Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups

We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions $\{T_t\}_{t>0}$ on $L_p(\Omega)$ for a fixed $1<p<\infty$. We prove that if a Banach space $X$ is of martingale cotype $q$, then there is a constant $C$ such that $$ \left\|\left(\int_0^\infty\big\|t\frac{\partial}{\partial t}P_t (f)\big\|_X^q\,\frac{dt}t\right)^{\frac1q}\right\|_{L_p(\Omega)}\le C\, \big\|f\big\|_{L_p(\Omega; X)}\,, \quad\forall\, f\in L_p(\Omega; X),$$ where $\{P_t\}_{t>0}$ is the Poisson semigroup subordinated to $\{T_t\}_{t>0}$. Let $\mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $\mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$\mathsf{L}^{P}_{c,q, p}(X)\lesssim \max\big(p^{\frac1{q}},\, p'\big) \mathsf{M}_{c,q}(X).$$ Moreover, the order $\max\big(p^{\frac1{q}},\, p'\big)$ is optimal as $p\to1$ and $p\to\infty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally $\{T_t\}_{t>0}$ is analytic on $L_p(\Omega; X)$, the semigroup $\{P_t\}_{t>0}$ in these results can be replaced by $\{T_t\}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one is more powerful in several aspects: a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; b) it yields the optimal orders of growth on $p$ for most of the relevant constants; c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood-Paley-Stein inequalities for symmetric submarkovian semigroups are better than the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and $\{P_t\}_{t>0}$ is the classical Poisson and heat semigroups on $\mathbb{R}^d$.

Lp(Ω) where {Pt} t>0 is the Poisson semigroup subordinated to {Tt} t>0 .Let L P c,q,p (X) be the least constant C, and let Mc,q(X) be the martingale cotype q constant of X.We show L P c,q,p (X) max p

