Quantitative estimates for bounded holomorphic semigroups

In this paper we revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood--Paley--Stein theory for symmetric diffusion semigroups.


Introduction
The theory of Banach valued martingales involving the notions of martingale type and cotype was introduced and studied in depth by Pisier (we refer to [10,11,12,13] and Section 4 for more information).His renorming theorem states that these are geometric properties of the underlying Banach space, characterized by the existence of an equivalent norm in the space which is uniformly convex of power type q (for the precise statement of this result see Theorem 4.7 in Section 4).
On the other hand, first Stein in [14, Chapter IV] proved the following result which extends the classical inequality on the Littlewood-Paley g-function: for every 1 < p < ∞ (1.1) where by {T t } t>0 we denote a symmetric diffusion semigroup (see Definition 4.1 in Section 4) and the equivalence constants depend only on p.
The aforementioned theory about martingale inequalities is closely related to the vector-valued Littlewood-Paley-Stein theory which was developed in [8,15,16].In this direction, a Littlewood-Paley theory was first developed in [15] for functions with values in uniformly convex Banach spaces.In particular, Xu obtained in [15] the one-sided vector-valued extension of (1.1) for the classical Poisson semigroup on the torus T.
Martínez-Torrea-Xu [8] characterized, in the vector-valued setting, the validity of the following one-sided inequality concerning the generalized Littlewood-Paley g-function associated with a subordinated Poisson symmetric diffusion semigroup {P t } t>0 by the martingale type and cotype properties of the underlying Banach space: for given Banach space X and 1 < q < ∞, X is of martingale cotype q iff for every 1 < p < ∞ (1.2) f Lp(Ω;X) , ∀f ∈ L p (Ω; X).
The converse inequality of (1.2) was also treated in [8] iff X is of martingale type q.Here the subordinated Poisson semigroup {P t } t>0 is defined by where {T t } t>0 is a symmetric diffusion semigroup.This {P t } t>0 is again a symmetric diffusion semigroup.Recall that if A denotes the negative infinitesimal generator of {T t } t>0 , then P t = e − √ At .The question whether (1.2) is also true for a general (not necessarily subordinated) diffusion semigroup {T t } t>0 in place of {P t } t>0 , or even just, in the special case of the heat semigroup on R n , was left open in [8] (see [8,Problem 2

on p. 447]).
This question concerning the heat semigroup on R n was answered in the affirmative by Naor and one of us [6], yielding good dependence of various bounds on the target Banach space X, but also with dependence on the dimension n of the domain R n .This was further elaborated by Xu [16], who extended the previous result to general diffusion semigroups {T t } t>0 and replaced a concrete bound for the derivative of the heat kernel used in [6] by an abstract analyticity result from [11].While giving a form of the Littlewood-Paley-Stein inequality entirely free of the domain's dimension n, this abstract argument relies on implicit unquantified bounds, and obscures the dependence of the bound on the target Banach space X, which was completely explicit in [6].
The main result of this work is Theorem 5.1 in Section 5 which provides a quantitative version of the main result of [16, Theorem 2 (i)], including explicit dependence on the target Banach space X and quantitative dimension free version of (1.3) not only for the heat semigroup on R n but also for the general diffusion semigroup {T t } t>0 (see also [17,18] for some related work).
The paper is organized as follows: in Sections 2, 3 and 4 we present the tools for proving our main result in Section 5. To be more precise, we provide quantitative versions of several classical results in the theory of bounded holomorphic semigroups, in particular their characterization due to Kato [7].Having these at our disposal, in Section 4 we quantify the dependence of the analyticity bound of [11,Remark 1.8(b)] (see also Corollary 4.16) for extensions of diffusion semigroups {T t } t>0 to uniformly convex spaces.As discussed above, this has direct implications to bounds in Littlewood-Paley-Stein inequalities (1.3) by [6,16].In addition, we provide an explicit dependence of sup t>0 t∂T t on the martingale cotype constant m q,X (see Corollary 4.17), which was the case for the related estimates in [6].Section 5 is devoted to the proof of Theorem 5.1.Finally, in Section 6 using Corollary 4.17 we obtain a quantitative version of one more result of Xu [17,Theorem 1.4] which we compare with Theorem 5.1.
Notation.Throughout the paper, given a, b ∈ (0, ∞), the notation a b means that a ≤ cb for some universal constant c ∈ (0, ∞).The notation a ≈ b stands for (a b) ∧ (b a).If we need to allow for dependence on parameters, we indicate this by subscripts.Moreover, given a Banach space X, we denote by L p (Ω; X) the usual L p space of strongly measurable functions from Ω to X and we will use the abbreviation ∂ = ∂/∂t.

