cid:13) ϕ ε ( tA ) f (cid:13)(cid:13) qY dtt . Z ∞ (cid:13)(cid:13) ψ ( tA ) f (cid:13)(cid:13) qY dtt = Z ∞ (cid:13)(cid:13) t ∂T t f (cid:13)(cid:13) qY dtt , where { T t } t> = { e − tA } t> is the semigroup already used at the beginning of the proof of Theo-rem 5. Thus by Lemma 13, Z ∞ (cid:13)(cid:13) ψ ( tA ) f (cid:13)(cid:13) qY dtt . (cid:13)(cid:13) f (cid:13)(cid:13) Y . . Combining all preceding inequalities, we finally get ∞ X n =1 n q − (cid:13)(cid:13) T n ( T − f (cid:13)(cid:13) qL q (Ω; X ) . f in the unit ball of Y . This yields (11) by homogeneity.Armed with (11), we can finish the proof of Theorem 6 (i) by Stein’s complex interpolationmachinery as in the continuous case. To that end, first recall that Lemma 14 is deduced byapproximation from its discrete analogue in [15]. Thus, although not explicitly stated there, thediscrete analogue of Lemma 14 is indeed obtained during the proof of [15, Theorem 2.3]. Then theinterpolation arguments in the previous section can be modified to the present discrete setting. We ECTOR-VALUED LITTEWOOD-PALEY-STEIN THEORY 13 refer the reader to [23] for the necessary ingredients. However, note that the presentation of [23]is quite brief, it is developed in detail in [9]. We leave the details to the reader. Thus the proof ofTheorem 6 is complete. (cid:3) Open problems
We conclude this article by some open problems. The first one concerns Theorem 6. Note thatin that theorem the contraction T is assumed to be the square of another symmetric Markovianoperator. Compared with the continuous case, this assumption is natural since every operator ina symmetric diffusion semigroup is automatically the square of a symmetric Markovian operator.However, a less restrictive assumption would be that T is a selfadjoint contraction on L (Ω) andits spectrum does not contain −
1. Under this assumption, T is analytic. If in addition T is acontraction on L p (Ω) for every 1 ≤ p ≤ ∞ , then T is also analytic on L p (Ω) for every 1 < p < ∞ . Problem 21.
Let T be a positive contraction on L p (Ω) for every ≤ p ≤ ∞ with T . Assumethat T is selfadjoint on L (Ω) and its spectrum does not contain − . (i) Let X be a uniformly convex Banach space. Is the extension of T to L p (Ω; X ) analytic forevery < p < ∞ ( or equivalently, for one < p < ∞ ) ? (ii) Let X be a Banach space of martingale cotype q and < p < ∞ . Does one have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n q − (cid:13)(cid:13) ( T n − T n − ) f (cid:13)(cid:13) qX ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X )?An affirmative answer to part (i) would imply the same for part (ii). In the spirit of [21], onecan ask a similar question as part (i) for K-convex X . In fact, we do not know whether Theorem 5holds for K-convex targets (see [12] for related results). This is the discrete analogue of Problem 11(i) of [26] for symmetric diffusion semigroups. Problem 22.
Does Theorem 5 remain true if X is assumed K-convex? Remark 23.
The answers to Problem 21 (i) and Problem 22 are both positive if X is a complexinterpolation space between a Hilbert space and a Banach space.This is the case if X is a K-convexBanach lattice thanks to [22]. More generally, let ( X , X ) be a compatible pair of Banach spaces,and let X = ( X , X ) θ with 0 < θ <
1. Assume that T is a contraction on both X and X , and T is analytic on X . Then T is analytic on X too.Indeed, since the semigroup { e ( T − t } t> is analytic on X , by Stein’s complex interpolation,it is analytic on X too. Thus by Lemma 18, it remains to show that as an operator on X , thespectrum of T intersects T at most at the point 1. The latter is equivalent to lim n →∞ k T n ( T −
1) : X → X k = 0, thanks to Katznelson and Tzafriri’s theorem [11]. Using the analyticity of T on X and interpolation, we get k T n ( T −
1) : X → X k . n θ . So we are done.Hyt¨onen [7] studied another variant of Stein’s inequality (1) in the vector-valued setting. Like[15], his main theorem deals with the Poisson semigroup subordinated to a symmetric diffusionsemigroup for a general UMD space X , except when X is a complex interpolation space betweena Hilbert space and another UMD space. In the same spirit of this article, one may ask whetherthe main result of [7] remains true for any symmetric diffusion semigroup and any UMD space X .It is easier to formulate this problem in the discrete case as follows. Let T be as in Theorem 6. Problem 24.