Introduction
This article pursues our investigation on the vector-valued Littlewood-Paley-Stein theory that was initiated in [64] and further carried out in [41,65,66].Our research in this domain has been profoundly influenced by Stein's monograph [55] and developed in two parallel directions.On the one hand, it deals with the Banach space valued case as in the just quoted articles as well as in the present one; and on the other hand, it extends Littlewood-Paley-Stein theory to the noncommutative setting (see [30,38] for maximal function inequalities and [29] for square function inequalities).
The last condition is the markovianity; the next to last is the symmetry.Thus such a semigroup is also called a symmetric markovian semigroup.A semigroup satisfying all the above conditions except markovianity is usually called a symmetric submarkovian semigroup (the submarkovianity means T t (1) ≤ 1).
It is a classical fact that the orthogonal projection F from L 2 (Ω) onto the fixed point subspace of {T t } t>0 extends to a contractive projection on L p (Ω) for every 1 ≤ p ≤ ∞.Then F is also positive and F L p (Ω) is the fixed point subspace of {T t } t>0 on L p (Ω).
The classical inequality corresponds to the case where {T t } t>0 is the Poisson semigroup on the torus T or the Euclidean space R d .Stein's inequality above is the core of [55] in which Stein developed a beautiful general theory.Later, Cowling [12] presented an elegant alternative approach to Stein's theory for symmetric submarkovian semigroups; Cowling's goal is to show that the negative generator of {T t } t>0 has a bounded holomorphic functional calculus, then to deduce the maximal inequality on {T t } t>0 which is another fundamental result of Stein.
In the present article we are concerned with the vector-valued case.Given a Banach space X let L p (Ω; X) denote the 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, throughout this article, we will denote T ⊗ Id X by T too.Thus {T t } t>0 is also a semigroup of contractions on L p (Ω; X) for any Banach space X with F L p (Ω; X) as its fixed point subspace.
The vector-valued Littlewood-Paley-Stein theory consists in investigating (1.1) for f ∈ L p (Ω; X) (with the absolute value on the right hand side replaced by the norm of X).It is not hard to show that the equivalence (1.1) continues to hold in the X-valued setting for the Poisson semigroup on T iff X is isomorphic to a Hilbert space (cf.[20,64]).However, if one requires only the validity of one of the two one-sided inequalities, the corresponding family of Banach spaces is much larger: the upper estimate corresponds to 2-uniformly smooth spaces while the lower one to 2-uniformly convex spaces (up to a renorming).
These geometrical properties of Banach spaces can be characterized by martingale inequalities.Recall that a Banach space X is of martingale cotype q (with 2 ≤ q < ∞) if there exists a positive constant c such that every finite X-valued L q -martingale (f n ) satisfies the following inequality where E denotes the underlying expectation.X is of martingale type q (with 1 < q ≤ 2) if the reverse inequality holds (with c −1 in place of c).The corresponding best constant will be denoted by M c,q (X) for the martingale cotype q and by M t,q (X) for the martingale type q.Pisier's famous renorming theorem asserts that X is of martingale type (resp.cotype) q iff X admits an equivalent norm that is q-uniformly smooth (resp.convex).We refer the reader to [51,53,54] for more information.
Note that in the study of one-sided inequalities in the vector-valued case, the index 2 on the right hand side of (1.1) plays no special role and can be replaced by 1 < q < ∞, q ≤ 2 for the upper estimate and q ≥ 2 for the lower.Now we can summarize the main results of [64,41,66] as follows.
Theorem A. Let X be a Banach space and 1 < q < ∞.
(i) X is of martingale cotype q iff for every symmetric diffusion semigroup {T t } t>0 and for every 1 < p < ∞ (equivalently, for some 1 < p < ∞) there exists a constant c such that Lp(Ω) ≤ c f Lp(Ω;X) , f ∈ L p (Ω; X).
(ii) X is of martingale type q iff for every symmetric diffusion semigroup {T t } t>0 and for every 1 < p < ∞ (equivalently, for some 1 < p < ∞) there exists a constant c such that Lp(Ω) , f ∈ L p (Ω; X).
Note that F(f ) does not contribute to the norm on the left hand side of the inequality in (i) above since ∂ ∂t T t (F(f )) = 0 for any t > 0; so if this inequality holds, it automatically holds with f replaced by f − F(f ) on the right hand side.In the sequel, when cotype inequalities will be considered, we will often use simply f instead f − F(f ) as in (i).However, for the type inequalities as in (ii), we must use f − F(f ) on the left hand side.
Both "if" parts in the above theorem are proved in [64]; for that purpose we need only the case where {T t } t>0 is the usual Poisson semigroup on T (or R d as in [41]).This is the easy direction thanks to the classical link between Poisson integral and Brownian motion.The other direction is harder.It is first proved in [64] for the Poisson semigroup on the unit circle, then in [41] for the Poisson semigroup subordinated to any symmetric diffusion semigroup {T t } t>0 .Left as an open problem in [41], the statement for {T t } t>0 itself as above was finally settled in [66].Note that like in [55], the key tool in [41,66] is Rota's martingale dilation of a symmetric diffusion semigroup that allows us to adapt the scalar Littlewood-Paley-Stein theory developed in [55].
The use of Rota's dilation prevented us from weakening the assumption on a symmetric diffusion semigroup.Cowling's approach in [12] does not use Rota's dilation but it requires the semigroup in consideration to be symmetric and submarkovian.It has been an open problem of establishing the results of [41] or [66] in Cowling's setting.In fact, since a long time it has been a desire to extend all previous results to more general semigroups.This was done in some special cases (Hermite, Laguerre and Bessel semigroups) by Betancor and coauthors (cf.[7,8,9,10]).
The objective of the present article is to resolve the above problems.We will develop a vectorvalued Littlewood-Paley-Stein theory for semigroups of regular operators on L p (Ω) for a single 1 < p < ∞, thereby going considerably beyond Stein-Cowling's setting.
Recall that an operator T on L p (Ω) (1 ≤ p ≤ ∞) is regular if there exists a constant c such that for all finite sequences {f k } k≥1 in L p (Ω).The least constant c is called the regular norm of T .Obviously, any positive operator is regular with regular norm equal to its operator norm.It is well known that, conversely, if T is regular, then there exists a positive operator S on L p (Ω) such that |T (f )| ≤ S(|f |) for any f ∈ L p (Ω) with S equal to the regular norm of T ; such a positive S is unique and called the absolute value of T and denoted by |T | (see [43,Chapter 1]).For presentation simplicity, in this article we will only consider contractively regular operators, i.e., those with regular norms less than or equal to 1, and will simply call these operators as regular operators with a light abuse of terminology.
It is well known (and easy to check) that if T is a contraction on L p (Ω) for every 1 ≤ p ≤ ∞, then T is regular on L p (Ω).Like positive operators, a regular operator T extends to a contraction on L p (Ω; X) for any Banach space X.This extension will be denoted by T too.Now let {T t } t>0 be a strongly continuous semigroup of regular operators on L p (Ω) with 1 < p < ∞.Extended to L p (Ω; X), {T t } t>0 remains to be a strongly continuous semigroup of contractions on L p (Ω; X).Let again F be the projection from L p (Ω) onto the fixed point subspace of {T t } t>0 .Then F is also regular, so extends to a contractive projection on L p (Ω; X).Note that F(L p (Ω; X)) coincides with the fixed point subspace of {T t } t>0 on L p (Ω; X).
Recall that if A denotes the negative infinitesimal generator of {T t } t>0 (i.e., T t = e −tA ), then A .Instead of the square root, one can, of course, consider other subordinated semigroups e −tA α for 0 < α < 1; but we will not deal with the latter here.
To proceed further, we need to introduce some notions.Define for f in the definition domain of A in L p (Ω; X).X is said to be of Luzin cotype q relative to {T t } t>0 if there exists a constant c such that for all f as above.The smallest c is denoted by L T c,q,p (X).Similarly, we define the Luzin type q of X by reversing the above inequality and changing c to c −1 , the corresponding type q constant is denoted by L T t,q,p (X).See section 4 below for more information.
Remark 1.1.The subordination formula (1.2) immediately implies the pointwise inequality G P q (f ) ≤ C G T q (f ) for any f , where C is an absolute positive constant.It then follows that L T c,q,p (X) ≥ C L P c,q,p (X) and L T t,q,p (X) ≤ C L P t,q,p (X).In [64], the Luzin type and cotype relative to the Poisson semigroup on the unit circle are shown to be equivalent to the martingale type and cotype, respectively.Theorem A above extends this to symmetric diffusion semigroups.
In the sequel, we will use the following convention: A B (resp.A ε B) means that A ≤ CB (resp.A ≤ C ε B) for some absolute positive constant C (resp. a positive constant C ε depending only on ε).A ≈ B or A ≈ ε B means that these inequalities as well as their inverses hold.The index p will be assumed to satisfy 1 < p < ∞ and p ′ will denote its conjugate index.
Below is our first principal result.
Theorem 1.2.Let X be a Banach space and 1 < p, q < ∞.Let {T t } t>0 be a strongly continuous semigroup of regular operators on L p (Ω) and {P t } t>0 its subordinated Poisson semigroup.
(i) If X is of martingale cotype q, then X is of Luzin cotype q relative to {P t } t>0 and L P c,q,p (X) max p 1 q , p ′ M c,q (X).
(ii) If X is of martingale type q, then X is of Luzin type q relative to {P t } t>0 and The two above inequalities can be reformulated in another (clearer) way, for instance, the first one reads as Remark 1.3.All the growth orders, except the one on L P t,q,p (X) as p → 1, are optimal since they are already so in the scalar case X = C.More precisely, (i) L P c,q,p (C) max p 1 q , p ′ for all 1 < p < ∞ when {P t } t>0 is the classical Poisson semigroup on R (see Proposition 8.5 below); (ii) L P t,q,p (C) p as p → ∞ when {P t } t>0 is the Poisson semigroup subordinated to a symmetric diffusion semigroup {T t } t>0 , as shown by Zhendong Xu and Hao Zhang [68]; in fact, they proved the stronger inequality L T t,q,p (C) p as p → ∞ for a symmetric diffusion semigroup {T t } t>0 .
Part (i) of the above theorem cannot hold for the semigroup {T t } t>0 itself without additional assumption (see Remark 4.4 below).It turns out that the missing condition is the analyticity of , where B(Y ) denotes the space of bounded linear operators on a Banach space Y .In this case, Theorem 1.4.Let X and p, q be as above.
(i) If X is of martingale type q, then X is of Luzin type q relative to {T t } t>0 and (ii) Assume additionally that {T t } t>0 satisfies (1.3).Let β q = β 0 min( p q , p ′ q ′ ).If X is of martingale cotype q, then X is of Luzin cotype q relative to {T t } t>0 and The two previous theorems considerably improve Theorem A. Firstly, the semigroup {T t } t>0 now acts on L p (Ω) for a single p.Secondly, the markovianity or submarkovianity is not assumed (in fact, T t (1) is even not defined if the measure on Ω is infinite).Thirdly, the symmetry is not needed either since the semigroup does not act on L 2 (Ω) if p = 2.
Another improvement concerns the precise estimates on the best constants, the present estimates are much better than all previously known ones, even in the scalar case (see section 8 below for historical comments).Moreover, except one case, they give the optimal orders of growth as p → 1 and as p → ∞, as already pointed out in Remark 1.3.This is perhaps a major novelty of our method.
The aforementioned optimality allows us to answer a question raised by Naor and Young about the optimal orders of L P c,q,p (X) and L T c,q,p (X) when {T t } t>0 is the heat semigroup on R d (see the appendix of [47]).In fact, we will show a much stronger result.Let ϕ : R d → C be an integrable function satisfying for some positive constants ε and δ.
We will also need ϕ to be nondegenerate in the sense that there exists another function ψ satisfying (1.4) such that This nondegeneracy allows us to use the Calderón reproducing formula.There exist plenty of functions satisfying these conditions, for instance, the kernel of t ∂ ∂t T t , where {T t } t>0 is either the heat or Poisson semigroup on R d , as well as any Schwartz function ϕ verifying that for any ξ ∈ R d \ {0} there is t > 0 such that ϕ(tξ) = 0.
Similarly, we introduce L ϕ t,q,p (X) for the reverse inequality (with c −1 instead of c).
(i) If X is of martingale cotype q, then L ϕ c,q,p (X) d,ε,δ max p 1 q , p ′ M c,q (X).
(ii) Assume additionally that ϕ is nondegenerate.If X is of martingale type q, then Let {H t } t>0 be the classical heat semigroup on R d whose convolution kernel is given by Its subordinated Poisson semigroup is the usual Poisson semigroup {P t } t>0 with the convolution kernel The above theorem implies the following corollary that resolves Naor and Young's problem.
Corollary 1.6.We have Moreover, L H c,q,p (X) L P c,q,p (C) max p 1 q , p ′ and L H c,q,q (X) L P c,q,q (X) M c,q (X).
Remark 1.7.It is worth to point out that the estimate on L P c,q,p (X) is independent of d thanks to Theorem 1.2 (i).It would be interesting to have a dimension free estimate for the heat semigroup too.This is related to another problem, whether the analyticity constant of {H t } t>0 on L p (R d ; X) relative to an appropriate angle can be controlled by a dimension free constant (see Example A.2 below).
Problem 1.8.(i) Does the second inequality in the first part of Corollary 1.6 hold with a constant independent of the dimension d?
(ii) It would be also interesting to determine the optimal orders of L P t,q,p (X) and L H t,q,p (X) as p → 1 or p → ∞.
Note that (i) above remains open even for X = C (see Problem 7 in [67]).It is also so for (ii) as p → ∞ (see section 8 below and [67] for more information).
Apart from the inequality G P q (f ) Lp(Ω) ≤ L P c,q,p (X) f Lp(Ω;X) , the following variant is also useful: when X is of martingale cotype q (see, for instance, [34]).Inequalities of this type are less delicate than the previous ones.It is well known that if X is of martingale cotype q, then L p (Ω; X) is of martingale cotype max(p, q), so the above inequality can hold only for r = max(p, q).We can, of course, consider similar variants in the situation of Theorem 1.4 and Theorem 1.5 as well as their reverse inequalities when X is of martingale type q; but we will concentrate on the above inequality and on the case 1 < p ≤ q for illustration.Theorem 1.2 (i) easily implies the following Corollary 1.9.Let {T t } t>0 and {P t } t>0 be as in Theorem 1.2.Assume that X is of martingale cotype q and 1 < p ≤ q.
for all f ∈ L p (Ω; X).Moreover, the constant on the right hand side is optimal as p → 1.
Like the martingale type and cotype, the Luzin type and cotype behave well with duality as shown by Theorem 4.5 below.This duality theorem allows us to deduce the type case from the cotype case.In contrast with the martingale case, the proof of Theorem 4.5 is much harder and depends on a bounded projection on a certain vector-valued radial tent space.Let R + be equipped with the measure dt t .The radial space is L p (Ω; L q (R + ; X)) whose elements h are functions of two variables ω ∈ Ω and t ∈ R + , i.e., h : (ω, t) → h t (ω).The desired projection maps L p (Ω; L q (R + ; X)) onto the subspace of all h of the form h t = t ∂ ∂t T t (f ) for some f ∈ L p (Ω; X); formally, it is given by Here the expression ∂ ∂t T t (h t ) is interpreted as ∂ ∂t T t (f ) with f = h t .Note that T (h) is well-defined for nice functions h ∈ L p (Ω; L q (R + ; X)), for instance, for all compactly supported continuous functions from R + to the definition domain of the generator of {T t } t>0 in L p (Ω; X).Similarly, we define P associated to the subordinated Poisson semigroup {P t } t>0 .
The following is the key result for the duality argument.
Theorem 1.10.Let {T t } t>0 be a strongly continuous semigroup of regular operators on L p (Ω) and {P t } t>0 its subordinated Poisson semigroup.Let X be a Banach space and 1 < p < ∞.
(i) The map P extends to a bounded projection on L p (Ω; L q (R + ; X)) with norm majorized by q ′ ) for any 1 ≤ q ≤ ∞. (ii) Assume additionally that 1 < q < ∞ and {T t } t>0 satisfies (1.3) for some 0 < β 0 ≤ π 2 .Then T extends to a bounded projection on L p (Ω; L q (R + ; X)) with norm majorized by It is remarkable that the first part of the theorem above holds for any Banach space X and any subordinated Poisson semigroup.Under the stronger assumption that {T t } t>0 be a symmetric diffusion semigroup, assertion (i) above is [41,Theorem 3.2].However, the proof in [41] contains a gap which lies in the reduction of Theorem 3.2 to Lemma 3.3 in [41] via Rota's dilation theorem.Recall that if {P t } t>0 is the Poisson semigroup on the torus, assertion (i) above was proved in [64] by using Calderón-Zygmund singular integral theory.
Remark 1.11.The analyticity assumption in Theorem 1.4 and Theorem 1.10 (ii) is unremovable.Recall that {T t } t>0 is analytic on L p (Ω; X) with 1 < p < ∞ in one of the following cases: • {T t } t>0 is a symmetric diffusion semigroup and X is superreflexive ( [52]); • {T t } t>0 is a convolution semigroup induced by symmetric probability measures on a locally compact abelian group and X is K-convex ( [52]); • {T t } t>0 is an analytic semigroup of regular operators on L p (Ω) and X is θ-Hilbertian, i.e., a complex interpolation space of a Hilbert space and another Banach space ( [65]).Many classical semigroups are analytic on L p (Ω; X) for any X (see Appendix A below).On the other hand, {T t } t>0 is analytic on L p (Ω; X) iff its adjoint semigroup {T * t } t>0 is analytic on L p ′ (Ω; X * ).Thus the class of Banach spaces X such that {T t } t>0 is analytic on L p (Ω; X) is stable under the passage to duals, subspaces and quotient spaces.
On the other hand, it is well known that the subordinated Poisson semigroup {P t } t>0 is always analytic on L p (Ω; X) for any Banach space X since its negative generator is a sectorial operator of type π 4 (see section 2 for more information).A summary of the main techniques and the contents seems to be in order.Our approach is different from all the previous ones.It is based on holomorphic functional calculus, which constitutes perhaps one of the major ideas of this article.In this regard, it shares some common points with Cowling's approach that deals with the bounded H ∞ functional calculus of the generator of a symmetric submarkovian semigroup and the related maximal inequality.We need, however, to adapt McIntosh's H ∞ functional calculus for our purpose.This is done in the preparatory section 2 in which we introduce a key notion of the article: the ℓ q -boundedness of a family of operators on L p (Ω; X); it gives rise to the definitions of ℓ q -sectorial operators and ℓ q -analytic semigroups.We transfer to this setting some well known results about sectorial operators and analytic semigroups.
After this preparation, we prove Theorem 1.10 in section 3.This projection theorem is a crucial ingredient for the duality studied in section 4. Theorem 4.5 establishes our duality result between the Luzin cotype of X and the Luzin type of the dual space X * ; this result is as nice as the corresponding one in the martingale case, except the links between the involved constants (compare the constants in Theorem 4.5 and those in (6.2) below).This section also contains some general properties of the Luzin type and cotype, in particular, a characterization by lacunary discrete differences (Theorem 4.7).As Rota's dilation is no longer available in the present situation, we use instead Fendler's dilation for semigroups of regular operators.Fendler's theorem transfers Theorem 1.2 (i) to the special case where {T t } t>0 is the translation group of R.This allows us to exploit techniques from harmonic analysis.Our strategy is built, in a crucial way, on Calderón-Zygmund singular integral theory and modern real-variable Littlewood-Paley theory.We present all this in the preparatory section 5 that will be needed for the proofs of Theorem 1.2 and Theorem 1.5.These proofs constitute the most heavy and technical part of the article.The proofs of Theorem 1.5 and Corollary 1.6 are done in section 6.We then use transference to show Theorem 1.2 in section 7. To that end, we first need to represent the g-function associated to the Poisson semigroup subordinated to the translation group of R as a singular integral operator.Theorem 1.4 will follow from Theorem 1.2 by functional calculus; Corollary 1.9 is an easy consequence of Theorem 1.2 (i).
An additional major significant aspect of the new approach is the fact that it improves the growth orders on p of the relevant best constants even in the scalar Littlewood-Paley-Stein inequalities, see section 8; moreover, except the case of the Luzin type constant as p → 1, it yields the optimal orders, which is not the case by the previous methods of Stein and Cowling (see the historical comments in section 8).This shows, to a certain extent, that our method is optimal.Section 8 also contains the optimality of the best constants in Corollary 1.6.We end the article by an appendix that gives some examples of semigroups.
The techniques developed in this article allow one to simplify and extend many recent results in the scalar Littlewood-Paley-Stein theory, in particular, those on positive operators on L 2 with kernels satisfying Gaussian upper estimates.On the other hand, they are also applicable to the noncommutative setting.We will carry out all this elsewhere.Throughout the article, X will be a Banach space, 1 < p < ∞ and 1 ≤ q ≤ ∞ (but 1 < q < ∞ most of time).Unless explicitly stated otherwise, {T t } t>0 will be a strongly continuous semigroup of regular operators on L p (Ω) and {P t } t>0 its subordinated Poisson semigroup.These semigroups are extended to L p (Ω; X).A will denote the negative generator of {T t } t>0 , so T t = e −tA and P t = e −t √ A .