Preliminaries
We begin by recalling several basic definitions and results from the semigroup theory: 2.1.Definition (Strongly Continuous Semigroup).Let X be a Banach space.A family (T (t)) t≥0 of bounded linear operators on X is called a strongly continuous (one-parameter semigroup) (or

2.2.
Remark.A consequence of Definition 2.1 and the uniform boundedness principle (see also [3,Proposition 5.5]) is the following exponential boundedness of a strongly continuous semigroup (T (t)): there exist constants M ≥ 1 and ω ∈ R such that T (t) ≤ M e ωt for all t ≥ 0.

Definition (Infinitesimal Generator
).Let (T (t)) be a C 0 -semigroup.The infinitesimal generator of (T (t)) is the linear operator defined by

Definition.
Let A : D(A) ⊂ X → X be a a closed, linear operator on some Banach space X.We call is, by the closed graph theorem, a bounded operator on X and will be called the resolvent of A at the point λ.(Hille-Yosida Generation Theorem) Let A be a linear operator on X and let M ≥ 1, ω ∈ R. The following conditions are equivalent: (a) A generates a C 0 -semigroup satisfying (b) A is closed and densely defined, the resolvent set ρ(A) contains the halfplane {λ ∈ C : Re λ > ω}, and for all n ∈ N.

Classical semigroup theory made quantitative
In this section we collect the additional results from semigroup theory that we will use in Section 4. Most of these results are in principle known, but difficult to find in the precise quantitative form that we need.For this reason, we also revisit most of the proofs in order to track the dependence on the various constants.Besides our application in Section 4, we hope that recording these quantitative formulations might have an independent interest and possible other applications elsewhere.In addition to the standard references [3,9], we have benefited from the nice presentation of the classical theory in [4].
3.1.Lemma ([3], Theorem 4.6; [9], Chapter 2, Theorem 5.2; [4], Theorem 1.1.23).Let (A, D(A)) be a linear operator on X. Suppose that A generates a bounded strongly continuous semigroup (T (t)) with T (t) ≤ M on X, where M ≥ 1, and there exists a constant C > 0 such that ) and the resolvent of A satisfies 1 − q for all q ∈ (0, 1) and all λ ∈ C such that | arg(λ)| ≤ π 2 + arctan( q C ). Proof.This is essentially contained in the proof of (b) ⇒ (a) of [4 thus | Re λ|/| Im λ| ≤ q/C.For any q ′ ∈ (q, 1), [4, page 22] obtains the estimates Taking the limit q ′ → q, we conclude that for all λ as in ( * ).By combining estimates (3.4) and (3.5) we deduce that For linear operators satisfying the conclusions of Lemma 3.1 and appropriate paths γ, the exponential function "e tA " can now be defined via the Cauchy integral formula.In particular, we write ), where C ≥ 1, and the resolvent of A satisfies Then, for all z ∈ Σ δ , the maps T (z) are bounded linear operators on X and ) and all u ∈ (0, 1).
We will verify that the integral in (3.6) defining T (z) converges uniformly in L(X) with respect to the operator norm.Since the functions e µz and R(µ, A) are analytic in Σ π 2 +δ , this integral, if it exists, is by Cauchy's integral theorem independent of the particular choice of γ.Hence, we may choose γ to be the positively oriented boundary of Σ π 2 +arctan( q C ) \ B 1 |z| (0).Now, we decompose γ in three parts given by the arc γ formed by the boundary of the disk and the two rays going to infinity (this follows by change of variables µ = re ±i( π 2 +arctan( q C )) ): Hence, where Combining this with (3.8) and (3.9), and denoting M := , we get We now choose q = 1+Cu 1+C ∈ (u, 1), so that Recalling that C ≥ 1, it follows that Together with (3.10), this shows that This shows that the integral defining (T (z)) converges in L(X) absolutely and uniformly for every z ∈ C such that | arg(z)| ≤ arctan( u C ), and proves the asserted bound.3.11.Remark.As shown in [3,Proposition 4.3] (see also [9, Chapter 2, Theorem 5.2]) the previous family of bounded linear operators T (z) also satisfies properties (i), (ii) and (iii) of Definition 2.7.
The following is a variant of [1, Corollary 3.7.12]and [9, Chapter 2, Theorem 5.2]; except for the quantitative bound, it is stated in the same form in [4, Lemma 1.1.28]:3.12.Theorem.Let A be the infinitesimal generator of a strongly continous semigroup T (t) with T (t) ≤ M , M ≥ 1. Suppose that there exists a constant C ≥ 1 such that is ∈ ρ(A) for |s| > 0 and Then A generates a bounded holomorphic semigroup T (z), defined at every
Proof.The proof follows the idea of the proof of [4, Lemma 1.1.28].Since a self-contained proof is not much longer than an indication of the relevant changes, we give it for the reader's convenience.The Hille-Yosida Theorem 2.5 shows that If r > 0 and s = 0, the resolvent equation (Theorem 2.8) provides the identity Taking the operator norm on both sides and applying (3.13) and (3.14), we deduce the bound where C := C(M + 1).Hence, A satisfies the assumptions and therefore the conclusions of Lemma 3.1.Moreover, by Lemma 3.7 we obtain where in the last step we chose u = 1 2 .Therefore A generates a bounded holomoprhic semigroup with the stated properties by Remark 3.11.
The following is a quantitative version of a theorem of Kato [7].We follow the proof given in [4, Lemma 1.2.2]: 3.15.Theorem.Let (T (t)) be a strongly continuous semigroup with T (t) ≤ M , M ≥ 1. Suppose that there is a complex number ζ with |ζ| = 1 and K ∈ (0, ∞) such that Then (T (t)) can be extended to a bounded holomorphic semigroup.In particular, , where Note that any constant K ∈ (0, ∞) in (3.16) must necessarily satisfy K ≥ 1 2 .Indeed, as t → 0, we have T (t)x → x, and hence Proof.We will borrow the results of some intermediate steps of the proof of the implication (c) ⇒ (a) of [4, Lemma 1.2.2].That proof is set up for a more general semigroup with a growth bound T (t) ≤ M e ωt , so we can simply take ω = 0.Moreover, (3.16) is only assumed for t ∈ (0, δ), so we can take δ = ∞, thus 1/δ = 0.This will simplify some of the formulas of [4].
3.17.Remark.A version of Theorem 3.15 is also true (see [4, Lemma 1.2.2] and [7]), where (3.16) is only assumed for 0 < t < δ, but we only treat the case stated, since it simplifies the quantitative conclusions and it is the only case that we need for our applications.
3.18.Corollary.Let (T (t)) be a strongly continuous semigroup with T (t) ≤ M , for some M ≥ 1.In addition, suppose that there exists 0 < ε ≤ 2 such that Then (T (t)) can be extended to a bounded holomorphic semigroup in Σ θ , where θ ≈ 1

