A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space \gamma(H,E) of all \gamma-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{\gamma(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation that \psi(x) = || x ||^p is Fr\'echet differentiable and its derivative satisfies the Lipschitz estimate || \psi'(x) - \psi'(y)|| \leq C(|| x || + || y ||)^{p-2} || x-y ||; the extension to 0<p<2 proceeds via Lenglart's inequality.


Introduction
Let (e tA ) t 0 be a C 0 -contraction semigroup on a 2-smooth Banach space E and let (W t ) t 0 be a cylindrical Brownian motion in a Hilbert space H. Let (g t ) t 0 be a progressively measurable process with values in the space γ(H, E) of all γ-radonifying operators from H to E satisfying T 0 g t 2 γ(H,E) dt < ∞ P-almost surely for all T 0. As is well known (see [6,15,16]), under these assumptions the stochastic convolution process X t = t 0 e (t−s)A g s dW s , t 0, is well-defined in E and provides the unique mild solution of the stochastic initial value problem dX t = AX t dt + g t dW t , X 0 = 0.
In order to obtain the existence of a continuous version of this process, one usually proves a maximal estimate of the form (1.1) The first such estimate was obtained by Kotelenez [11,12] for C 0 -contraction semigroups on Hilbert spaces E and exponent p = 2. Tubaro [19] extended this result to exponents p 2 by a different method of proof which applies Itô's formula to the C 2 -mapping x → x p . The case p ∈ (0, 2) was covered subsequently by Ichikawa [10]. A very simple proof, still for C 0 -contraction semigroups on Hilbert spaces, which works for all p ∈ (0, ∞), was obtained recently by Hausenblas and Seidler [9]. It is based on the Sz.-Nagy dilation theorem, which is used to reduce the problem to the corresponding problem for C 0 -contraction groups. Then, by using the group property, the maximal estimate follows from Burkholder's inequality. This proof shows, moreover, that the constant C in (1.1) may be taken equal to the constant appearing in Burkholder's inequality. In particular, this constant depends only on p.
The maximal inequality (1.1) has been extended by Brzeźniak and Peszat [4] to C 0 -contraction semigroups on Banach spaces E with the property that, for some p ∈ [2, ∞), x → x p is twice continuously Fréchet differentiable and the first and second Fréchet derivatives are bounded by constant multiples of x p−1 and x p−2 , respectively. Examples of spaces with this property, which we shall call (C 2 p ), are the spaces L q (µ) for q ∈ [p, ∞). Any (C 2 p ) space is 2-smooth (the definition is recalled in Section 2), but the converse doesn't hold: Proposition 17]. On the other hand, the norm of ℓ 2 (F ) is twice continuously Fréchet differentiable away from the origin if and only if F is a Hilbert space [14,Theorem 3.9]. Thus, for q ∈ (2, ∞), ℓ 2 (ℓ q ) and ℓ 2 (L q (0, 1)) are examples of 2-smooth Banach spaces which fail property (C 2 p ) for all p ∈ [2, ∞). To the best of our knowledge, the general problem of proving the maximal estimate (1.1) for C 0contraction semigroups on 2-smooth Banach space remains open. The present paper aims to fill this gap: Let (e tA ) t 0 be a C 0 -contraction semigroup on a 2-smooth Banach space E, let (W t ) t 0 be a cylindrical Brownian motion in a Hilbert space H, and let (g t ) t 0 be a progressively measurable process in then the stochastic convolution process X t = t 0 e (t−s)A g s dW s is well-defined and has a continuous version. Moreover, for all 0 < p < ∞ there exists a constant C, depending only on p and E, such that For p 2, the proof of Theorem 1.2 is based on a version of Itô's formula (Theorem 3.1) which exploits the fact (proved in Lemma 2.1) that in 2-smooth Banach spaces the function ψ(x) = x p is Fréchet differentiable and satisfies the Lipschitz estimate The extension to exponents 0 < p < 2 is obtained by applying Lenglart's inequality (see (4.1)). We conclude this introduction with a brief discussion of some developments of the inequality (1.1) into different directions in the literature. Seidler [18] has proved the inequality (1.1) with optimal constant C = O( √ p) as p → ∞ for positive C 0 -contraction semigroups on the (2-smooth) space E = L q (µ), q 2. He also proved that the same result holds if the assumption 'e tA is a positive contraction semigroup' is replaced by '−A has a bounded H ∞ -calculus of angle strictly less than 1 2 π'. The latter result was subsequently extended by Veraar and Weis [20] to arbitrary UMD spaces E with type 2. In the same paper, still under the assumption that −A has a bounded H ∞ -calculus of angle strictly less than 1 2 π, the following stronger estimate is obtained for UMD spaces E with Pisier's property (α): with a constant C depending only on p and E. If, in addition, E has type 2, then the mapping f ⊗ (h ⊗ x) → (f ⊗ h)⊗ x extends to a continuous embedding L 2 (0, T ; γ(H, E)) ֒→ γ(L 2 (0, T ; H), E) and (1.2) implies (1.1).