The ℓ q -boundedness
This section is the preparatory part of the article.We will introduce the notion of ℓ q -boundedness that is the direct extension to the vector-valued setting of the R q -boundedness introduced by Weis [59].In fact, though not explicitly stated, this notion appeared before in harmonic analysis with regard to vector-valued inequalities of classical operators.Most results below are the ℓ qboundedness analogues of well known results or those of Kunstmann and Ullmann [33] in the scalar case.I learnt the existence of [59,33] after the submission of this article for publication, and I thank Emiel Lorist for pointing out these references to me.
We start with a brief introduction to holomorphic functional calculus in order to fix notation (see [13] for more information).Recall that a densely defined closed operator B on a Banach space Y is called a sectorial operator of type α with 0 ≤ α < π if C \ Σ γ is contained in the resolvent set of B for any γ > α and where Σ γ is the open sector z ∈ C : |arg(z)| < γ in the complex plane.Let β > γ and H ∞ (Σ β ) denote the space of bounded analytic functions in Σ β and H ∞ 0 (Σ β ) its subspace consisting of all ϕ satisfying |ϕ(z)| ≤ c|z| δ 1 + |z| 2δ for some c > 0 and δ > 0.
Let Γ be the boundary of Σ β , oriented in the positive sense.Then for any ϕ ∈ H ∞ 0 (Σ β ) the integral defines a bounded operator on Y , where the integral absolutely converges in B(Y ).
The following resolution of the identity will be useful later.
for all finite sequences {A k } ⊂ F and {f k } ⊂ L p (Ω; X), with the usual modification for q = ∞ in the above inequality.
Remark 2.2.It is clear that the sums in the above definition can be replaced by integrals without changing the constant c.On the other hand, it is easy to show that the absolutely convex hull of an ℓ q -bounded family is again ℓ q -bounded with the same constant.
Accordingly, we introduce the ℓ q -boundedness versions of sectoriality of operators and analyticity of semigroups.Recall that a semigroup {S t } t>0 on a Banach space Y is said to be analytic if it extends to an analytic function from Σ β to B(Y ) for some 0 < β ≤ π 2 and bounded in any smaller sector.In this case, we call {S t } t>0 an analytic semigroup of type β.
The following is the ℓ q -boundedness analogue of a classical characterization of analytic semigroups.
Proposition 2.4.Let {S t } t>0 be a strongly continuous bounded semigroup on L p (Ω; X) and B its negative generator.Then the following statements are equivalent: Proof.The proof is a straightforward adaptation of the classical argument (cf.e.g. the proof of [50,Theorem 5.2]).As we want to track the links between the different constants involved, we give below an outline. ( Then for any z = re iθ / ∈ Σ γ and z = 0, we have If C ν denotes the ℓ q -boundedness constant of the family (ii) ⇒ (iii).Let α < γ < π 2 and Γ be the boundary of Σ γ oriented in the positive sense.Then Thus {tBS t } t>0 is ℓ q -bounded with constant To show the ℓ q -boundedness of {S t } t>0 we need to slightly modify the contour Γ.Let Γ ′ be the union of the part of Γ with |λ| ≥ 1 and the arc in C \ Σ γ of the circle with the origin as center and radius 1.Then we have The change of variables ζ = tλ yields where tΓ ′ is the union of the part of Γ with |λ| ≥ t and the arc in C \ Σ γ of the circle with the origin as center and radius t.However, the Cauchy formula insures that we can go back to Γ ′ : This implies that {S t } t>0 is ℓ q -bounded with constant The function t → S t is infinitely derivable in R + and for any positive integer n Then {S t } t>0 becomes an ℓ q -analytic semigroup of type β thanks to the following formula and for any ν < β the family S z : z ∈ Σ ν is ℓ q -bounded with constant The proof is thus complete.
The following is again the ℓ q -boundedness version of an elementary result on sectorial operators.The case used later concerns only √ B.
Decomposing the last integral into three parts corresponding to |ζ| close to 0, 1 and ∞, respectively, we get Hence by Remark 2.2, we deduce the desired assertion.Now we return back to our distinguished semigroup {T t } t>0 of regular operators on L p (Ω).We have extended {T t } t>0 to L p (Ω; X).Recall our convention that the regular operators considered in this article are all assumed contractively regular.Also recall the fact that T is regular on [40]).Consequently, T is regular on L p (Ω) iff its adjoint T * is regular on L p ′ (Ω).In particular, {T * t } t>0 is a strongly continuous semigroup of regular operators on L p ′ (Ω).
The family Proof.The celebrated theorem of Akcoglu [2] asserts that {M t } t>0 satisfies the following maximal ergodic inequality sup see also [31,Theorem 5.2.5].The regularity of {M t } t>0 insures that this inequality remains valid for any f ∈ L p (Ω; X).Thus for any finite sequences This means that {M t } t>0 is ℓ ∞ -bounded on L p (Ω; X) with constant p ′ .On the other hand, {M t } t>0 is bounded, so ℓ p -bounded on L p (Ω; X) with constant 1.Thus by complex interpolation, {M t } t>0 is ℓ q -bounded on L p (Ω; X) with constant (p ′ ) 1− p q for q > p.The case of q < p is treated by duality.Applying the previous discussion to the adjoint semigroup is ℓ 1 -bounded on L p (Ω; X) with constant p.The assertion for q < p then follows by complex interpolation once more.
Remark 2.7.The above lemma and the subordination formula (1.2) imply that the Poisson It is a classical result that the negative generator A of {T t } t>0 on L p (Ω; X) is a sectorial operator of type π 2 .The following shows that it is moreover ℓ q -sectorial.Proposition 2.8.The negative generator A of {T t } t>0 is an ℓ q -sectorial operator of type π 2 on L p (Ω; X).More precisely, the family Thus by Lemma 2.6 and Remark 2.2, we deduce that ).This implies the assertion on A with the constant C α given by The second part on √ A then follows from Proposition 2.5.
Proposition 2.9.Assume that {T t } t>0 satisfies (1.3) for some 0 < β 0 ≤ π 2 .Let 1 < q < ∞ and β q = β 0 min( p q , p ′ q ′ ).Then {T t } t>0 is an ℓ q -analytic semigroup of type β q on L p (Ω; X).More precisely, for any 0 < β < β q the family T z : z ∈ Σ β is ℓ q -bounded on L p (Ω; X) with constant majorized by Proof.Define where the integral is taken along the segment [0, z].Clearly, M is analytic in Σ β0 .By Lemma 2.6, {M t : t > 0} is ℓ ∞ -bounded (resp.ℓ 1 -bounded) on L p (Ω; X) with constant p ′ (resp.p).On the other hand, (1.3) means that {M z : z ∈ Σ β0 } is ℓ p -bounded on L p (Ω; X) with constant T β0 .Then by complex interpolation, we see that {M z : z ∈ Σ βq } is ℓ q -bounded on L p (Ω; X) with constant T p q β0 (p ′ ) 1− p q for p < q and T p ′ q ′ β0 p 1− p ′ q ′ for p > q.We use the identity T z = M z + zM ′ z to pass from M z to T z , so it remains to show that {zM ′ z : z ∈ Σ β } is ℓ q -bounded.To this end, let δ = 1 2 (β + β q ).For any z = re iθ ∈ Σ β let C be the circle with center z and radius r sin(δ − |θ|).Note that one of the two rays limiting Σ δ is a tangent of C. By the Cauchy integral formula, we have The ℓ q -boundedness of {M z : z ∈ Σ δ } and Remark 2.2 imply that {zM ′ z : z ∈ Σ β } is ℓ q -bounded on L p (Ω; X) with constant majorized by The last part on the ℓ q -sectoriality of A follows from the proof of the implication (i) ⇒ (ii) of Proposition 2.4, β and ν there being respectively β q and β now.

