DIFFERENTIABILITY IN INFINITE DIMENSION AND THE MALLIAVIN CALCULUS

. In this paper we study two notions of diﬀerentiability introduced by G. Cannarsa and G. Da Prato (see [20]) and L. Gross (see [41]) in both the framework of inﬁnite dimensional analysis and the framework of Malliavin calculus.


Introduction
The problem of differentiability along subspaces arises in a natural way in the study of differential equations for functions of infinitely many variables.Over the years, regularity properties along subspaces, such as Lipschizianity and Hölderianity, have become increasingly central in the theory of infinite dimensional analysis.Various authors have introduced many definitions for these regularity properties, the most widely used are two.One introduced by L. Gross in [41] and systematically presented by V. I. Bogachev in [19], the other one introduced by P. Cannarsa and G. Da Prato in [20] and later developed by E. Priola in his Ph.D. thesis, see [57].The main purpose of this paper is to compare the Gross and Cannarsa-Da Prato notions of differentiability.We will begin by relating the two different notions of gradients along subspaces, introduced in [20] and [41], when these operators act on a class of sufficiently smooth functions.We will then turn to the specific case where the subspace along which to differentiate is the Cameron-Martin space associated to a given Gaussian measure.In this framework, it is possible to extend such operators to spaces of less regular functions, i.e., Sobolev spaces with respect to the reference Gaussian measure.Such extensions are called Malliavin derivatives.The central result of this paper will be to rigorously show that the Gross and the Cannarsa-Da Prato Malliavin derivatives are two different operators (although linked by a relationship that we will clarify) that still have the same Sobolev space as their domain.This work should therefore be understood as a review of existing results with the specific purpose of relating them through rigorous proofs.Moreover, we will also provide the proofs of some results that, to the best of our knowledge, are not present in the literature.
More in details, in Section 2 we recall the notions of differentiability given in [20] and [41] and investigate their relation.In Subsection 2.1, given a separable Hilbert space H 0 continuously embedded in a separable Hilbert space H, we recall the definition of differentiability along H 0 presented by Gross in [41] for functions with values in a Banach space Y .Over the years, this notion has become essential to prove many regularity results for stationary and evolution equations for functions of infinitely many variables both in spaces of continuous functions and in Sobolev spaces, see for instance [1,3,4,7,6,8,10,12,13,14,23,25,51].In Subsection 2.2, given a linear bounded self-adjoint operator R : H → H we define the differentiability along the directions of H R := R(H) presented by Cannarsa and Da Prato in [20].This notion of differentiability has also been widely used over the years, see for instance [2,5,11,28,29,31,32,42,52,53,57,58,59].When H 0 = H R one can compare the two above mentioned notions of differentiability, this is done in Subsection 2.3 where we provide the explicit relation between the n-order Gross derivatives ∇ n HR and the n-order Cannarsa-Da Prato derivatives ∇ n R .We highlight that a first comparison between these two derivatives has been already presented in [56] but in the specific case of injective operators R. To conclude this part, Subsection 2.4 is devoted to the comparison of the above mentioned notions of differentiability with the classical notions of Gateaux and Fréchet differentiability.
The results of Section 2 lay the ground for the comparison of the Malliavin derivatives that naturally appear in the setting considered by Gross and Cannarsa-Da Prato when a Gaussian framework comes into play.This is the content of Section 3. On the space (H, B(H)) one considers a centered (that is with zero mean) Gaussian measure γ with covariance operator Q, with Q : H → H a linear, self-adjoint, non-negative and trace class operator.The subspace along which to differentiate is the Cameron-Martin space associated to the Gaussian measure, that is It is classical to prove (see e.g.[19] and [27]) that the gradient operators ∇ H Q 1/2 and ∇ Q 1/2 , in the sense of Gross and Cannarsa-Da Prato, respectively, are closable operators in L p (H, B(H), γ), p ≥ 1.Their extensions are called Malliavin derivatives and the domain of their extension is a Sobolev space with respect to the measure γ.We refer to these two Malliavin derivatives as the Malliavin derivative in the sense of Gross and the Malliavin derivative in the sense of Cannarsa-Da Prato, respectively.In Subsections 3.3, 3.4 and 3.5 we recall the construction of these two Malliavin derivatives and prove that they are indeed two different operators linked by the relation . Nevertheless these two derivatives, although different, have the same Sobolev space as their domain.
In order to make a rigorous comparison between the Malliavin derivative in the sense of Gross and Cannarsa-Da Prato, it is convenient to approach the Malliavin calculus from a more abstract point of view, as done for example in [55].We briefly recall this approach to Malliavin calculus in Appendix A. What we will show is that the Malliavin derivatives in the sense of Gross and Cannarsa-Da Prato can be interpreted as two particular examples of the general notion of Malliavin derivative given in [55].This will make it possible, in a rather simple way, to understand the relationship between the two derivatives.
