Space regularity for evolution operators modeled on H\"ormander vector fields with time dependent measurable coefficients

We consider a heat-type operator L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, multiplied by a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if Lu is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives satisfy a 1/2-H\"older continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.

Let G = R N , •, D λ a Carnot group and let X 1 , ..., X q be the generators of its Lie algebra, so that the canonical sublaplacian q i=1 X 2 i and the corresponding heat operator are hypoelliptic in R N and R N +1 , respectively (precise definitions will be given in Section 1). Let us now consider where {a ij (t)} q i,j=1 is a real symmetric matrix of bounded measurable coefficients, uniformly positive: for every ξ ∈ R q , a.e. t ∈ (0, T ). We want to prove a regularity result for L in the space variables, that is, roughly speaking: if u ∈ W 1,2 (0, T ) , L 2 loc R N is a weak solution to Lu = F , u (0, ·) = 0 and F is smooth, with respect to the space variables, in some domain (0, T ) × Ω, then the same is true for u, with quantitative regularity estimates on u in terms of Lu. Also, we will prove that, if F is smooth w.r.t. the space variables, then u and every space derivative ∂ α x u are 1 2 -Hölder continuous with respect to t. See Theorems 2.14 and 2.15 for the precise statements. This kind of regularity is the best we can hope, even for a uniformly parabolic operator Lu = u t − a (t) u xx as soon as a is only L ∞ (see Example 2.16). The above regularity result can be extended also to distributional solutions belonging to W 1,2 (0, T ) , D ′ R N (see Theorem 3.3 for the precise statement). This can be seen as a kind of Hörmander's theorem with respect to the space variables.
Results of this kind have been proved by Krylov [14], who considered operators where the functions σ ik (t, x) are assumed to have x-derivatives of every order uniformly bounded for x ∈ R N and t ∈ (0, 1), and the vector fields L 0 , L 1 , ..., L q for every fixed t satisfy Hörmander's condition in R N . Now, every operator (0.1) can be rewritten as Since the coefficients b ji (x) of the generators on a Carnot group are polynomials, the functions |D α x b ji (x)| are not globally bounded on R N . Therefore, although the class of operators that we consider is strictly contained in the class considered by Krylov as to their structure, the assumption on σ ik (t, x) made in [14] is not satisfied in our situation.
Actually, the technique employed in this paper is very different from that in [14]. In [14], following the classical approach introduced by Kohn [13] and Oleȋnik-Radkevič [17], pseudodifferential operators and Sobolev spaces of fractional order are used. Here, instead, we adapt to the evolutionary case the technique introduced in [3] to give a proof of Hörmander's theorem for sublaplacians on Carnot groups. The main idea consists in measuring the regularity of solutions of an equation Lu = f , where L is a left invariant operator, in terms of Sobolev spaces induced by right invariant vector fields. Since a right invariant and a left invariant operator always commute, this approach greatly simplifies the proof of higher order estimates. We handle Sobolev norms with respect to vector fields by means of equivalent norms defined in terms of finite difference operators, in the directions of the vector fields X 1 , ..., X q . This feature of our argument is reminiscent of the original proof of Hörmander's theorem given in [12], although in the richer framework of Carnot groups the proof becomes much simpler.
Let us now give some motivation for the present research and describe some related literature. The regularity result proved in [14] has been applied by the same Author in [15] to prove an analogous result for stochastic PDEs, and in [16], in the context of filtering problems. We refer to [15] for motivations to prove this result without any continuity assumption on the coefficients with respect to time.
Hyperbolic operators of the kind a ij (t) u xixj with merely bounded measurable a ij have been studied by many authors, see for instance [9], [8], [11] and references therein. In particular, [11] gives some physical motivation to study this class of operators under no regularity condition on a ij (t).
Operators of the kind satisfying (0.2) have been studied by several Authors, assuming the coefficients a ij (t, x) either Hölder continuous or with vanishing mean oscillation, and proving a priori estimates and regularity results in the scale of Hölder or Sobolev spaces induced by the vector fields {X i } q i=1 and the distance they induce. See for instance [4], [6], [7] and references therein. In [6], for the operator L with Hölder continuous coefficients, a heat kernel has been constructed and shown to satisfy sharp Gaussian estimates, which also imply a scale invariant Harnack inequality.
The operators (0.1) studied in the present paper can also be seen as model operators to study the more general class (0.3) with the coefficients satisfying some moderate regularity assumtpion in x, but only L ∞ with respect to time, an area of research that we plan to attack in the future.