Proof of Theorem 1.10
Armed with the preparation in section 2, we will follow the proof of [29,Theorem 4.14].In the sequel, we will use the abbreviation that ∂ = ∂ ∂t .Recall that R + is equipped with the measure dt t .Also recall our convention that {T t } t>0 is a strongly continuous semigroup of regular operators on L p (Ω) and {P t } t>0 its subordinated Poisson semigroup.
We first show part (i) concerning the subordinated Poisson semigroup Recall that the map P is defined by (1.7) (with {P t } t>0 instead of {T t } t>0 there).It can be rewritten as Let Γ be equipped with the measure |dz| |z| .We define three maps as follows: Then P = Φ 2 ΦΦ 1 .Thus it remains to show that the three newly defined maps are all bounded.Consider first the case q < ∞.By the Hölder inequality, we have On the other hand, for any t > 0 We then deduce that Thus The same upper estimate holds for Φ 2 too.Finally, the boundedness of Φ is just a reformulation of the ℓ q -boundedness of 2 πi zR(z) : z ∈ Γ \ {0} .Thus by Proposition 2.8, This finishes the proof of the first assertion for q < ∞ (choosing α close to π 4 ).The boundedness of P for q = ∞ is obtained from that for q = 1 by duality.
Let us show that P is a projection.Let h ∈ L p (Ω; L q (R + ; X)) be given by h t = t∂P t (f ) for some f ∈ L p (Ω; X).Then by (3.2) Thus P(h) = h, so P is a projection.This shows the first assertion (i).
Assertion (ii) on the semigroup {T t } t>0 itself is proved exactly in the same way.Indeed, letting α q = π 2 − β q , by Proposition 2.9, A is ℓ q -sectorial of type α q .Let β = βq 2 and α = π 2 − β.Then α q < α < π 2 and Thus using the estimate on the ℓ q -sectoriality constant of A and repeating the above argument, we show that T is bounded with the announced norm estimate.

Luzin type and cotype
In this section we study Banach spaces that are of Luzin coype or type.Before proceeding we briefly discuss the projection F onto the fixed point subspace of {T t } t>0 (equivalently, of {P t } t>0 ).By the mean ergodic theorem, F is given by Thus F is also regular, so extends to a contractive projection on L p (Ω; X).Then the above formula remains valid for f ∈ L p (Ω; X) and F(L p (Ω; X)) coincides with the fixed point subspace of {T t } t>0 on L p (Ω; X).It follows that L p (Ω; X) admits the following direct sum decomposition: On the other hand, ker F is the closure of (Id By the paragraph before Lemma 2.6, we know that the adjoint semigroup {T * t } t>0 is regular on L p ′ (Ω) too.Thus the above discussion also applies to the semigroup {T * t } t>0 that is extended to L p ′ (Ω; X * ) again.Consequently, (4.1) and (4.2) transfer to this dual setting.We should draw the reader's attention to the fact that L p ′ (Ω; X * ) is in general not the dual of L p (Ω; X) but an isometric subspace.With this in mind, we have .
A similar formula holds for the negative generator A of {T t } t>0 on L p (Ω; X) and the negative generator A * of {T * t } t>0 on L p ′ (Ω; X * ), that is, the restriction to L p ′ (Ω; X * ) of the adjoint of the former coincides with the latter.Moreover, F * L p ′ (Ω;X * ) is the fixed point projection associated to {T * t } t>0 on L p ′ (Ω; X * ).All this allows us to use duality arguments without any problem as when L p (Ω; X) * = L p ′ (Ω; X * ) (which is the case for reflexive X).
According to [64], we introduce the following definition already mentioned before Theorem 1.2.
(ii) X is said to be of Luzin cotype q relative to {T t } t>0 if there exists a constant c such that The smallest c is denoted by L T c,q,p (X).(iii) X is said to be of Luzin type q relative to {T t } t>0 if there exists a constant c such that The smallest c is denoted by L T t,q,p (X).In (4.3), f is implicitly assumed to belong to the definition domain of A in order to guarantee the derivability of ) is trivially verified for such f ).On the other hand, the above definition implicitly depends on p, but this dependence is not essential thanks to the fact that in most cases, if (4.4) or (4.5) holds for one p, so it does for any allowed p.Thus to lighten the terminology, we have decided to not explicitly mention p in the above notions; anyway, this dependence on p is reflected in the constants L T c,q,p (X) and L T t,q,p (X).
Remark 4.2.Without additional assumption on {T t } t>0 , the definition may be insignificant.For instance, if {T t } t>0 is the translation group of R, it is easy to check that for any 1 ≤ q ≤ ∞ and for any f ∈ L p (R) with f ′ not identically zero.Thus C is not of Luzin cotype q for any q relative to the translation group of R.
This remark shows that to have a meaningful theory of Luzin type and cotype some minimal condition should be imposed to {T t } t>0 .It turns out that this minimal condition is the analyticity of {T t } t>0 on L p (Ω; X).As shown in section 2, the subordinated Poisson semigroup {P t } t>0 always satisfies this condition.