At first glance it might seem strange to refer to Malliavin derivatives that are different, since one usually speaks of the Malliavin derivative.We point out that in fact it would be more appropriate to speak of a (choice of) Malliavin derivative rather than the Malliavin derivative.In fact, as explained in details in Appendix A, one can construct infinitely many different Malliavin derivative operators.On the other hand, it turns out that all these Malliavin derivatives have the same domain when somehow the Gaussian framework is the same.In a sense, the results of Section 3 can be considered as an example of this general fact in a concrete situation: we deal with two particular Malliavin derivatives that naturally appear in the literature for the study of different problems.However, these Malliavin derivatives are just two possible choices among the infinite possible ones that would be possible to consider in that specific Gaussian framework.
We emphasize here that the general framework for Malliavin calculus considered in [55] not only proves useful in understanding the relationship between different Malliavin derivatives that appear in the literature in different contexts, but also turns out to be particularly flexible for dealing with different types of problems in the study of stochastic partial differential equations.We mention, for example, the study of the regularity of the image law of solutions of stochastic partial differential equations (see e.g.[16,17,38,55] and the therein references), the study of ergodic problems, (see among all [46]), or even the study of integrations by parts formulas on level sets in infinite-dimensional spaces (see e.g.[7,15,18,30]).
Section 4 should be read as an application of the results of Section 2; there, working under the assumption that ker R = {0}, we prove an interpolation result (see Theorem 4.11).This result has already been presented in [20,Proposition 2.1] in the sense of the Cannarsa-Da Prato differentiability.Exploiting the results in Subsection 2.3 we immediately infer that Theorem 4.11 holds true also for Gross differentiability.Theorem 4.11 improves the result in [14,Section 3].This improvement is possible due to some regularity results of Lasry-Lions type approximants finer than those found in the literature (see, for example, [14,20]).These results can be found in Section 4.2, and they are of interest regardless of the interpolation result we are going to prove.Finally we recall that interpolation theorems are useful for Schauder regularity results for Ornstein-Uhlenbeck type operators in infinite dimension, see for instance [14,24,25,26,59].
Notations.In this section we recall the standard notations that we will use throughout the paper.We refer to [35] and [60] for the notation and basic results about linear operators and Banach spaces.Throughout the paper, all Banach and Hilbert spaces are real.
Let K 1 and K 2 be two Banach spaces equipped with the norms • K1 and • K2 , respectively.Let H be a Hilbert space equipped with the inner product •, • H and associated norm • H .
We denote by B(K 1 ) the family of the Borel subsets of K 1 .B b (K 1 ; K 2 ) is the set of the bounded and Borel measurable functions from is the space of bounded and continuous (uniformly continuous, respectively) functions from For any k ∈ N, let L (k) (K 1 ; K 2 ) be the space of continuous multilinear maps from K k 1 to K 2 endowed with the norm We denote by Id K1 the identity operator from K 1 to itself.
We say that be a non-negative and self-adjoint operator.We say that Q is a trace class operator, if Qe n , e n H < +∞, for some (and hence for all) orthonormal basis {e n } n∈N of H.We recall that the trace is independent of the choice of the orthonormal basis.

Differentiability along subspaces
In this section we present the notions of differentiability along subspaces considered in [20] and [41].Throughout the paper we will consider two separable Hilbert spaces H and H 0 equipped with the inner products •, • H and •, • H0 and associated norms • H0 and • H0 , respectively.We assume H 0 to be continuously embedded in H, namely there exists C > 0 such that We also fix a self-adjoint operator R ∈ L(H).We denote by ker R the kernel of R and by (ker R) ⊥ its orthogonal subspace in H.By P ker R we denote the orthogonal projection on ker R. We denote by H R := R(H) the range of the operator R. In order to provide H R with a Hilbert structure, we recall that the restriction R | (ker R) ⊥ is a injective operator, and so is bijective.Hence, we can define the pseudo-inverse of R as see [48,Appendix C].We introduce the scalar product with its associated norm x HR := R −1 x H .With this inner product H R is a separable Hilbert space and a Borel subset of H (see [44,Theorem 15.1]).A possible orthonormal basis of H R is given by {Re k } k∈N , where {e k } k∈N is an orthonormal basis of (ker R) ⊥ .We recall that it holds ) Thus, when H 0 = H R the constant C appearing in (2.1) is given by R L(H) .Moreover we recall that ker R = {0} if, and only if, R(H) is dense in H (see [33,Lemma VI.2.8]).
In Subsection 2.1, we present the notion of differentiability along the subspace H 0 first considered by L. Gross in [41] for vector valued functions.This notion is widely used in the literature, see for example [10,12,13,14,19,25,45].For the sake of clarity, in Subsubsection 2.1.1 we rewrite some definitions of Subsection 2.1 in the special case of real-valued functions.In Subsection 2.2 we recall the notion of differentiability along H 0 given by P. Cannarsa and G. Da Prato in [20] and later revised by E. Priola in [57, Sections 1.2 and 1.3] in the particular case H 0 = H R .This approach is also widely used in the literature, see for instance [36,37,42,52,53].Subsection 2.3 focus on the comparison between the two above mentioned notions of differentiability.Finally, in Subsection 2.4 we make clear their relation with the classical notions of Fréchet and Gateaux differentiability.