C
and C = M 2 ε > 0. In particular, and Proof.We choose ζ = −1.Since we deduce from Theorem 3.15 with K = 1 ε ≥ 1 2 that (T (t)) extends to a bounded holomorphic semigroup which satisfies the following estimate: , where C = M 2 ε > 0. By Cauchy's integral formula we get where γ is the boundary of the disk B t sin(θ) (t) for t > 0 and θ ≈ 1

C
is the angle of the sector Σ θ in which T (z) is holomorphic.Using (3.22) and (3.23) we deduce that Hence, where in the last step sin(θ) 3.24.Remark.The first part of Corollary 3.18 was proved by Kato in [7] under the assumption of (3.19) for 0 < t < δ.The novelty here is the estimate (3.20) and (3.21).

Quantitative analyticity of diffusion semigroups in uniformly convex spaces
In our discussions below we follow [16] but we provide new details concerning the quantitative bounds.Let us recall the following definitions about symmetric diffusion semigroups, uniform convexity, martingale type and cotype: 4.1.Definition.Let (Ω, A, µ) be a σ-finite measure space.An operator T on (Ω, A, µ) is a symmetric Markovian operator if it satisfies the following properties: (1) T is a linear contraction on L p (Ω) for every 1 ≤ p ≤ ∞; (2) T is positivity preserving and T 1 = 1; and (3) T is a self-adjoint operator on L 2 (Ω).