Let T be as in Theorem 6, X be a UMD space and < p < ∞ . Does one have E (cid:13)(cid:13) ∞ X n =1 ε n √ n ( T n − T n − ) f (cid:13)(cid:13) L p (Ω; X ) ≈ (cid:13)(cid:13) f − F ( f ) (cid:13)(cid:13) L p (Ω; X ) , ∀ f ∈ L p (Ω; X )?Here { ε n } is a sequence of symmetric random variables taking values ± E is the corresponding expectation. Acknowledgements.
This work is partially supported by NSFC grants No. 11301401 and theFrench project ISITE-BFC (ANR-15-IDEX-03) and IUF.
References [1] J. Bergh, and J. L¨ofstr¨om.
Interpolation spaces.
Springer-Verlag, Berlin, 1976.[2] S. Blunck Maximal regularity of discrete and continuous time evolution equations.
Studia Math.
146 (2001),157-176.[3] M. Cowling, I. Doust, A. McIntosh, and A. Yagi. Banach space operators with a bounded H ∞ functionalcalculus. J. Aust. Math. Soc.
60 (1996), 51-89.[4] N. Dunford, and J.T. Schwartz.
Linear Operators. I. General Theory . Applied Mathematics, Vol. 7. IntersciencePublishers, Inc., New York, 1958.[5] G. Hong, and T. Ma. Vector valued q -variation for ergodic averages and analytic semigroups. J. Math. Anal.Appl.
437 (2016) 1084-1100.[6] G. Hong, and T. Ma. Vector valued q -variation for differential operators and semigroups I. Math. Z.
286 (2017)89-120.[7] T.P. Hyt¨onen. Littlewood-Paley-Stein theory for semigroups in UMD spaces.
Rev. Mat. Iberoamericana
J. Euro. Math. Soc.
To appear.[9] M. Junge, and Q. Xu. Noncommutative maximal ergodic inequalities.
J. Amer. Math. Soc.
20 (2007), 385-439.[10] T. Kato. A characterization of holomorphic semigroups.
Proc. Amer. Math. Soc.
25 (1970), 495-498.[11] Y. Katznelson, and L. Tzafriri. On power bounded operators.
J. Funct. Anal.
68 (1986), 13-328.[12] F. Lancien, and C. Le Merdy. The Ritt property of subordinated operators in the group case
J. Math. Anal.Appli.
462 (2018),191-209.[13] C. Le Merdy. The Weiss conjecture for bounded analytic semigroups.
J. London Math. Soc.
67 (2003) 715-738.[14] C. Le Merdy, and Q. Xu. Maximal theorems and square functions for analytic operators on L p -spaces. J. LondonMath. Soc.
86 (2012), 343-365.[15] T. Mart´ınez, J. L. Torrea, and Q. Xu. Vector-valued Littlewood-Paley-Stein theory for semigroups.
Adv. Math.
203 (2006), 430-475.[16] A. McIntosh, and A. Yagi. Operators of type ω without a bounded H ∞ functional calculus. Miniconference onOperators in Analysis, Proceedings of the Centre for Mathematics and its Applications
24 (Australian NationalUniversity, Canberra, 1989) 159-172.[17] O. Nevanlinna.
Convergence of iterations for linear equations.
Birk¨auser, Basel,1993.[18] A. Pazy.
Semigroups of linear operators and applications to partial differential equations.
Springer-Verlag NewYork, 1983.[19] G. Pisier. Martingales with values in uniformly convex spaces,
Israel J. Math. , 20 (1975), 326-350.[20] G. Pisier. Probabilistic methods in the geometry of Banach spaces.
Springer Lect. Notes in Math.
Ann. Math.
115 (1982), 375-392.[22] G. Pisier. Some applications of the complex interpolation method to Banach lattices.
J. Analyse Math.
Proc. Nat. Acad. Sci. U.S.A.
47 (1961), 1894-1897.[24] E.M. Stein.
Topics in harmonic analysis related to the Littlewood-Paley theory.
Ann. Math. Studies, Princeton,University Press, 1970.[25] E.C. Titchmarsh.
The theory of functions.
Oxford University Press, 1939.[26] Q. Xu. Littlewood-Paley theory for functions with values in uniformly convex spaces.
J. Reine Angew. Math.
504 (1998), 195-226.
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China;and Laboratoire de Math´ematiques, Universit´e de Bourgogne Franche-Comt´e, 25030 Besanc¸on Cedex,France; and Institut Universitaire de France
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