Preliminaries about Carnot groups
Let us recall some standard definitions and results that will be useful in the following. For the proofs of these facts the reader is referred to [10], [1,Chap.1]. A homogeneous group (in R N ) is a Lie group R N , • (where the group operation • will be thought as a "translation") endowed with a one parameter family {D λ } λ>0 of group automorphisms ("dilations") which act this way: for suitable integers 1 = α 1 α 2 ... α N . We will write G = R N , •, D λ to denote this structure. The number such that, for some constant c > 0 and every x, y ∈ G, We will always use the symbol · , without any subscript, to denote a homogeneous norm in G. Examples of homogeneous norms are the following: It can be proved that any two homogeneous norms on G are equivalent. We say that a smooth function Given any differential operator P with smooth coefficients on G, we say that P is left invariant if for every x, y ∈ G and every smooth function f Analogously one defines the notion of right invariant differential operator. Also, for every smooth function f , λ > 0 and x ∈ G.
A vector field is a first order differential operator Let g be the Lie algebra of left invariant vector fields over G, where the Lie bracket of two vector fields is defined as usual by Let us denote by X 1 , X 2 , . . . , X N the canonical base of g, that is for i = 1, 2, ...., N , X i is the only left invariant vector field that agrees with ∂ xi at the origin. Also, X R 1 , X R 2 , . . . , X R N will denote the right invariant vectors fields that agree with ∂ x1 , ∂ x2 , ..., ∂ xN (and hence with X 1 , X 2 , . . . , X N ) at the origin.
We assume that for some integer q < N the vector fields X 1 , X 2 , . . . , X q are 1-homogeneous and the Lie algebra generated by them is g. If s is the maximum length of commutators .., q} required to span g, then we will say that g is a stratified Lie algebra of step s, G is a Carnot group (or a stratified homogeneous group) and its generators X 1 , X 2 , . . . , X q satisfy Hörmander's condition at step s in G. Under these assumptions, by Hörmander's theorem (see [12]), the canonical sublaplacian Analogously, the corresponding heat operator We will make use of the Sobolev spaces W k,p X (G), W k,p X R (G) induced by the systems of vector fields respectively. More precisely, given an open subset Ω of R N , we say that f ∈ W 1,2 X (Ω) if f ∈ L 2 (Ω) and there exist, in weak sense, X j f ∈ L 2 (Ω) for j = 1, 2, ..., q. Inductively, we say that f ∈ W k,2 (Ω) and any weak derivative of order k − 1 of f , X j1 X j2 ...X j k−1 f , belongs to W 1,2 X (Ω). We set The space W k,2 X R (Ω) has a similar definition. We will also use local Sobolev spaces. For example, we will say that f ∈ W k,2 X,loc (Ω) if for every ϕ ∈ C ∞ 0 (Ω), we have ϕf ∈ W k,2 X (Ω). For homogeneity reasons, the generators X 1 , ..., X q satisfy the simple relation X * i = −X i (where X * stands for the transposed operator of X). In other words, 2) whenever f ∈ W 1,2 X,loc (G) and g ∈ C 1 0 (G). The validity of Hörmander's condition at step s implies the following important: Under the above assumptions we have: 1.
2. For any positive integer k and any Ω ′ ⋐ Ω there exists a constant c > 0 such that, for every u ∈ W ks,2 X (Ω) we have where W k,2 (Ω ′ ) denotes the standard Sobolev space.
Let us point out a relation between left and right invariant operators which will be very useful in the following. Proposition 1.2 (see [3,Prop. 2.2]) Let L, R be any two differential operators on G with smooth coefficients, left and right invariant, respectively. Then L and R commute: for any smooth function f.
For every given couple of measurable functions ϕ, ψ : G → R we define whenever the integral makes sense. One can prove the following: For every couple of measurable functions f, ψ defined on G such that the following convolutions are well defined, we have i) if P is a left invariant differential operator then ii) if P is a right invariant differential operator then whenever Pψ exists at least in weak sense.
We are going to define several function spaces on G T = (0, T ) × G that we will use in the following. The definitions of the spaces L 2 ((0, T ) , X), W 1,2 ((0, T ) , X), C 0 ([0, T ] , X) when X is a Banach space are standard. For instance, we will often use the spaces and the analogous spaces L 2 (0, T ) , W k,2 X R (G) .
Definition 2.1 We say that a function u belongs to L 2 (0, T ) , C ∞ Ω if u ∈ L 2 (0, T ) , C k Ω for every k = 0, 1, 2, ... Explicitly, this implies that We say that a function u belongs to
Proof. For u ∈ H we have, recalling that In particular, for u vanishing on t = 0 we get (2.1).
In the following of this section we will recall and adapt several definitions and arguments taken from [3]. The reader is referred to that paper for some details.