2.1.Differentiability in the sense of Gross.Here we introduce the notion of Gross differentiability.We thought it appropriate to prove some results concerning differentiability in the sense of Gross in a rather general setting, since these results are widely used in many papers [3,4,10,12,13,14,22,23,25,51].Throughout this section Y will denote a Banach space endowed with the norm • Y .
Let us start by recalling the notions of H 0 -continuity and H 0 -Lipschitzianity.
Definition 2.1.We say that a function ϕ : ϕ is H 0 -continuous if it is H 0 -continuous at any point x ∈ H.We say that ϕ : H → Y is H 0 -Lipschitz if there exists a positive constant L H0 such that for any x ∈ H and h ∈ H 0 it holds The best constant that verifies (2.5) is called H 0 -Lipschitz constant of ϕ.
We now introduce the notions of Fréchet and Gateaux differentiability along H 0 .
Definition 2.2.We say that a function ϕ : The operator L x is unique and it is called H 0 -Fréchet derivative of ϕ at x ∈ H.We set In a similar way, for any k ∈ N we introduce the notion of k-times H 0 -Fréchet differentiability of ϕ and we denote by D k H0 ϕ(x) its k-order H 0 -Fréchet derivative at x ∈ H.In particular D k H0 ϕ(x) belongs to L (k) (H 0 ; Y ).We say that ϕ is k-times H 0 -Fréchet differentiable if it is k-times H 0 -Fréchet differentiable at any point x ∈ H.
Definition 2.3.We say that a function ϕ : The operator L x is unique and it is called H 0 -Gateaux derivative of ϕ at x ∈ H.We set D G,H0 ϕ(x) := L x .For any k ∈ N (in an analogous way of the k-times H 0 -Fréchet differentiability) we can define the notion of k-times H 0 -Gateaux differentiability of a function ϕ and we denote by If H 0 = H Definitions 2.2 and 2.3 are the classical notions of Fréchet and Gateaux differentiability, respectively, and in this case we will use the notation Dϕ and The following result provides a sufficient condition for the equivalence of H 0 -Fréchet and H 0 -Gateaux differentiability.

. Corollary 1] for the case H
Since ϕ is H 0 -continuous, g x : H 0 → Y is continuous.By the H 0 -Gateaux differentiability of ϕ, for any x ∈ H and h, k ∈ H 0 , we infer Corollary 1] we thus infer that g x : H 0 → R is Fréchet differentiable at 0 and D G g x (0) = Dg x (0).To conclude, we observe that for any x ∈ H In the next propositions we collect some basic properties of the H 0 -Fréchet and H 0 -Gateaux differentiability.In the literature the following results are widely used in different situations, see for instance [10,12,13,14,19,25,45], here we decided to prove them in an abstract setting.We start with the chain rule.Proposition 2.5.Let Z be a Banach space equipped with the norm Proof.Let h ∈ H 0 and let (t n ) n∈N be an infinitesimal sequence of positive real numbers.We consider the sequence (z n ) n∈N ⊆ Z defined as and We write Proof.The proof is standard, we give it for the sake of completeness.Let φ : [0, 1] → H be defined as φ(t) := x + th and let Ψ(t) := ϕ(φ(t)), for any t ∈ [0, 1].Observe that Ψ is derivable in (0, 1) in fact for t ∈ (0, 1) Furthermore, observe that since ϕ is H 0 -Fréchet differentiale, then Ψ is continuous in [0, 1].By the mean value theorem there exists t 0 ∈ (0, 1) such that Ψ(1) − Ψ(0) = Ψ ′ (t 0 ).Thus This concludes the proof.
Moreover if H 0 is dense in H and ϕ is a continuous function, then ϕ is constant.
Proof.By Proposition 2.6 we get that ϕ(x+ h) = ϕ(x), for every x ∈ H and h ∈ H 0 .Now assume that H 0 is dense in H and that ϕ is a continuous function.Let x 0 ∈ H 0 and let (h n ) n∈N ⊆ H be a sequence converging to x 0 , it holds This conclude the proof.
The following result clarifies the relation between the classical notion of Fréchet differentiability and the notion of H 0 -Fréchet differentiability.

Proof. By (2.1) we infer that if
The proof concludes taking the limit as h H0 approaches zero in (2.6).
The converse implication of Proposition 2.8 is not true in general, see the following example.
Example 2.9.Let ϕ : H → R defined as ϕ is not Fréchet differentiable (it is not continuous), but it is H 0 -Fréchet differentiable and it holds 2.1.1.Gross differentiability for real-valued functions.In this subsection we rewrite Definitions 2.2 and 2.3 for real-valued functions, as they will be the main focus of this paper from here on.Let k ∈ N and L ∈ L (k) (H; R), by the Riesz representation theorem there exists a unique l ∈ L (k−1) (H) (where we set Notice that ∇f and ∇ G f are the standard Fréchet and Gateaux gradient of f in x ∈ H, respectively.Now we introduce some natural functional spaces associated to the notion of H 0differentiability. For any k ∈ N, the space BUC k H0 (H) endowed with the norm is a Banach space.We conclude this subsection noting that, for real-valued functions Theorem 2.4 reads as follows.