4.2.
Definition.Let (Ω, A, µ) be a σ-finite measure space.By a symmetric diffusion semigroup on (Ω, A, µ) in Stein's sense [14, Section III.1], we mean a family {T t } t>0 of linear maps where each member of this family is a symmetric Markovian operator and satisfies the following two additional properties: (4)

Definition.
A Banach space X is said to be uniformly convex if for every ε ∈ (0, 2] there exists δ ∈ (0, 1] such that, for any vectors x, y in the closed unit ball (i.e.x ≤ 1 and y ≤ 1) with x − y ≥ ε, one has x + y ≤ 2(1 − δ).In addition, (X, • ) is said to be uniformly convex of power type q with 2 ≤ q < ∞ and positive constant δ if the following inequality holds: (4.4) x + y 2 Examples of uniformly convex spaces that satisfy (4.4) with δ = 1 are the following: (1) X is a Hilbert space and q = 2.By the Parallelogram law for inner product spaces it follows: (2) X = L q (Ω) and 2 ≤ q < ∞.We have the following Clarkson's inequality (see [5,Corollary 2.29]): 4.5.Definition.Let 1 < q < ∞.A Banach space X is of martingale cotype q if there exists a positive constant C such that every finite X valued L q -martingale (f n ) defined on some probability space satisfies the following inequality: where E denotes the expectation on the underlying probability space.We then must have q ≥ 2. X is of martingale type q if the reverse inequality holds.

4.7.
Theorem ([10], Theorem 3.1).Let q be such that 2 ≤ q < ∞ and let X be a Banach space.Assume that X is of martingale cotype q, namely: Then there exists an equivalent norm | • | on X such that: In particular, (X, | • |) is uniformly convex of power type q with constant δ = m −q q,X .4.9.Remark.The above result states that if X has martingale cotype q, then it admits an equivalent norm that satisfies (4.8).The converse to this is also true and it can be found in [10, Remark 3.1] and [13, Theorem 10.6 (i) =⇒ (iii)].In particular, if we assume that X is uniformly convex of power type q with constant δ, then X has martingale cotype q with m q,X ≤ 2δ − 1 q .As it can be seen below in our applications, we could work under the assumption of uniformly convex of power type q spaces which would imply estimates in terms of the constant δ.
We will need two results in order to derive bounds for the family of operators {t∂T t } t>0 on Y = L p (Ω; X), where 1 < p < ∞, X is a uniformly convex Banach space or a Banach space of martingale cotype 2 ≤ q < ∞.The first one is the following Rota's dilation theorem for positive Markovian operators, which we state following the formulation in [16, Lemma 10]: 4.10.Lemma ( [14], Chapter IV and [16], Lemma 10).Let T = S 2 with S a symmetric Markovian operator on (Ω, A, µ).Then there exist a larger measure space ( Ω, A, µ) containing (Ω, A, µ), and a σ-subalgebra B of A such that, for every p ∈ [1, ∞) and every Banach space X, where E A is the conditional expectation relative to A (and similarly for E B ).
The fact that Rota's theorem remains valid for functions taking values in an arbitrary Banach space, as stated above, has been observed e.g. in [8, after Theorem 2.5]) and [16, the beginning of the proof of Lemma 9].We indicate a proof for the reader's convenience.
Proof.For X ∈ {R, C} (i.e., scalar-valued functions), a more general version of this lemma is formulated and proved in [14, Chapter IV, Theorem 9], and restated in the form above as [16,Lemma 10].
Let then X be a general Banach space.
, where f k ∈ L p (Ω, A, µ) are scalar-functions and x k ∈ X, the identity follows from the fact that it holds for each f k by linearity of both sides of (4.11).
For the general case, we note that functions of the form just discussed are dense in L p (Ω, A, µ; X).On the other hand, both symmetric Markovian operators (by Definition 4.1) and conditional expectations (by their wellknown basic properties) are positive linear operators (i.e., they map nonnegative functions to nonnegative functions).Such operators, initially defined on scalar-valued L p spaces, have canonical bounded linear extensions to the corresponding L p spaces of X-valued functions, for any Banach space X (for this result, see e.g.[5,Theorem 2.1.3]).Since (4.11) holds for all f in a dense subspace of L p (Ω, A, µ; X), and both sides depend continuously on f ∈ L p (Ω, A, µ; X), the identity remains valid for all f as stated.
The following lemma is well known and immediate from the definition: 4.14.Lemma.If (X, • ) is uniformly convex of power type q with 2 ≤ q < ∞ and constant δ > 0 appearing in (4.4), then so is The following lemma is a quantitative elaboration of [16, Lemma 9]: 4.15.Lemma.Let T = S 2 with S a symmetric Markovian operator on (Ω, A, µ) and suppose that X is uniformly convex of power type q with 2 ≤ q < ∞ and δ > 0 appearing in (4.4).Then where ε := 2δ 1+2δ 1 q ∈ (0, 1).Proof.Notice that by Lemma 4.14, Y = L q (Ω; X) is also uniformly convex of power type q with constant δ > 0 and T is a contraction on Y .By Rota's dilation lemma (its X-valued version as formulated in Lemma 4.10), we have On the other hand, observe that E A acs as the identity on Y .Hence, for any λ ∈ C, we have where we abbreviated P := E B .