It is sometimes convenient to have a discrete version of G
for some constant T max (see [38]).A similar inequality holds for the adjoint semigroup {T * t } t>0 , the relevant constant being denoted by T * max .
Proposition 4.3.Assume that {T t } t>0 is analytic on L p (Ω).Let 1 ≤ q ≤ ∞ and a > 1.Then for any f ∈ L p (Ω; X) where c T,q,a = q − 1 q (a q − 1) max ) .Similar inequalities hold for {P t } t>0 in place of {T t } t>0 without any additional assumption on {T t } t>0 , the corresponding constants being given by c P,q,a = C −1 q − 1 q (a q − 1) Proof.Using (4.6), its adjoint version and repeating the proof of Lemma 2.6, we show that {T t } t>0 is ℓ q -bounded on L p (Ω; X) with constant max(T Using ∂T t+s = T s ∂T t , we have ∂T Lp(Ω) = q − 1 q (a q − 1) .
For the converse inequality, we write As above, we then deduce The second part on {P t } t>0 is just a particular case with P max = Cp ′ and P * max = Cp by virtue of Remark 2.7.
Recall the classical fact that {T t } t>0 is analytic on L p (Ω; X) iff {t∂T t : t > 0} is uniformly bounded on L p (Ω; X).Thus the following remark immediately follows from the above result; it shows in particular that the analyticity of {T t } t>0 on L p (Ω; X) is necessary for X to be of Luzin cotype q relative to {T t } t>0 for some q.
Remark 4.4.Assume that {T t } t>0 is analytic on L p (Ω).If X is of Luzin cotype (resp.type) q relative to {T t } t>0 , then X is of Luzin cotype (resp.type) r relative to {T t } t>0 for any r > q (resp.r < q).Moreover, if X is of Luzin cotype ∞ relative to {T t } t>0 , then {T t } t>0 must be analytic on L p (Ω; X).
The following is one of the main results of this section.Theorem 4.5.Let X be a Banach space and 1 ≤ q ≤ ∞.
(i) X is of Luzin cotype q relative to {P t } t>0 iff X * is of Luzin type q ′ relative to {P * t } t>0 .Moreover, the relevant constants satisfy Then X is of Luzin cotype q relative to {T t } t>0 iff X * is of Luzin type q ′ relative to {T * t } t>0 .Moreover, the relevant constants satisfy with β q = β 0 min( p q , p ′ q ′ ).Proof.(i) Assume that X is of Luzin cotype q.Let g ∈ L p ′ (Ω; X * ) with F * (g) = 0. Let f ∈ L p (Ω; X).We want to estimate f, g , where the duality bracket is that between L p (Ω; X) and L p ′ (Ω; X * ).By (4.1) and its dual version, we can assume F(f ) = 0, which, together with (4.2), implies that f ∈ ker F = im √ A. With F (z) = −ze −z and by (2.1) we have Thus by the Hölder inequality and the Luzin cotype q of X f, g = 4 Taking the supremum over f with norm 1, we show that X * is of Luzin type q ′ with L P * t,q ′ ,p ′ (X * ) ≤ 4 L P c,q,p (X).To show the converse implication, let f ∈ L p (Ω; X) and h ∈ L p ′ (Ω; L q ′ (R + ; X * )) (recalling that R + is equipped with dt t ).We have Applying Theorem 1.10 (i) to {P * t } t>0 on L p ′ (Ω; X * ), we have Combining the above inequalities, we get which implies the Luzin cotype q of X with The proof of this part is similar by using Theorem 1.10 (ii).
Corollary 4.6.Any Banach space X is of Luzin type 1 relative to {P t } t>0 , so relative to Proof.Indeed, let f ∈ L p (Ω; X) such that F(f ) = 0. Then by (4.2) and (2.1) we have . Thus X is of Luzin type 1 relative to {P t } t>0 , hence also relative to {T t } t>0 by virtue of Remark 1.1.Passing to duality by means of Theorem 4.5, we see that X is of Luzin cotype ∞ relative to {T t } t>0 under the analyticity assumption of {T t } t>0 on L p (Ω; X).
The following formulation of the Littlewood-Paley function G P q (f ) by discrete lacunary differences is of interest in its own right.Theorem 4.7.Let X be a Banach space, 1 ≤ q ≤ ∞ and a > 1.
(i) X is of Luzin cotype q relative to {P t } t>0 iff there exists a constant c such that for all 1 ≤ t ≤ a and f ∈ L p (Ω; X).Moreover, the best c and L P c,q,p (X) are linked by (ii) X is of Luzin type q relative to {P t } t>0 iff there exists a constant c such that for all f ∈ L p (Ω; X).Moreover, the best c and L P t,q,p (X) are linked by (iii) Similar statements hold for {T t } t>0 under the additional assumption that {T t } t>0 be analytic on L p (Ω; X) and 1 < q < ∞.
Proof.(i) We have Thus if X is of Luzin cotype q relative to {P t } t>0 , then (4.7) holds with c ≤ (log a) 1 q ′ L P c,q,p (X) .To show the converse implication, let b = 1 2 (1 + a).We use an idea from [27] (see also [66]) to write By Propositions 2.4 and 2.8, the family {t∂P t } t>0 is ℓ q -bounded with constant C max((p ′ ) 1− p q , p 1− p ′ q ′ ).Therefore, To estimate the norm on the right hand side, we write Note that the function is continuous from R + to L p (Ω), so there exists We then deduce the Luzin cotype q of X from (4.7) with L P c,q,p (X) (log a) (ii) The above argument yields the following discretization of G P q (f ): ≤ (a − 1) 1 q (log a) . This immediately implies assertion (ii).
(iii) is proved similarly by virtue of Propositions 2.4 and 2.9.
We have seen that the proofs of Theorem 4.5 and Theorem 1.10 are based on functional calculus for the special function F (z) = −ze −z .It is known that functional calculus allows us to use more general functions.Definition 4.8.Let B be an ℓ q -sectorial operator of type α on L p (Ω; X) with α < π.Let β > α and ϕ ∈ H ∞ 0 (Σ β ) be a nonzero function.Define The following result is a variant of [42, Theorem 5] (see also [66,Lemma 20]).
Proposition 4.9.Let ϕ and ψ be two nonzero functions in Combined with (2.1), this implies Let α < γ < β and Γ be the boundary of Σ γ .We then deduce These equalities are the analogues of (3.1) and (3.2) with h t = ψ(tB)(f ).It remains to repeat the proof of Theorem 1.10 to conclude where C B,γ is the ℓ q -boundedness constant of z(z − B) In particular, combining the previous proposition with the results in section 2, we obtain the following Corollary 4.10.Let X be a Banach space and 1 ≤ q ≤ ∞.
We conclude this section by some remarks on general ℓ q -sectorial operators on L p (Ω; X) for which we have defined the g-function in Definition 4.8.In fact, what we have done so far for semigroups can be developed for these operators too.Definition 4.11.Let B be an ℓ q -sectorial operator of type α on L p (Ω; X) with α < π.
(i) X is said to be of Luzin cotype q relative to B if there exists a constant c such that for every f ∈ imB and some nonzero ϕ ∈ H ∞ 0 (Σ β ) with β > α. (ii) X is said to be of Luzin type q relative to B if there exists a constant c such that for every f ∈ imB and some nonzero ϕ ∈ H ∞ 0 (Σ β ) with β > α.Proposition 4.9 shows that the above definition is independent of the choice of ϕ.Assume additionally that B admits a dual operator B ′ on L p ′ (Ω; X * ) in the sense of [13], namely, Assume further that B ′ is also ℓ q ′ -sectorial of type α on L p ′ (Ω; X * ).Proposition 4.12.Under the above assumption, X is of Luzin cotype q relative to B iff X * is of Luzin type q ′ relative to B ′ .Proof.Noting that Theorem 1.10 transfers to the present setting, we can repeat the proof of Theorem 4.5, and so omit the details.
It would be interesting to investigate the Luzin type and cotype relative to B as above.Guided by Theorems 1.2 and 1.4, one would like to know those operators B such that the Luzin type or cotype relative to B is implied by the martingale type or cotype.It seems that more structure should be imposed to B in order to get significant results.We have seen that this is indeed the case if B is the negative generator of {P t } t>0 or {T t } t>0 on L p (Ω; X).On the other hand, we have the following proposition that is contained (essentially) in [58].Note that [58] can be viewed as the particular case of our discussion where Ω is a singleton (the ℓ q -boundedness then simply becomes the usual boundedness).
The notion of type and cotype referred in the next proposition is the usual Rademacher type and cotype.Proposition 4.13.Assume that B has a bounded H ∞ functional calculus.If X is of cotype (resp.type) q, then X is of Luzin cotype (resp.Luzin type ) q relative to B.
Proof.As in [58], this is a simple consequence of Kalton-Weis' theorem on the unconditionality of bounded H ∞ functional calculus (see Theorem 10.4.6 or its discrete version, Theorem 10.4.4. in [28]).
Remark 4.14.The assumption on the H ∞ functional calculus of B seems too strong since it implies that X is a UMD space in many cases, for instance, if B is the negative generator of {T t } t>0 on L p (R; X) when {T t } t>0 is the heat or Poisson semigroup on R.

Dyadic martingales and singular integrals
The proofs of Theorem 1.2 (i) and Theorem 1.5 heavily rely on tools from harmonic analysis, notably from modern real-variable Littlewood-Paley theory.This section is a preparation for using these tools.We will mainly follow Wilson's beautiful treatment in [62] (see also [60,61]) for the part on Littlewood-Paley theory.5.1.Dyadic martingales.All cubes in R d considered in the sequel are bounded and with sides parallel to the axes.|Q| and ℓ(Q) denote respectively the volume and side length of a cube Q; tQ stands for the cube with the same center as Q and ℓ(tQ) = tℓ(Q) for t > 0.
Let D be the family of all dyadic cubes of R d , D k ⊂ D the subfamily of all cubes with side length 2 −k for k ∈ Z.Let A k be the σ-algebra generated by D k and E k the associated conditional expectation.For f ∈ L p (R d ; X), S q (f ) is the q-variant of the usual martingale square function of f .It is useful to note that d k (f ) is of vanishing mean on every We will need dyadic-like families of cubes that Wilson calls good families.F is such a family if a) for Q ∈ F , all its 2 d immediate dyadic subcubes belong to For F a dyadic-like family of cubes, we define the associated S q,F : It is easy to see that given a finite number of cubes in F , we can bring the subfamily consisting of those cubes in F that are contained in one of the given cubes to a subfamily of D after appropriate translation and rescaling.Thus we have the following.
Lemma 5.1.Let F be a dyadic-like family of cubes.If X is of martingale type q, then for all f ∈ L q (R d ; X) supported by cubes from F .
An important case needed later concerns the family {3Q} Q∈D .The following is due to Wilson [60]. ) with j, k ∈ Z and s ∈ {0, 1, 2}.Let F k s be the collection of all such intervals for given k and s.Then the desired union is ∪ 2 s=0 ∪ k∈Z F k 2 |k| s mod 3 .Let F be a dyadic-like family of cubes and δ > 0. Consider a family {a Q } Q∈F of X-valued functions satisfying the following conditions: These are smooth atoms.Let {λ Q } Q∈F be a finite family of complex numbers and The following is the adaptation of a lemma due to Wilson to the present setting.We include its proof for the convenience of the reader.Lemma 5.3.Under the above assumption, we have Proof.Without loss of generality, we assume that F = D, so S q,F (f Thus by the last condition of (5.1), It then follows that This gives the desired assertion.

Singular integrals.
Given ε > 0 and δ > 0, let H ε,δ be the class of all integrable functions ϕ on R d satisfying (1.4).Let ϕ ∈ H ε,δ .We consider the vector-valued kernel K defined by With a slight abuse of notation, we use K to denote the associated singular integral too: Lemma 5.4.The kernel K satisfies the regularity properties: This lemma is proved.

5.3.
A quasi-orthogonal decomposition.Beside H ε,δ introduced in the previous subsection, we will need its subclass of functions supported in the unit ball.More precisely, let H 0 δ be the class of integrable functions ϕ on R d such that Here B(x, t) denotes the ball of R d with center x and radius t.
Any function in H ε,δ can be decomposed into a series of functions in H 0 δ thanks to the following lemma due to Uchiyama [57] (see also [61]).
Lemma 5.5.Let ϕ ∈ H ε,δ .Then there exist a positive constant C ε,δ and a sequence of functions Proof.The proof is elementary.Let η be a smooth function supported in {x ∈ R d : and Then the desired decomposition is given by By our convention that R + is equipped with the measure dt t , the upper half space R d+1 + is equipped with the product measure dx dt t .Consistent with our convention before, we write a function h : R d+1 + ; X) with compact support.Consider the following function dy dt t .
We will decompose g into a series of smooth atoms where the a i 's satisfy (5.1) relative to {3Q} Q∈D and the λ i 's are reals such that This is the so-called the quasi-orthogonal decomposition of g.First, using Lemma 5.5, we reduce our problem to the case where ϕ is supported in the unit ball: Note that h 2 −k • has the same norm as h in L q (R d+1 + ; X).Thus it suffices to do the decomposition for each ψ (k) in place of ϕ.
In the following, we will assume that ϕ itself belongs to H 0 δ .The argument below is classical. Clearly, Since ϕ is supported in the unit ball and of vanishing mean, we see that a Q is supported in 3Q and of vanishing mean too.On the other hand, since ϕ is in the Hölder class H 0 δ , by the Hölder inequality, Thus a Q is a smooth atom.This yields the desired quasi-orthogonal decomposition.
Combining the above discussion with Lemmas 5.1 -5.3, we get the following Lemma 5.6.Keep the above notation and assume that X is of martingale type q.Then