Theorem 2.12.Let ϕ : 2.2.Differentiability in the sense of Cannarsa and Da Prato.We introduce here the notion of R-differentiability considered by Cannarsa and Da Prato in [20] dropping the assumption ker R = {0} considered in that paper.
Definition 2.13.We say that a function f : We set ∇ R f (x) := l x .We say that a function is R-differentiable if it is R-differentiable at any x ∈ H.We say that f is twice R-differentiable at x ∈ H if it is R-differentiable and there exists a unique B x ∈ L(H) such that for any v ∈ H we have Remark 2.14.In [20] the authors introduce a weaker notion of twice R-differentiability: a function ϕ : We introduce some natural functional spaces associated to the notion of R-differentiability.
Definition 2.15.For any k ∈ N, we denote by The space BUC k R (H) equipped with the norm is a Banach space.The following result will be useful throughout the paper.Notice that if ker R = {0} (as in [20]) the next proposition is trivial.
Proposition 2.16.Let k ∈ N and let f : where we set ) Proof.The case k ≤ 2 is an immediate consequence of (2.7) and (2.8) by taking v ∈ ker R. For k > 2 we proceed by induction.Assume the assertion to hold true for k and let us prove it for k + 1.Let f : H → R be a (k + 1)-times R-differentiable function.By the inductive hypothesis and Definition 2.13 we infer = 0, hence (2.9) holds true and (2.10) holds true when v n ∈ ker R. Moreover for any v 1 , . . ., v n ∈ H we have thus the inductive hypothesis yields (2.10).

2.3.
Comparisons between R-differentiability and H R -differentiability.We aim to compare the notion of R-differentiability of Section 2.2 with the notion of H 0 -differentiability of Section 2.1.1,if Proposition 2.17.A function ϕ : In particular, for any Proof.Assume that ϕ is R-differentiable.By (2.3), Definition 2.13 and Proposition 2.16, for every Assume now that ϕ is H R -Gateaux differentiable.Recalling that h = Rv, by Definitions 2.3 and 2.10, and (2.4) for every x ∈ H, v ∈ H it holds Since, by (2.2), R −1 ∇ G,HR ϕ(x) ∈ (ker R) ⊥ and h = Rv, by (2.3) we obtain Hence ϕ is R-differentiable and (2.11) is verified.
Bearing in mind Definitions 2.11 and 2.15, we now show that BUC k HR (H) = BUC k R (H) for any k ∈ N. We need the following preliminary result.Lemma 2.18.For any n ∈ N the map T n : is a linear isometry and an isomorphism.We recall that for n = 0 we let L (0) ((ker R) ⊥ ) := (ker R) ⊥ and L (0) (H R ) := H R and we set (2.12) Proof.For any n ∈ N, since R | (ker R) ⊥ : (ker R) ⊥ → R(H) is linear and bijective, it follows that T n is linear.By definitions of R and H R , for any A ∈ L (n) ((ker R) ⊥ ) we have with T n−1 as in Lemma 2.18.
Proof.We proceed by induction.We start by proving the base case n = 1 , we obtain (2.13) and BUC n R (H) ⊆ BUC n HR (H) .The inclusion BUC n HR (H) ⊆ BUC n R (H) follows in a similar way using T −1 n .
In view of the above result, from here on we will use the space BUC k R (H) to represent both BUC k HR (H) and BUC k R (H).

A Comparison with the classical notions of differentiability.
We focus here on the relationship between the R-differentiability and H R -differentiability, and the classical Fréchet and Gateaux differentiability.
Proposition 2.20.For any n ∈ N, if ϕ : H → R is n-times Gateaux differentiable, then ϕ is n-times R-differentiable and for any x ∈ H and n ≥ 2 Proof.We proceed by induction.Let ϕ : H → R be a Gateaux differentiable function and let x, v ∈ H.By Definition 2.3 with H 0 = H we have Thus the thesis follows for n = 1.Now we assume that the statements hold true for n and we prove it for n + 1.Let ϕ : H → R be a (n + 1)-times Gateaux differentiable function and let x, v 1 , . . ., v n ∈ H.By the inductive hypothesis we have for any s ∈ R \ {0} To conclude it is enough to take the limit as s approaches zero in (2.14 ∇ R ϕ(x) = R∇ϕ(x).
We conclude this section with a useful characterization of BUC Proof.We start by proving the following claim: if a function f : H → H belongs to BUC(H; H), then (f • R −1 ) |H R belongs to BUC(H; H).Indeed, for any ε > 0 there exists δ > 0 such that for any v, w ∈ H it holds that if v − w H ≤ δ, then Thus ϕ is R-differentiable for any x ∈ H R and (2.17) By the claim proved at the beginning of the proof it holds (∇(ϕ • R) Thus, by the continuity of the map H) and (2.16) holds true.