Let us finally mention that, for p > 2, a weaker version of (1.1) for arbitrary C 0 -semigroups on Hilbert spaces has been obtained by Da Prato and Zabczyk [5]. Using the factorisation method they proved that with a constant C depending on p, E, and T . The proof extends verbatim to C 0 -semigroups on martingale type 2 spaces. This relates to the above results for 2-smooth spaces through a theorem of Pisier [17,Theorem 3.1], which states that a Banach space has martingale type p if and only if it is p-smooth.

The Fréchet derivative of
It is known (see [17,Theorem 3.1]) that E is q-smooth if and only if there exists a constant K 1 such that for all x, y ∈ E, x + y q + x − y q 2 x q + K y q .
(2.1) Lemma 2.1. Let E be a Banach space and let 1 < q 2 be given. For p q set ψ p (x) := x p .
1. E is q-smooth if and only if the Fréchet derivative of ψ q is globally (q − 1)-Hölder continuous on E.
2. If E is q-smooth, then for p > q the Fréchet derivative of ψ p is locally (q − 1)-Hölder continuous on E. Moreover, for all p q and x, y ∈ E we have where C depends only on p, q and E.
Proof. If the Fréchet derivative of ψ q is (q − 1)-Hölder continuous on E, then by the mean value theorem we can find 0 θ, ρ 1 such that for all x, y ∈ E, Hence the Banach space E is q-smooth. Suppose now that the norm of E is q-smooth. Then for all x, y ∈ E with x , y = 1 and all t > 0 we have which by [7, Lemma I.1.3] means that · is Fréchet differentiable on the unit sphere. Hence, by homogeneity, · is Fréchet differentiable on E\{0}. Let us denote by f x its Fréchet derivative at the point x = 0. We begin by showing the (q − 1)-Hölder continuity of x → f x on the unit sphere of E, following the argument of [7, Lemma V.3.5]. We fix x = y ∈ E such that x , y = 1 and h ∈ E with h = x − y and x − y + h = 0. Since the norm · is a convex function, Similarly, we have By using above inequalities and the linearity of the function f x , we have where we also used (2.3). Since the roles of x and y may be reversed in this inequality, this implies This proves the (q − 1)-Hölder continuity of the norm · on the unit sphere.
We proceed with the proof of (2.2); the (q − 1)-Hölder continuity of ψ q as well as the local (p− 1)-Hölder continuity of ψ p follow from it. For all x, y ∈ E with x = 0 and y = 0 we have ψ ′ If q p 2, then by the inequality |t r − s r | |t − s| r , valid for 0 < r 1 and s, t ∈ [0, ∞), we have If p > 2, by applying the mean value theorem, for some θ ∈ [0, 1] we have The above lemma will be combined with the next one, which gives a first order Taylor formula with a remainder term involving the first derivative only.
Lemma 2.2. Let E and F be Banach spaces, let 0 < α 1, and let ψ : E → F be a Fréchet differentiable function whose Fréchet derivative ψ ′ : E → L (E, F ) is locally α-Hölder continuous. Then for all x, y ∈ E we have Proof. Pick w ∈ E such that w 1 and consider the function f : For all x * ∈ F * , f ′ , x * is locally α-Hölder continuous. To see this, note that for |θ 1 |, |θ 2 | R and x R we have x + θ 1 w , x + θ 2 w 2R, so by assumption there exists a constant C 2R such that Applying Taylor's formula and [1, Lemma 1, Theorem 3] to the function f, x * we obtain Now let x, y ∈ E be given and set t = y − x and w = y−x y−x . With these choices we obtain Since x * ∈ F * was arbitrary, this proves the lemma.
3. An Itô formula for · p From now on we shall always assume that E is a 2-smooth Banach space. We fix T 0 and let (Ω, F , P) be a probability space with a filtration ( The space of all such ξ which satisfy where Π = {0 = t 0 < t 1 < · · · < t n = T } is a partition of the interval [0, T ] and the random variables A i are F ti -measurable and take values in the space of all finite rank operators from H to E, we define the random variable I(ξ) ∈ L 0 (Ω, F T ; E) by where C depends on p and E only. It follows that I has a unique extension to a bounded linear operator An excellent survey of the theory of stochastic integration in 2-smooth Banach spaces with complete proofs is given in Ondreját's thesis [16], where also further references to the literature can be found.