Proofs of Theorem 1.5 and Corollary 1.6
With the preparation in section 5, we are in a position to show Theorem 1.5.For clarity, we divide the proof into several steps.X will be assumed of martingale cotype q in the first four steps, and of martingale type q in the last step.
Step 1: A weighted norm inequality.Let ϕ be a function satisfying (1.4), i.e., ϕ ∈ H ε,δ .Beside the g-function defined by (1.6), we will need the Luzin integral function for nice f : R d → X.The key of this proof is the following weighted norm inequality: For any locally integrable nonnegative function w on R d and any f ∈ L q (R d ; X) (6.1) where M (w) denotes the Hardy-Littlewood maximal function of w: First consider the unweighted case, i.e., w ≡ 1.By the Fubini theorem, we have Let h : R d+1 + → X * be a compactly supported smooth function such that where Recall that it is well known (and easy to check) that X is of martingale type q iff X * is of martingale cotype q ′ with the following relation between the relevant constants: Thus applying Lemma 5.6 to h and g with (X * , q ′ ) in place of (X, q) there, we get Then taking the supremum over all h in the unit ball of L q ′ (R d+1 + ; X * ), we deduce S q,ϕ (f ) Namely, the unweighted version of (6.1) holds.
We will deduce the weighted version by a trick from [11].By Lemma 5.5, we can assume that ϕ is supported in the unit ball of R d .Given a weight w, we write (recalling that B(y, t) denotes the ball of center y and radius t) w(x)dx dy dt t . Let Thus using the unweighted version already proved (applied to f 1 E k ), we deduce Thus (6.1) is proved.
Step 2: Another weighted norm inequality.We need to show that (6.1) remains valid for G q,ϕ instead of S q,ϕ : To this end, we have to control the g-function by the Luzin area function.If ϕ is the Poisson kernel, this is a classical fact thanks to harmonicity.In the present setting, we need a little bit more efforts.
If additionally all partial derivatives of ϕ with order up to d belong to H ε,δ , then we can show where The proof of this inequality is elementary (see [63,Lemma 4.3]).Thus (6.3) holds for such ϕ.
For a general ϕ, we need to adapt the arguments of [61] to the present setting by introducing the vector-valued q-variants of Wilson's intrinsic square functions: One can show, quite easily, that S q,ε,δ (f )(x) ≈ d,ε,δ G q,ε,δ (f )(x) for every x ∈ R d (see [61] for more details).
On the other hand, for a compactly supported smooth function h : R d+1 Then by adapting the arguments in subsection 5.3 to the present situation and by repeating step 1, one can estimate the integral This implies (6.1) with S q,ψ replaced by S q,ε,δ by the passage from the unweighted case to the weighted one.Then the pointwise equivalence S q,ε,δ (f ) ≈ d,ε,δ G q,ε,δ (f ) shows that (6.3) holds for G q,ε,δ instead of G q,ϕ , whence (6.3) for every ϕ ∈ H ε,δ .
Step 3: Proof of Theorem 1.5 (i) for p ≥ q.We can now easily prove part (i) of Theorem 1.5 for p ≥ q.Indeed, the case p = q is just the unweighted version of (6.3).For p > q, let w be a nonnegative function on R d with L r -norm equal to 1, where r is the conjugate index of p q .Then for f ∈ L p (R; X), . Taking the supremum over all w, we get Step 4: Proof of Theorem 1.5 (i) for p < q.We deal with the case p < q by singular integrals.
Let K be the singular integral associated to ϕ as in subsection 5.2.We reduce to showing that K is bounded from L p (R d ; X) to L p (R d ; L q (R + ; X)).The previous step insures this boundedness for p = q.On the other hand, Lemma 5.4 shows that K is a regular Calderón-Zygmund kernel.Thus K satisfies the assumption of [16,Theorem V.3.4].Note that [16, Theorem V.3.4] is formulated for kernels satisfying the regularities in Lemma 5.4 with δ = 1; however, it is well known that [16, Theorem V.3.4]remains valid for any kernel as in Lemma 5.4 with the same proof.Therefore, K is of weak type (1, 1), so by the vector-valued Marcinkiewicz interpolation theorem (see [4,Theorem 1.3.1]and its proof), K is bounded from L p (R d ; X) to L p (R d ; L q (R + ; X)) with norm controlled by C d,ε,δ p ′ M c,q (X) for 1 < p < q.This finishes the proof of Theorem 1.5 (i).
Step 5: Proof of Theorem 1.5 (ii).In this last step we show part (ii) by duality.Let In the scalar case, this Calderón reproducing formula is proved by taking Fourier transforms of both sides.Then by linearity, the formula extends to the vector-valued case too when both f and g take values in finite dimensional subspaces, which can be assumed by approximation.Therefore, Since we are in part (ii) of Theorem 1.5, X is of martingale type q, so X * is of martingale cotype q ′ and M c,q ′ (X * ) ≤ 2M t,q (X) by (6.2).Thus by part (i) already proved, we have Taking the supremum over all g with g L p ′ (R d ;X * ) ≤ 1, we deduce as desired.So the proof of Theorem 1.5 is complete.
Proof of Corollary 1.6.The first part of this corollary immediately follows from Theorem 1.5.The first two inequalities of the second part are consequences of Remark 1.1 and Proposition 8.5 below.Finally, the last inequality L P c,q,q (X) M c,q (X) is obtained by combining [41] and [64].
7. Proofs of Theorem 1.2, Theorem 1.4 and Corollary 1.9 In this section we will first prove Theorem 1.2.Theorem 1.4 and Corollary 1.9 will then follow quite easily.Our strategy for the proof of Theorem 1.2 is to reduce part (i), via transference, to the special case of the translation group to which we can apply Theorem 1.5.
Proof of Theorem 1.2.Again, we divide this proof into several steps.X will be assumed to be of martingale cotype q in the first two steps, and of martingale type q in the last step.
Step 1: The case of the translation group.For any t ∈ R let τ t be the translation by t on L p (R), i.e., τ t (f )(s) = f (s + t).Then {τ t } t∈R is a strongly continuous group of positive isometries on L p (R).As usual, {τ t } t∈R extends to a group of isometries on L p (R; X) too.Let {P τ t } t>0 be the associated Poisson subordinated semigroup.Our aim in this step is to show (7.1)L P τ c,q,p (X) max p 1 q , p ′ M c,q (X).
We need to express t∂P τ t as a convolution operator: Elementary computations show that φ is the function with Fourier transform We are going to show that φ belongs to the class H 1 2 ,1 introduced in subsection 5.2.More precisely, φ satisfies the following estimates: , x ∈ R.
Since ξ k φ(ξ) is integrable on R for any nonnegative integer k, φ is of class C ∞ with bounded derivatives of any order.Thus it suffices to prove the estimates for |x| ≥ 1.
Let η be a C ∞ even function on R, supported in {ξ : Let m j (ξ) = φ(ξ)η(2 −j ξ) and φ (j) be defined by φ (j) = m j .Then Using (7.3), one easily shows This implies Let x ∈ R with |x| > 1.We consider j according to two cases, I: 2 j |x| ≤ 1 and II: In Case I, we must have j ≤ −1.Using (7.5) for k = 1, we then have On the other hand, if j ∈ II, we use (7.5) for k = 3 to get Hence, the first estimate of (7.4) is proved.
The second is shown in a similar way.Indeed, since φ ′ (ξ) = 2πi ξ φ(ξ), we have It then remains to repeat the above argument with φ replaced by φ ′ .
Thus by (7.2), Theorem 1.5 implies (7.1).Let us note that for the kernel φ here, we can avoid Wilson's intrinsic square functions considered in step 2 of the proof of Theorem 1.5 since φ ′ ∈ H 1 2 ,1 too.Indeed, repeating the proof of (7.4), we show .
Consequently, as pointed out in step 2 of the proof of Theorem 1.5, we have The proof of this inequality is very easy.It suffices to consider x = 0. Let y ∈ R such that |y| < t.We write Then Integrating both sides against dydt t 2 over the cone {(y, t) ∈ R 2 + : |y| < t}, we deduce the desired inequality.
Step 2: Transference.We now use the transference principle to bring the general case to the special one of the translation group.To that end, we first need to dilate our semigroup {T t } t>0 to a group of isometries.Fendler's dilation theorem is at our disposal for this purpose.It insures that there exist another larger measure space ( Ω, µ), a strongly continuous group { T t } t∈R of regular isometries on L p ( Ω), a positive isometric embedding D from L p (Ω) into L p ( Ω) and a regular projection P from L p ( Ω) onto L p (Ω) such that This theorem is proved in [17] for positive T t and then extended to regular T t in [18].
To prove part (i) of Theorem 1.2, it suffices to show (7.6)L P c,q,p (X) ≤ L P τ c,q,p (X).By the above dilation, we can assume that {T t } t>0 itself is a group of regular isometries on L p (Ω).So its extension to L p (Ω; X) is a group of isometries too.Recall that {M t } t>0 denote the ergodic averages of {T t } t>0 in Lemma 2.6.We use {M τ t } t>0 to denote the corresponding averages of the translation group {τ t } t>0 .By (1.2), we have Thus where Let f ∈ L p (Ω; X) be an element of norm 1.Let a > 0 (large).Then Thus for any t > 0 where Let b be another large number.Then where . Denote the first term on the right hand side by I. Using the fact that {T t } is a group of isometries on L p (Ω; X), we introduce an additional variable u in the integrand of I: , u > 0.
Remark 7.1.The step 1 of the previous proof of Theorem 1.2 can be largely shortened for the case p ≥ q.This alternate proof does not rely on the heavy Littlewood-Paley theory.Its key point is to show the boundedness of G P q on L q (Ω; X), i.e., for p = q (see the following remark).Assuming this boundedness and showing that K is bounded from L ∞ (R d ; X) to BM O(R d ; L q (R + ; X)) (the latter boundedness is quite easy to get), we can then use the singular integral as in the proof of the step 4 of Theorem 1.5 to conclude the case p > q.Unfortunately, this proof yields p as the order of L P c,q,p (X) instead of the optimal p 1 q .
Remark 7.2.The boundedness of G P q on L q (Ω; X) can be proved by using Theorem 4.7 and the following inequality from [22]: For a Banach space X of martingale cotype q, we have k∈Z By the discussion in the previous remark, the validity of the above inequality characterizes the martingale cotype q of X.More generally, let 1 < p < ∞ and 2 ≤ q < ∞.Then X is of martingale cotype q iff there exists a constant c such that k∈Z See [23] for related results.
Assertion (ii) is an easy consequence of Theorem 1.2 and Corollary 4.10.But we will use Proposition 4.9 in order to explicitly track the relevant constants.Let ϕ(z) = −ze −z and ψ(z . By Proposition 2.9 and the notation there, A is ℓ q -sectorial of type α q = π 2 − β q .Let 0 < β < β q and α = π 2 − β.Then by Proposition 4.9 and the estimates obtained in its proof, we get , where q ′ .We then deduce Theorem 1.4 (ii) from Theorem 1.2 (i).
Proof of Corollary 1.9.By [44,45], We view f as a function from Ω to L p (Ω; X) by ω → f (ω)f 0 .Applying Theorem 1.2 (i) to this function with X replaced by L p (Ω; X) and p = q (noting then that max q 1 q , q ′ ≈ 1 since q ≥ 2), we get The left hand side is equal to Combining the above estimates we get the desired inequality of the corollary.
To show the optimality of the constant, we consider the special case where X = C and the classical Poisson semigroup {P t } t>0 on R. Let f = P 1 .Then We then deduce the announced optimality.