Littlewood-Paley-Stein inequalities: First approach
In this section we will use the quantitative analyticity bounds of the previous section to obtain the following new quantitative version of [16, Theorem 2 (i)]: 5.1.Theorem.Let X be a Banach space and k a positive integer.If X has martingale cotype q with 2 ≤ q < ∞, then for every symmetric diffusion semigroup {T t } t>0 and for every 1 < p < ∞ we have p,q k k−1 B k+1 m q,X f Lp(Ω;X) , for all f ∈ L p (Ω; X), where B := q 2 (m q,X ) 2q+1 (1 + log(q) + q log(m q,X )) is the constant appearing in (4.19) of Corollary 4.17.
If p = q and k = 1, then we have the sharper bound Bm q,X f Lq(Ω;X) , for all f ∈ L q (Ω; X) and the implied constant is absolute.

5.4.
Remark (see also [16], Remark 3).Applied to the heat semigroup {H t } t>0 of R n , the above theorem implies a dimension free estimate for the g-function associated to {H t } t>0 : when X is of martingale cotype q.In addition, if p = q the same theorem implies Compare this with the following estimate which was obtained in [6,Theorem 17]: when Y is a Banach space that admits an equivalent norm with modulus of uniform convexity of power type q.We note that the constant B can be very large, so that the second bound is typically sharper in moderate dimensions n, but becomes inferior as n → ∞.
In order to prove Theorem 5.1 the following lemma, due to Naor and one of us [6, Lemma 24] will play a crucial role in our argument: 5.5.Lemma ([6], Lemma 24).Fix q ∈ [2, ∞), α ∈ (1, ∞) and a Banach space (X, • X ) of martingale cotype q with constant m q,X .Suppose that {T t } t>0 is a symmetric diffusion semigroup on a measure space (Ω, X).Then for any f ∈ L q (Ω; X) we have The following lemma shows Theorem 5.1 in the case p = q and k = 1, 2, but the bound provided by the lemma is worse than that asserted in the theorem when k > 2. Thus, some more work will be needed further below to obtain Theorem 5.1 as stated.5.6.Lemma (quantitative version of [16], Lemma 13).Let X be a Banach space of martingale cotype 2 ≤ q < ∞ with cotype constant m q,X and k be a positive integer.Moreover, suppose that {T t } t>0 is a symmetric diffusion semigroup on a measure space (Ω, X).Then for any f ∈ L q (Ω; X) we have where B := q 2 (m q,X ) 2q+1 (1 + log(q) + q log(m q,X )) is the constant appearing in (4.19) of Corollary 4.17.

5.8.
Remark.Later on we will only use this lemma in the case k = 1 and we will get a different bound form the one appearing in (5.7) for k > 1.
Proof.We will use the idea of the proof of [6,Theorem 17].By the semigroup identity ∂T t+s = ∂T t T s and the convergence Then by the triangle inequality we can estimate (5.10) We are now in a position of using Corollary 4.17 as follows: (5.11) where B := q 2 (m q,X ) 2q+1 (1 + log(q) + q log(m q,X )).
Combining the above inequalities together with Lemma 5.5, we derive Bm q,X f Lq(Ω;X) .This is (5.7) for k = 1.To handle a general k, by the semigroup identity T t+s = T t T s , we have This implies that (5.12) . Thus, by (5.11), (5.12) and the already proved inequality, we obtain (5.12) The lemma is thus proved.

Littlewood-Paley-Stein inequalities: Second approach
In this section we obtain an alternative estimate for the constant appearing in (5.3) of Theorem 5.1 by combining our Corollary 4.17 with another result of Xu [17,Theorem 1.4].Before we proceed we need to recall some notions from [17].
Recall that an operator T on L p (Ω) (1 ≤ p ≤ ∞) is regular (more precisely, contractively regular) if for all finite sequences {f k } k≥1 in L p (Ω).