In what follows we fix p 2 and set ψ(x) := ψ p (x) = x p . Since we assume that E is 2-smooth, this function is Fréchet differentiable. Following the notation of Lemma 2.2 we set We have the following version of Itô's formula.  The process s → ψ ′ (X s )g s is progressively measurable and belongs to M 1 ([0, T ]; H), and for all t ∈ [0, T ] we have with convergence in probability, for any sequence of partitions Π n = {0 = t n 0 < t n 1 < · · · < t n m(n) = T } whose meshes Π n := max 0 i m(n)−1 |t n i+1 − t n i | tend to 0 as n → ∞. Moreover, there exists a constant C and, for each ε > 0, a constant C ε , both independent of a and g, such that The proof shows that we may take C ε = C ′ (ε 1− 2 p + 1) for some constant C ′ independent of a, g, and ε. Before we start the proof of the theorem we state some lemmas. The first is an immediate consequence of Burkholder's inequality (3.3).  Proof. By the identity ψ ′ (x) = p x p−1 and Hölder's inequality, Proof. By similar estimates as in the previous lemma, The progressively measurability is again clear. To prove the identity we first assume that g is a simple adapted process of the form where Π = {u = t 0 < t 1 < · · · < t n = t} is a partition of the interval [0, T ] and the random variables are F ti -measurable and take values in the space of all finite rank operators from H to E. Then, For general progressively measurable g ∈ L p (Ω; L 2 ([0, T ]; γ(H, E))), the identity follows by a routine approximation argument.
Proof of Theorem 3.1. The proof of the theorem proceeds in two steps. All constants occurring in the proof may depend on E and p, even where this is not indicated explicitly, but not on T . The numerical value of the constants may change from line to line.
Step 1 -Applying Lemma 2.2 to the function ψ(x) = x p and the process X, we have, for every t ∈ [0, T ], We shall prove the identity (3.4) by showing that with convergence in probability. In view of the definition of X t , it is enough to show that where we used the continuity of the process X in the last line. Next, by Lemma 3.4 and the inequalities (3.2) and (2.2), Recall that the localized stochastic integral is continuous from M ([0, t]; γ(H, E))) into L 0 (Ω, F t ; E). Hence, in order to prove that the right-hand side converges to 0 in probability, it suffices to prove that For this, in turn, it suffices to observe that P-almost surely by the path continuity of X.
Step 2 -In this step we prove the estimate (3.5). By (2.2), for all x, y ∈ E and r ∈ [0, 1] we have Combining this with (2.5) we obtain We shall estimate the two terms on the right hand of (3.7) side separately. For the first term, using the inequality |a + b| 2 2|a| 2 + 2|b| 2 we obtain Next we estimate the second term in (3.7). We have Collecting terms, for any ε > 0 we obtain the estimate In the proof of Theorem 1.2 we will also need the following simple observation.
Proof. Fix t ∈ (0, T ] and let k(n) be the unique index such that t ∈ (t n k(n) , t n k(n)+1 ]. Then Now (3.8) follows by taking the limes inferior for n → ∞ and using path continuity.

Proof of Theorem 1.2
We proceed in four steps. In Steps 1 and 2 we establish the estimate in the theorem for g ∈ M p ([0, T ]; γ(H, E)) with 2 p < ∞. In order to be able to cover exponents 0 < p < 2 in Step 3, we need a stopped version of the inequalities proved in Steps 1 and 2. For reasons of economy of presentations, we therefore build in a stopping time τ from the start. In Step 4 we finally consider the case where g ∈ M ([0, T ]; γ(H, E)).
We shall apply (a special case of) Lenglart's inequality [13, Corollaire II] which states that if (ξ t ) t∈[0,T ] and (a t ) t∈[0,T ] are continuous non-negative adapted processes, the latter non-decreasing, such that Eξ τ Ea τ for all stopping times τ with values in [0, T ], then for all 0 < r < 1 one has Step 1 -Fix p 2 and suppose first that g ∈ M p ([0, T ]; γ(H, D(A)). As is well known (see [16]), under this condition the process X t = t 0 e (t−s)A g s dW s is a strong solution to the equation dX t = AX t dt + g t dW t , t 0; X 0 = 0.
In other words, X satisfies  X s∧τ p + C ε E Hence X n t → X t in L p (Ω; E). Therefore,X is a modification of X. This concludes the proof for p 2.
Step 4 -Finally, the existence of a continuous version for the process X under the assumption g ∈ M ([0, T ]; γ(H, E)) follows by a standard localisation argument.