Malliavin calculus in Wiener spaces
We put ourselves in a Gaussian framework, that is we consider on (H, B(H)) a centered (that is with zero mean) Gaussian measure γ with covariance operator Q.Here Q ∈ L(H) is a selfadjoint non-negative and trace class operator.It is classical to prove (see e.g.[19] and [27]) that the gradient operators ∇ H Q 1/2 and ∇ Q 1/2 introduced in Definitions 2.10 and 2.13 are closable operators in L p (H, B(H), γ), p ≥ 1.Their extensions are called Malliavin derivatives and the domain of their extension is a Sobolev space with respect to the measure γ.We will refer to these two Mallavin derivatives as the Malliavin derivative in the sense of Gross and the Malliavin derivative in the sense of Cannarsa-Da Prato, respectively.
In this Section we briefly recall the construction of these two Malliavin derivatives operators (mainly referring to the books [19] and [27]).Then we show that they can be interpreted as two (different) examples of the general notion of Malliavin derivative that is briefly recalled in Appendix A.
Referring to Appendix A, to make a comparison of the Malliavin derivatives in the sense of Gross and in the sense of Cannarsa-Da Prato we need to identify the Gaussian Hilbert space and to characterize it in terms of a suitable choice of a separable Hilbert space and a unitary operator; this is done in Subsection 3.1.In Subsection 3.2 we write the integration by parts formulae that are the crucial ingredients to show the closability of the gradient operators.In the remaining subsections we reinterpret the two constructed Malliavin derivatives in the abstract framework of Section A. In particular, we show that the the Malliavin derivative in the sense of Gross and in the sense of Cannarsa-Da Prato are different operators but with the same domain.

The Gaussian Hilbert space H *
γ .Let us recall that we denote by H * , with norm • H * , the topological dual of H consisting of all linear continuous functions f : H → R, we denote by H ′ the algebraic dual of H consisting of all linear functions f : H → R. The space H * is included in L 2 (H, B(H), γ) and the inclusion map j : ), when endowed with the scalar product of L 2 (H, γ), is a Gaussian Hilbert space (see e.g.[19,Lemma 2.2.8]).We introduce the covariance operator R γ : R γ is injective and its range is contained in H (see e.g.[49,Proposition 2.3.6]).We define the Cameron-Martin space K (for the measure γ) as K := R γ (H * γ ) ⊆ H. K inherits a structure of separable Hilbert space through R γ (see e.g.[19,Lemma 2.4.1]), that is introducing the map As proved in [49,Theorem 4.2.7], the Cameron-Martin space coincide with the Hilbert space H Q 1/2 = Q 1/2 (H) and its inner product is given by h From the very definition of the Cameron-Martin space it follows that the map • := R −1 γ is a unitary operator and this yields that where every ĥ ∈ H * γ is a centered Gaussian random variable with variance ĥ 2 L 2 (H,γ) = h 2 K .On the other hand, when the measure γ is non degenerate (that is ker Q = {0}), the Cameron-Martin space turns out to be dense in H (see e.g.[27,Lemma 2.16]).In this case, see [27, Section 2.5.2], the map ) can be uniquely extended to a linear isometry W • defined as H, γ).In the literature the map W • is usually called white noise map.So W • is a unitary operator and it holds 3.2.Sobolev spaces.We denote by ∇ H Q 1/2 and ∇ Q 1/2 the gradient operators introduced in Definitions 2.10 and 2.13, respectively, with the choice R = Q 1/2 .In Section 2 we analyzed the relations between this two operators; we summarize them in the following result.
Lemma 3.1.Let Q ∈ L(H) be a self-adjoint non-negative and trace class operator with ker Q = {0}.For any ϕ ∈ BUC 1 (H), In particular, The following integration by parts formula with respect to γ is well known (see e.g.[19,Theorem 5.1.8]) and in [19,Section 2] it is used to prove that the operator Moreover, the integration by parts formula (3.3) holds for any ϕ belonging to W 1,p [49,Proposition 9.3.10]).
When γ is non degenerate, Lemma 3.1 provides the following equivalent form of the integration by parts formula (3.3) where we used ĥ = W z for h = Q 1/2 z.The integration by parts formula (3.5) is the one used in [27] to prove that the operator γ) is defined as the domain of the closure of the operator ∇ Q 1/2 , denoted by M .It is a Banach space with the norm Moreover, the integration by parts formula (3.5) holds for any ϕ ∈ W 1,p Q 1/2 (H, γ) and z ∈ H.In the following sections we show that the gradient operators ∇ H Q 1/2 and M , can be thought as Malliavin derivative operators.For this purpose, referring back to Section A, it will be enough to identify the choices of the probability space (Ω, F, P), the Gaussian Hilbert space H 1 , the Hilbert space H and the unitary operator W .