The scalar case revisited and optimality
The approach previously presented gives new insights even in the scalar case with regard to the involved best constants.Let L T c,p = L c,2,p (C) and L T t,p = L t,2,p (C).Thus L T c,p and L T t,p are the best constants in the following inequalities Let us restate Theorems 1.2 and 1.4 for X = C and q = 2. Theorem 8.1.Let {T t } t>0 be a semigroup of regular contractions on L p (Ω) with 1 < p < ∞ and {P t } t>0 its subordinated Poisson semigroup.
(i) We have and L T t,p max(p, p ′ ).
For symmetric diffusion semigroups we have the following more precise orders than those in part (ii) above.We are very grateful to the anonymous referee for pointing out the references [32,37] that allow us to improve our previous estimate on L T c,p based on Stein's classical analyticity angle of {T t } t>0 on L p (Ω).
Corollary 8.2.Let {T t } t>0 be a semigroup of contractions on L p (Ω) for every 1 ≤ p ≤ ∞.Assume that {T t } t>0 is strongly continuous on L 2 (Ω) and each T t is a selfadjoint operator on L 2 (Ω).Then Proof.First note that T t is a regular contraction on L p (Ω) for any 1 ≤ p ≤ ∞ and any t > 0.
The selfadjointness of T t implies that {T t } t>0 is an analytic semigroup of type π 2 with constant 1 on L 2 (Ω).Then by [37,Corollary 3.2]    (iv) If {T t } t>0 is a symmetric submarkovian semigroup, Cowling [12] proved that the negative generator A of {T t } t>0 has a bounded holomorphic functional calculus whose relevant constant is of polynomial growth on p as p → 1 and p → ∞.Using the equivalence between bounded holomorphic functional calculus and square function inequalities, one can then deduce a polynomial growth of L T c,p and L T t,p too, the resulting orders are less good than those in Corollary 8.2.
Remark 8.3.The orders on L P c,p in Theorem 8.1 are optimal both as p → ∞ and p → 1 for they are already optimal for the classical Poisson semigroup on R (see Proposition 8.5 below).Zhendong Xu and Hao Zhang [68] proved that L T t,p p as p → ∞ when {T t } t>0 is a symmetric diffusion semigroup, so L P t,p p as p → ∞ for the subordinated Poisson semigroup {P t } t>0 too.This shows that our method is optimal.However, at the time of this writing, we are unable to determine the optimal orders of L P t,p as p → 1 even when {T t } t>0 is a symmetric diffusion semigroup.Problem 8.4.It would be interesting to determine the optimal orders of L P t,p as p → 1 when {T t } t>0 is a symmetric submarkovian (or markovian) semigroup.In particular, does there exist a constant C (possibly depending on {T t } t>0 ) such that The dual version of the above inequality is related to the BMO space considered in [19].It is true when {P t } t>0 is the Poisson or heat semigroup on R d .
We conclude this section by the proof of the optimality of the growth orders of the best constants in Corollary 1.6 in the scalar case, i.e., X = C (see [67] for more related results).We will denote L P c,q,p (C) and L H c,q,p (C) simply by L P c,q,p and L H c,q,p , respectively.It suffices to consider R. Proposition 8.5.Let 1 < p, q < ∞.Then L P c,q,p max p 1 q , p ′ and L H c,q,p max p 1 q , p ′ .Proof.By Remark 1.1, it suffices to show the assertion on the Poisson semigroup.
Let us first consider the case p ≤ q.Fix s > 0 and let f = P s .Then For x ≥ 6s, we have x . Thus On the other hand, Hence, L P c,q,p p ′ .
Unfortunately, the above simple argument does not apply to the case p > q.Our proof for the latter is much harder.By periodization, it is equivalent to considering the torus T (equipped with normalized Haar measure).The g-function relative to the Poisson semigroup on T is defined by where P r denotes the corresponding Poisson kernel: It is shown in [64] that the inequality G P q (f ) Lp(T) ≤ L P c,q,p f Lp(T) is equivalent to the corresponding dyadic martingale inequality on Ω = {−1, 1} N .It is well known that the relevant constant in the latter martingale inequality for q = 2 is of order √ p as p → ∞.To reduce the determination of optimal order of L P c,q,p to the martingale case, we need to refine an argument in the proof of [64,Theorem 3.1].
Keeping the notation there, let M = (M k ) 0≤k≤K be a finite dyadic martingale and where (ε k ) are the coordinate functions of Ω.The transformation ε k = sgn(cos θ k ) establishes a measure preserving embedding of Ω into T N .Accordingly, define Given (n k ) a rapidly increasing sequence of positive integers, put The functions f (n) , a k,(n) and b k,(n) are viewed as functions on T for each (θ 1 , • • • , θ K ) arbitrarily fixed.Furthermore, by approximation, we can assume that all a k and b k are polynomials.Then, if the sequence (n k ) rapidly increases, Lemmas 3.4 and 3.5 of [64] imply Therefore, The discussion so far comes from [64].Now we require a finer analysis of the g-function G P q (b k,(n) ).To this end we write the Fourier series of the function b = sgn(cos θ): b(e iθ ) = 2 π ∞ j=0 (−1) j 2j + 1 e i(2j+1)θ + e −i(2j+1)θ . Then (−1) j r 2n k j e i(2j+1)(θ k +n k θ) .

Elementary computations show
Here and below, c, C denote absolute positive constants.Thus Now lifting both sides of (8.1) to power p, then integrating the resulting inequality over T K with respect to (θ 1 , • • • , θ K ), we get For each fixed θ, the change of variables being a measure preserving transformation of T K , we deduce Letting n 1 → ∞, we get . Now we consider an elementary example where M is simple random walk stopped at ±2, namely Note that the probability of the event {τ = j} is zero for odd j and 2 − j 2 for even j.On the other hand, recalling ε k = sgn(cos θ k ) and letting we easily check that the probability of A j is 8 − j 2 for even j.Thus for K ≥ j Noting that |M K | ≤ 2 and combining all the previous inequalities together, we finally obtain This completes the proof.

Appendix A. Examples
There exist plenty of examples of semigroups to which the results of this article apply.Many second order differential operators in analysis generate such semigroups.In the following we will only discuss the cotype case since the type case can be dealt with by duality.Note that it is obvious that if X is of Luzin cotype q relative to {T t } t>0 , it is so relative to the subordinated Poisson semigroup {P t } t>0 of {T t } t>0 .
A main task in the study of the vector-valued Littlewood-Paley-Stein theory would be the following Problem A.1.Determine the family of semigroups {T t } t>0 such that a Banach space X is of Luzin type (resp.cotype) q relative to {T t } t>0 or its subordinated Poisson semigroup {P t } t>0 iff X is of martingale type (resp.cotype) q.
Example A.2. Laplacian operators.The classical heat semigroup on R d is given by H t = e t∆ , where ∆ is the Laplacian operator.It is well known (and easy to check) that {H t } t>0 is analytic of angle π 2 on L p (R d ; X) for any 1 ≤ p < ∞ and any Banach space X; the relevant constant as in (1.3) with β 0 = π 2 depends only on d.By [27,66], if X is of martingale cotype q, then X is of Luzin cotype q relative to {H t } t>0 .Conversely, suppose that X is of Luzin cotype q relative to {H t } t>0 , then X is also of Luzin cotype q relative to the classical Poisson semigroup {P t } t>0 , thereby X is of martingale cotype q by virtue of [41].Thus the Luzin cotype relative to the classical heat semigroup is equivalent to the martingale cotype.
Example A.3.Schrödinger operators.Let Ω be a region in R d equipped with Lebesgue measure.Let a(x) = (a ij (x)) 1≤i,j≤d be a positive matrix whose entries are locally integrable real functions on Ω such that α(x) ≤ a(x) ≤ β(x) for two positive continuous functions α and β on Ω.We consider the following elliptic operator Given V a nonnegative locally integrable function on Ω, define A(f ) = L(f ) + V f .It is well known that −A generates a symmetric submarkovian (markovian for V = 0) semigroup {T t } t>0 on Ω (cf. [15, Theorem 1.8.1]).In particular, {T t } t>0 is analytic on L p (Ω) for any 1 < p < ∞.Thus if X is of martingale cotype q, then it is of Luzin cotype q relative to the Poisson semigroup {P t } t>0 subordinated to {T t } t>0 on L p (Ω; X) for any 1 < p < ∞.Assume in addition that L is uniformly elliptic, namely, the above two functions α and β are constant.Then the integral kernel K 0 t (x, y) of e −tL satisfies the following Gaussian upper bound (cf.[15, Theorem 3.2.8])

