A note on Malliavin smoothness on the L\'evy space

We consider Malliavin calculus based on the It\^o chaos decomposition of square integrable random variables on the L\'evy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying L\'evy process is a compound Poisson process on a finite time interval.

The wide interest in Malliavin calculus for Lévy processes in stochastics and applications motivates the study of an accessible characterization of differentiability and fractional differentiability. Fractional differentiability is defined by real interpolation between the Malliavin Sobolev space 1,2 and L 2 (È) and we recall the definition in Section 4 of this paper. Geiss and Geiss [5] and Geiss and Hujo [9] have shown that Malliavin differentiability and fractional differentiability are in a close connection to discrete-time approximation of certain stochastic integrals when the underlying process is a (geometric) Brownian motion. Geiss et al. [6] proved that this applies also to Lévy processes with jumps. These works assert that knowing the parameters of fractional smoothness allow to design discretization time-nets such that the optimal approximation rate can be achieved. For details, see [5], [9] and [6].
Steinicke [17] and Geiss and Steinicke [8] take advantage of the fact that any random variable Y on the Lévy space can be represented as a functional Y = F (X) of the Lévy process X, where F is a real valued measurable mapping on the Skorohod space of right continuous functions. Let us restrict to the case that F (X) only depends on the jump part of X. Using the corresponding result from Solé, Utzet and Vives [16] and Alòs, León and Vives [1] on the canonical space, Geiss and Steinicke [8] show that the condition F (X) ∈ 1,2 is equivalent with where ν is the Lévy measure of X. On the other hand one gets from Mecke's formula [12] that for any nonnegative measurable F and any N is the Poisson random measure associated with X as in Section 2. These results raise the following questions: when can Malliavin differentiability be described using a weight function such as N(A), and is there a weight function for fractional differentiability? In this paper we search for weight functions Λ and measurability conditions on Y such that the criteria describes the smoothness of Y . We begin by recalling the orthogonal Itô chaos decomposition on L 2 (È) and the Malliavin Sobolev space in Section 2. Then, in Section 3, we obtain an equivalent condition for Malliavin differentiability. The assertion is that whenever Y is measurable with respect to F A , the completion of the sigmaalgebra generated by {N(B) : Section 4 treats fractional differentiability and our aim is to adjust the weight function Λ so that the condition (1.1) describes a given degree of smoothness. We recall the K-method of real interpolation which we use to determine the interpolation spaces (L 2 (È), 1,2 ) θ,q for θ ∈ (0, 1) and q ∈ [1, ∞]. These spaces are intermediate between 1,2 and L 2 (È). We show that when Y is F A -measurable and [N(A)] < ∞, then Y has fractional differentiability of order θ for q = 2 if and only if

Preliminaries
Consider a Lévy process X = (X t ) t≥0 with càdlàg paths on a complete probability space (Ω, F , È), where F is the completion of the sigma-algebra generated by X. The Lévy-Itô decomposition states that there exist γ ∈ Ê, σ ≥ 0, a standard Brownian motion W and a Poisson random measure N on xN(ds, dx) + when 0 ∈B. The triplet (γ, σ, ν) is called the Lévy triplet.
We let I n denote the multiple integral of order n defined by Itô [10] and It is then extended to a linear and continuous operator I n : For the multiple integral we have for all f n ∈ L 2 (Ñ ⊗n ) and g k ∈ L 2 Ñ ⊗k .
We recall the definition of the Malliavin Sobolev space 1,2 based on the Itô chaos decomposition. We denote by 1,2 and the operator is extended to {I n (f n ) : f n ∈ L 2 (Ñ ⊗n )} by linearity and continuity. For Remark 2.1. Note that also for any u ∈ L 2 (Ñ ⊗ È) one finds a chaos ) .
The following theorem implies that if We denote by S the set of random variables Y such that there exists (a) S is dense in 1,2 and L 2 (È).
Proof. Assume first that Y ∈ S. Then also Y 2 = ∞ n=0 I n (g n ) ∈ S. Letting h(t, x) = 1 x ½ A (t, x) we have I 1 (h) = N(A) − [N(A)] and we get using (2.2) and (2.3) that Using Hölder's inequality we get Taking the square root yields to the double inequality (3.3). Using Lemma 3.1 (a) we find for any bounded Y a uniformly bounded sequence (Y k ) ⊂ S such that Y k → Y a.s. Since inequality (3.3) holds for all random variables Y k − Y m ∈ S, they are uniformly bounded and Y k − Y m → 0 a.s. as k, m → ∞, we have by dominated convergence that The mapping u has a representasion u = ∞ n=0 I n (h n+1 ) (see Remark 2.1), where for all n ≥ 0 we have that as k → ∞. We obtain (3.3) for the random variable Y using dominated and the fact that (3.3) holds for all random variables Y k .
where the norms may be infinite.
Proof. The inequalities (3.1) and (3.2) give the relation The claim follows from .

From Lemma 3.3 we obtain the inequalities
for c = .
Using the triangle inequality and the fact that for all ω ∈ Ω, y ∈ Ê and a ≥ 0 we obtain from (4.2) the lower bound cK(Y, s) .
We have shown that (4.1) holds. From (4.1) we get We finish the proof by computing the integral using first Fubini's theorem. We get

Concluding remarks
From Theorem 3.1 assertion 2. we can conclude that a higher integrability than square integrability can imply Malliavin differentiability. For example, all the spaces L p (Ω, F A , È) are subspaces of 1,2 when p > 2 and [N(A)] < ∞ as we can deduce from the following corollary.
for some β ∈ Ê. The process (N t ) t≥0 , with N t = N((0, t] × Ê 0 ) a.s., is the Poisson process associated to X. Let T ∈ (0, ∞) and F T be the completion of the sigma-algebra generated by (X t ) t∈