, under specific compatibility assumptions between R and Q it is possible to prove that R∇ is closable, we call generalized gradient the closure of it (see, for example, [13,40]).The Sobolev space W 1,p R (H, γ) is the domain of the closure of the operator R∇.See also [9] for the problem of equivalence of Sobolev norms.

3.3.
Malliavin derivative in the sense of Gross.In [41] (see also [19]), the reference probability space is (Ω, F, P) = (H, B(H), γ), with γ a centered Gaussian measure.The Gaussian Hilbert space H 1 is H * γ , the space H is the Cameron-Martin space K = H Q 1/2 and the unitary operator W is the operator • = R −1 γ .With these identifications, by comparing the integration by parts formula ∀ϕ ∈ D 1,2 = Dom(D), h ∈ H, we immediately see that the Malliavin derivative in [41] (see also [19]) is the gradient operator 3.4.Malliavin derivative in the sense of Cannarsa-Da Prato.In [27] the reference probability space is (Ω, F, P) = (H, B(H), γ), with γ a centered non degenerate Gaussian measure.The Gaussian Hilbert space H is H * γ , the space H is H itself and the unitary operator W is the white noise map W • .With these identifications, by comparing the integration by parts formula with (A.4), we immediately see that the Malliavin derivative in [27] is the gradient operator M .
On the other hand, the domain of the two derivatives is the same, that is In fact, thanks to Lemma 3.1, the closure of the space C 1 b (H) with respect to the norm (3.4) is the same as its closure with respect to the norm (3.6).This should not be surprising in light of the general results of Section A: in Sections 3.3 and 3.4 the reference Gaussian Hilbert space H 1 is the same, that is H * γ ; thus Proposition A.6 ensures the two Malliavin derivatives and M to have the same domain.What changes in Sections 3.3 and 3.4 is how the space H 1 is characterized.In Section 3.3 we considered the unitary operator • = R −1 γ between H Q 1/2 and H 1 and obtain the characterization (3.1), whereas in Section 3.4 we considered the unitary operator W • between H and H 1 and obtain the characterization (3.2).This naturally leads to different Malliavin derivatives, having chosen different Hilbert spaces H and unitary operators W .

Application: Lasry-Lions approximation and an interpolation result
We consider the same framework of Section 2. We introduce here the notions of H 0 -Hölder and R-Hölder functions.We prove this notions to be equivalent when H 0 = H R .We thus prove an interpolation type result for the space of H R -Hölder functions.A key role in the proof is played by Lasry-Lions type approximations along the space H R (see Subsection 4.2).4.1.Hölder functions along subspaces.We recall here the notions of H 0 -Hölderianity and R-Hölderianity and show that they are equivalent when H 0 = H R .Definition 4.1.We say that ϕ : H → R is a H 0 -Hölder function of exponent α ∈ (0, 1) if there exists a positive constant L α,H0 such that for any x ∈ H and h ∈ H 0 it holds We call H 0 -Hölder constant of ϕ the best constant L α,H0 that verifies (4.1).
It is trivial to see that a H 0 -Hölder function ϕ : H → R is H 0 -continuous.When H 0 = H we recover the classical definition of Hölder continuous function from H to R.Moreover, by (2.1), if ϕ is Hölder continuous, then ϕ is H 0 -Hölder.The converse is not true as shown by the following example.
Definition 4.5.We denote by BUC R (H) the subspace of BUC(H) of the R-Hölder functions of exponent α.
BUC α R (H) is a Banach space for any α ∈ (0, 1), when endowed with the norm where Let us compare the above definitions in the specific case H 0 = H R .
In view of the above result, from here on we will use the space BUC α R (H) to represent both BUC α HR (H) and BUC α R (H).We state now a useful characterization of the space BUC α R (H) whenever ker R = {0}.Proposition 4.7.Assume that ker R = {0} and let α ∈ (0, 1).A function ϕ : H → R belongs to BUC α R (H) if, and only if, the function ϕ • R belongs to BUC α (H).Furthermore it holds Proof.Let us start by noticing that H R is dense in H, since ker R = {0}.We begin to prove that Lasry-Lions type approximations.From here on we will assume that ker R = {0} since we will need Proposition 2.22, that holds under this assumption, to prove what follows.We recall the classical Lasry-Lions approximating procedure introduced in [47].
We now recall a modification of the Lasry-Lions approximating procedure presented in [20]: given f ∈ BUC(H) and t > 0 one defines the function with the convention that R −1 y = +∞ if y / ∈ R(H).We will consider a slight modification of (4.3) obtained via a change of variables Proposition 4.9.Assume that ker R = {0} and let f ∈ BUC(H).For any t > 0, it holds Proof.We start by proving (4.5) Moreover, since f • R ∈ BUC(H), by Theorem 4.8 we infer that for any t > 0 the map S(t)(f • R) belongs to BUC 1 (H).By Proposition 2.22 and (4.5) we conclude that, for any t > 0, the map S R (t)f belongs to BUC 1 R (H).The following proposition summarize some of the properties of {S R (t)f } t≥0 that we will use throughout this section.Proposition 4.10.Assume that ker R = {0} and let f ∈ BUC α R (H), for some α ∈ (0, 1).Let {S R (t)f } t≥0 be the family of functions introduced in (4.4).There exists c α > 0 such that for every t > 0 and x ∈ H it holds Proof.Let us start by proving (4.6).