By the Trotter formula
we deduce that the integral kernel K t (x, y) of e −tA is majorized by K 0 t (x, y): K t (x, y) ≤ K 0 t (x, y).Thus K t (x, y) satisfies the same Gaussian upper bound as K 0 t (x, y).Let z ∈ C with Rez > 0. By [15, Theorem 3.4.8], the complex time heat kernel of e −zA satisfies Then we easily show that {T t } t>0 extends to an analytic semigroup of type π 2 on L p (Ω; X) for any X and any 1 < p < ∞ (p can be equal to 1 too).It then follows that X is of Luzin cotype q relative to {T t } t>0 whenever X is of martingale cotype q.
As in the case of Laplacian operators, it is likely that the Luzin cotype relative to the semigroups generated by Schrödinger operators characterizes the martingale cotype.This is indeed the case if Ω = R d with d ≥ 3, L = −∆ and the potential V satisfies a reverse Hölder inequality (see [1]).Let us formulate the general case explicitly as a conjecture.
Conjecture A.4.Let {P t } t>0 be the Poisson semigroup subordinated to the heat semigroup {T t } t>0 generated by a Schrödinger operator as above.If a Banach space X is of Luzin cotype q relative to {P t } t>0 , then X is of martingale cotype q.The same is conjectured for {T t } t>0 itself when the underlying differential operator L is uniformly elliptic.
Example A.5. Laplace-Beltrami operators.The preceding examples can be extended to the setting of Riemannian manifolds.Let M be a complete d-dimensional Riemannian manifold with metric g.Let a(x) = (a ij (x)) 1≤i,j≤d be a positive matrix smoothly depending on x ∈ M .The associated second order elliptic operator L is represented as in local coordinates.Then T t = e −tL extends to a symmetric diffusion semigroup on M for all 1 ≤ p ≤ ∞ (cf.[14]).Thus our previous results apply to the associated subordinated Poisson semigroup.The most important case is the one where −L = ∆ is the Laplace-Beltrami operator.
Then the celebrated theorem of Li and Yau [39] asserts that the integral kernel of e t∆ has a Gaussian upper bound under the additional assumption that the Ricci curvature be nonnegative (see also [15,Theorem 5.5.6]).Thus as in the Euclidean case, the heat semigroup {e t∆ } t>0 extends to an analytic semigroup on L p (M ; X) for any Banach space X and 1 < p < ∞.It would be interesting to determine whether the Luzin cotype of X relative to {e t∆ } t>0 characterizes the martingale coptye of X.
Example A.6.Hermite operators.The Hermite operator on R d is a particular Schrödinger operator: A = −∆ + |x| 2 .The associated semigroup {T t } t>0 is a symmetric submarkovian semigroup on R d .The integral kernel of T t is given by |x − y| 2 coth t + |x + y| 2 tanh t .
Using the Trotter formula, we see that K t (x, y) is less than or equal to the heat kernel: This Gaussian upper bound can be deduced from the above explicit formula of K t .It then follows that {T t } t>0 is an analytic semigroup of type π 2 on L p (R d ; X) for any Banach space X and 1 < p < ∞.Betancor et al showed in [7] that X is of Luzin cotype q relative to {T t } t>0 iff X is of martingale cotype q (see also [9] for related results).
Example A.7. Laguerre operators.For simplicity, we only consider the Laguerre semigroup on R + , the multi-dimensional case can be treated by tensor product.In this example, R + is equipped with Lebesgue measure, contrarily to our usual convention.Let α > −1 and ) , x > 0.
For every f ∈ L 2 (R + ), setting we consider the operator A formally defined by Note that A(f ) = A(f ) if f is compactly supported and smooth.Then −A generates a symmetric semigroup {T t } t>0 of positive contractions on L 2 (R + ): with kernel given by where I α is the modified Bessel function of the first kind and order α: .
On the other hand, [7] shows that for α > − 1 2 , a Banach space X is of Luzin cotype q relative to {T t } t>0 iff X is of martingale cotype q; as a byproduct, [7] also shows that for the same range of α, {T t } t>0 is analytic on L p (R + ; X) for any X.
In the remaining case of −1 < α < − 1 2 , it is quite easy to show that {T t } t>0 is a bounded semigroup on L p (R + ) for p α < p < p ′ α .Let us outline the argument for the convenience of the reader.By dilation invariance via the change of variables u = x/ √ sinh t and v = y/ √ sinh t, the kernel K t is brought to ϕ(x, y) = √ xy I α (xy) exp − 1 2 (x 2 + y 2 ) cosh t .
Let Φ be the associated integral operator: ϕ(x, y)f (y)dy.
To estimate the norm of Φ in B(L p (R + ; X)) for p α < p < p ′ α , we appeal to the following estimates of the Bessel function (cf.[35,Chapter 5]) as z → 0 and I α (z) ≈ e z √ πz as z → ∞.
In particular, {T t } t>0 is a symmetric semigroup of positive contractions on L 2 (R + ), so analytic.Applying the previous sections to the associated subordinated Poisson semigroup {P t } t>0 , we recover the result of [9] that X is of Luzin cotype q relative to {P t } t>0 on L 2 (R; X) whenever X is of martingale cotype q.Moreover, [9] shows that the converse is also true.Note that [9] also extends this result to all p ∈ (p α , p ′ α ).We do not know, however, to determine the analyticity of {T t } t>0 on L p (R; X) for the range −1 < α < − 1 2 .Remark A.8.We would like to point out an interesting phenomenon revealed by this example.It is easy to get a semigroup of contractions on L 2 thanks to spectral theory.If the contractions are further positive (or regular), the results of the previous sections apply.In many concrete examples, one can then extrapolate L 2 to L p using tools from harmonic analysis.This is indeed the case for all previous examples.Example A.9. Ornstein-Uhlenbeck semigroup.Now R d is equipped with the canonical Gaussian measure γ d .Let {T t } t>0 be the Ornstein-Uhlenbeck semigroup on R d whose negative generator is given by A = −∆ + x • ∇.This is again a symmetric diffusion semigroup.By [52], {T t } t>0 is analytic on L p (R d , γ d ; X) iff X is K-convex (a property weaker than the finite martingale cotype).On the other hand, by [41], X is of Luzin cotype q relative to the Poisson semigroup subordinated to {T t } t>0 iff X is of martingale cotype q.We then deduce that the Luzin cotype q of X relative to {T t } t>0 itself on L p (R d , γ d ; X) characterizes the martingale cotype q of X.It is worth noting that in contrast to [41], all estimates obtained by the present method or by [66] are dimension free.
Example A.10. Walsh semigroup.Let Ω = {−1, 1} N be the dyadic group equipped with normalized Haar measure.The coordinate functions {ε n } n≥1 on Ω form an independent sequence of symmetric random variables (Rademacher functions).We introduce the Walsh system (w A ): for any finite subset A ⊂ N let w A = k∈A ε k .
If A = ∅, w ∅ = 1.All such w A 's form an orthonormal basis of L 2 (Ω).Any f ∈ L 2 (Ω) admits the following Fourier expansion: Then {T t } t>0 is a symmetric diffusion semigroup on Ω, it can be viewed as a baby model of the Ornstein-Uhlenbeck semigroup.Again, by [52], {T t } t>0 is analytic on L p (Ω; X) iff X is K-convex.
Remark A.11.Let {T t } t>0 be the semigroup in the above example.Then the Luzin cotype relative to {T t } t>0 characterizes the martingale cotype.
Indeed, assume that X is of Luzin cotype q relative to {T t } t>0 .Then by an argument via the central limit theorem as in [3], we can show that X is of Luzin cotype q relative to the Ornstein-Uhlenbeck semigroup too, so by the previous example, X is of martingale cotype q.
It would be interesting to show the analogue of Corollary 1.6 for Ornstein-Uhlenbeck or Walsh semigroup.
Problem A.12.Let {T t } t>0 be the Ornstein-Uhlenbeck or Walsh semigroup as above and X be of martingale cotype q.Does one have L T c,q,p (X) max p 1 q , p ′ M c,q (X)?
It would, of course, suffice to consider the Walsh case.On the other hand, it is likely that in the Ornstein-Uhlenbeck setting one could get a dimension dependent estimate L T c,q,p (X) d max p 1 q , p ′ M c,q (X) by standard techniques on the Ornstein-Uhlenbeck semigroup.However, here a dimension free estimate is more important than the corresponding one for the heat semigroup in R d in view of analysis in Wiener space.
Example A.13.Translation semigroup.We have already used the translation semigroup {τ t } t>0 in in the proof of Theorem 1.2.It is not analytic on L p (R) for any p.By (7.6), if X is of Luzin cotype q relative to the subordinated Poisson semigroup {P τ t } t>0 on L p (R; X), so is X relative to the Poisson semigroup subordinated to any semigroup of regular contractions {T t } t>0 on L p (Ω; X).Consequently, the Luzin cotype relative to {P τ t } t>0 is equivalent to the martingale cotype.
Example A.14. L 2 -theory.Let A be a positive densely defined operator on L 2 (Ω).Then −A generates an analytic semigroup {T t } t>0 of contractions on L 2 (Ω).Being positivity preserving can be characterized by means of the Dirichlet form associated to A (cf. [15,Theorem 1.3.2]).Many classical examples are built in this way.Note, however, that it can happen that {T t } t>0 extends to a semigroup of bounded operators on L p (Ω) for p only in a small symmetric interval around 2 as shown by the Laguerre semigroup.Even worse, it can happen that {T t } t>0 does not extrapolate to L p (Ω) for any p = 2.

Added in proof
After the submission of this article for publication, a few related works have appeared.For instance, A.K. Lerner, E. Lorist and S. Ombrosi [36] found a new proof of Theorem 1.5 (i) by the sparse domination principle, T.P. Hytönen and S. Lappas [26] obtained an estimate close to that appearing in Theorem 1.4 (ii), and G. Hong, Z. Xu and H. Zhang [24] partially resolved Problem 1.8, Problem A.1 and Conjecture A.4.

Lemma 5 . 2 .
The family {3Q} Q∈D is a disjoint union of 3 d dyadic-like families.It suffices to consider the case d = 1.Then every 3Q can be written in the form [ 3j+s 2 k , 3(j+1)+s 2 k