Let us now prove (4.7).By (4.4), for every η > 0 there exists From the above inequality we get the estimate HR + 2tη, and the Young inequality yields, for every c > 0, (4.12) Combining (4.11) and (4.12) we obtain Since the above estimate holds for every η > 0, by choosing η small, we get (4.7).
Let us now prove (4.8).First notice that by (4.4) for every σ > 0 there exists Thus from (4.7) we obtain By (4.13) we get Since the above inequality holds for every σ > 0 taking the infimum we get A standard argument concludes this proof.

4.
3. An interpolation result for the space BUC α R (H).We have now all the ingredients to prove an interpolation result for the space BUC α R (H).We shall use the K method for real interpolation spaces (see [50,61]).Let K 1 and K 2 be two Banach spaces, with norms • K1 and • K2 , respectively.If K 2 ⊆ K 1 with a continuous embedding, then for every r > 0 and x ∈ K 1 we define For any ϑ ∈ (0, 1), we set The following result can be found in [21] for the case R = Id H and a similar result can be found in [14], where the space BUC 1 R (H) is substituted by another space.Theorem 4.11.Assume that ker R = {0} and let α ∈ (0, 1).Up to an equivalent renorming, it holds For any element ϕ ∈ (BUC(H), BUC 1 R (H)) α,∞ and any r, t > 0 there exist f r,t ∈ BUC(H) and g r,t ∈ BUC 1 R (H) such that ϕ(x) = f r,t (x) + g r,t (x), x ∈ H; Thus, the mean value theorem and (4.16) yield Now letting t tend to zero and setting r = h HR we get This proves the continuous embedding (BUC(H), BUC For every t > 0 let S R (t)ϕ be the function defined in (4.4).For r ∈ (0, 1) we consider the functions f r : H → R and g r : H → R defined by so that ϕ = f r + g r with f r ∈ BUC R (H) and g r ∈ BUC 1 R (H) in virtue of Proposition 4.9.By (4.7) we get that there exists a constant k 1 = k 1 (α, ϕ) > 0 such that f r ∞ ≤ k 1 r α .By (4.6) and (4.8), there exist a constant Thus, bearing in mind (4.14), for every r ∈ (0, 1) we get K(r, ϕ) ≤ (k 1 + k 2 )r α .Notice that the previous estimate is trivial if r > 1. Keeping in mind (4.15) we get the thesis.
A.1.Gaussian Hilbert spaces.Let (Ω, F, P) be a probability space, we denote by E the expectation under P. Let H be a real separable Hilbert space with inner product •, • H and corresponding norm • H . Definition A.1.A Gaussian linear space is a real linear space of random variables, defined on (Ω, F, P), such that each variable in the space is centered Gaussian.A Gaussian Hilbert space is a Gaussian linear space which is complete, i.e. a closed subspace of L 2 (Ω, F, P) consisting of centered Gaussian random variables.We denote it by H 1 .
We recall that a linear isometry between Hilbert spaces is a linear map that preserves the inner product.Linear isometries that are onto are called unitary operators.
Proposition A.2. Let H 1 be a separable Gaussian Hilbert space and H be a separable Hilbert space with the same dimension of H 1 .There exist infinitely many unitary operators W : H → H 1 such that H 1 = W (H). In particular, for any h, k ∈ H, Proof.Let {ξ i } i∈I be an orthonomal basis in H 1 , that is a collection of independent standard normal variables, defined on (Ω, F, P).Let H be a separable real Hilbert space and let {e i } i∈I be an orthonormal basis in H with the same index set I (the cardinality of I can be finite or countably infinite).Every element h ∈ H can be written as h = i∈I h, e i H e i .It is easy to verify that the map H ∋ h → W (h) := i∈I h, e i H ξ i provides an onto isometry from H to H 1 .
In [55] the unitary operators W , introduced in Proposition A.2, are called isonormal Gaussian processes.The space H and the operator W play the role to index the elements in H 1 .In the concrete situations, plainly, one will select an Hilbert space H and a unitary operator W that are well adapted to the specific problem at hand.A.2. Wiener Chaos Decomposition.Every Gaussian Hilbert space induces an orthogonal decomposition, known as the Wiener Chaos Decomposition, of the corresponding L 2 (Ω, σ(H 1 ), P) space of square integrable random variables that are measurable with respect to the σ-field generated by the Gaussian Hilbert space, that we denote by σ(H 1 ).For n ≥ 0 we introduce the linear space P n (H 1 ) := {p(ξ 1 , . . ., ξ m ) | p is a polynomial of degrees ≤ n, ξ 1 , . . ., ξ m ∈ H 1 , m ∈ N} .
Let P n (H 1 ) be the closure of P n (H 1 ) in L 2 (Ω, F, P).For n ≥ 0 the space H n := P n (H 1 ) ⊖ P n−1 (H 1 ) = P n (H 1 ) ∩ P n−1 (H 1 ) ⊥ is called n-th Wiener Chaos (associated to H 1 ).We remark that H 0 = R.The following result is usually called Wiener chaos decomposition, its proof can be found in [43,Theorem 2.6].
Theorem A.3.The spaces H n , n ≥ 0, are mutually orthogonal, closed subspaces of L 2 (Ω, F, P) and For n ≥ 0 let us denote by J n the orthogonal projection of L 2 (Ω, F, P) onto H n ; in particular, J 0 (X) = E [X].Theorem A.3 yields that every random variable X ∈ L 2 (Ω, σ(H 1 ), P) admits the unique expansion J n (X), with the series converging in L 2 (Ω, σ(H 1 ), P).
A.3.The Malliavin derivative operator and the Sobolev spaces.From here on we fix a probability space (Ω, F, P) and an infinite dimensional separable Gaussian Hilbert space H 1 .We assume F to be the σ-field generated by H 1 .Moreover, according to Proposition A.2, we fix an infinite dimensional separable Hilbert space H and a unitary operator W : H → H 1 ⊆ L 2 (Ω, σ(H 1 ), P), so that we characterize H 1 = {W (h) | h ∈ H}, (A.1) and every W (h) ∈ H 1 is a centered Gaussian random variable with variance W (h) 2  L 2 (Ω,σ(H1),P) = h 2 H . Let us denote by S(H 1 ) the set of smooth random variables, i.e. random variables of the form F = f (W (h 1 ), . . ., W (h m )) for some m ≥ 1 and h 1 , . . ., h m ∈ H, where f is a C ∞ (R m ) function such that f and all its partial derivatives have at most polynomial growth.According to Proposition A.5 the operator D admits a closed extension (still denoted by D) with domain D 1,p .We call this extension Malliavin derivative and we call D 1,p the domain of D in L p (Ω, σ(H 1 ), P).For any p ≥ 1 the space D 1,p endowed with the norm (A.3) is a Banach space, for p = 2 the space D 1,2 is a Hilbert space with the scalar product It is not difficult to prove that the integration by parts formula (A.2) extends to elements in D 1,2 , that is (A.4) The space D 1,2 is characterized in terms of the Wiener chaos expansion (see [55, Proposition 1.2.2]).
Proposition A.6.Let F ∈ L 2 (Ω, σ(H 1 ), P) with Wiener chaos expansion F = ∞ n=0 J n (F ).Then F ∈ D 1,2 if and only if Let us emphasize that, once we have fixed the reference probability space (Ω, F, P) and the Gaussian Hilbert spaces H 1 , different (infinitely many) choices of the separable Hilbert space H and the unitary operator W lead to different (infinitely many!) Malliavin derivative operators.On the other hand, in view of Proposition A.6, all these Malliavin derivatives have the same domain D 1,2 when the Gaussian Hilbert space H 1 is the same.In fact the characterization of D 1,2 is given in terms of the Wiener chaos decomposition that relies only on the Gaussian Hilbert space H 1 (and not on the choices of H and W ).

Definition A. 4 .
The derivative of a random variable F ∈ S(H 1 ) of the form (A.1) is the Hvalued random variableDF = m i=1 ∂f ∂x i (W (h 1 ), . . ., W (h m ))h i .The space S(H 1 ) turns out to be dense L p (Ω, σ(H 1 ), P) for any p ∈ [1, ∞), see e.g.[54, Lemma 3.2.1].This, along with the following integration by parts formula (see e.g.[55, Lemma 1.2.1]):E [ DF, h H ] = E [W (h)F ] , h ∈ H, F ∈ S(H 1 ), (A.2)is the crucial ingredient to extend the class of differential random variables to a larger class.For a proof of the following proposition see[54, Proposition 2.3.4].Proposition A.5.For any p ≥ 1 the operator D : S(H 1 ) ⊆ L p (Ω, σ(H 1 ), P) → L p (Ω, σ(H 1 ), P; H), introduced in Definition A.4, is closable as an operator from L p (Ω, σ(H 1 ), P) to L p (Ω, σ(H 1 ), P; H).For any p ≥ 1 we denote with D 1,p the closure of S(H 1 ) with respect to the norm F p D 1,p = E [|F | p ] + E DF p H . (A.3) Thus the case n = 1 follows by Theorem 2.12.Now we prove the induction step.Assume the thesis to be true for an integer n ≥ 2. Letϕ ∈ BUC n+1 R (H), x ∈ H and v n ∈ H R \{0} such that v n = Rh n with h n ∈ (ker R) ⊥ .By Proposition 2.16 and Lemma 2.18 we infer lim s→0 . Let ϕ : H → R; by Proposition 2.17 the map x → ∇ G,HR ϕ(x) belongs to BUC(H; H R ) if, and only if, the map x → ∇ R ϕ(x) belongs to BUC(H; H).