Definition 2.4 (Finite difference operators)
For every h ∈ G and function f defined in G, let us define the operators: Whenever the function f also depends on t, we will simply write and analogously for ∆ h f (t, x) .
Then, for α > 0 and f ∈ L 2 (G T ) we define the semi-norms We also set for convenience The relations between the above seminorms and Sobolev norms with respect to vector fields are contained in the following two results, which can be derived by [3,Thm. 3.11,Prop.3.13] simply integrating in t. (2.5) The following bound instead links the L 2 (0, T ) , W 1,2 X (G) norm with the operators ∆ h : (Recall that s is the step of the Lie algebra).
Proof. It is enough to apply to u (t, ·) the computations made in [3, Prop.3.7, Lemma 3.8] for functions in W 1,2 X (G) and then integrate on (0, T ). If u ∈ H, u (t, ·) is supported in some bounded domain Ω for every t ∈ [0, T ] and u (0, ·) = 0, then by the previous Proposition and (2.1) we get (2.6) Notation 2.9 Henceforth, we will write We have the following analog of Theorem 3.15 in [3]: whenever the right hand side is finite.
Proof. We can repeat the proof of Theorem 3.15 in [3] applying (2.6) to the function ζ 0 u ∈ H, since u ∈ H 0 , and exploiting the identity 8) and the fact that the operators ∂ t and ∆ h commute, so that L and ∆ h still commute. Also Proposition 3.16 in [3] (Marchaud inequality on Carnot groups) still holds, with L 2 (G) norms replaced with L 2 (G T ) norms, and this implies the following analog of Corollary 3.17 in [3].

Corollary 2.11
Let u ∈ H, u (0, ·) = 0, and assume that for ε ∈ (0, 1) and some integer m > 1 the seminorm |u| R m,1+ε is finite. Then We are now in position to state the first step of our regularity estimate: Proof. Applying to ζ 0 u Corollary 2.11 and Theorem 2.10 with m = s + 1 and ε = 1/s we can write: From this inequality, by Propositions 2.7 and 2.6 we conclude the desired result.
To iterate this result to higher order derivatives, we first need a regularization result allowing to apply (2.9) to functions u satisfying weaker assumptions.
and for every ζ, ζ 1 ∈ C ∞ 0 (G) with ζ ≺ ζ 1 the following estimate holds: Proof. Let us define the ε-mollified u ε of u as follows.
Now the function u ε is smooth with respect to x (as can be seen computing and, for any couple of domains K ⋐ K ′ ⋐ G and ε small enough, Here we have used Young's inequality in the form for K ⋐ K ′ , and ε small enough, since φ ε is compactly supported. Also, hence u ε ∈ H 0 and we can apply to u ε the estimate proved in Proposition 2.12: for a.e. t and a.e. x. This is not trivial since Lu just exists in the above weak sense, hence we cannot simply write L (u ε ) = (Lu) ε . However, for every ϕ ∈ we can write: and (2.14) follows. By known properties of the mollifiers, as ε → 0 we have φ ε * u → u in L 2 R N as soon as u ∈ L 2 R N . Also, for every left invariant differential operator L we can write L (φ ε * u) = φ ε * Lu as soon as Lu exists in L 2 R N . Therefore as soon as F ∈ L 2 (0, T ) , W k,2 X,loc (G) . To prove convergence in L 2 (0, T ) , W s,2 X R ,loc (G) we make the following rough estimates: In the first inequality we have bounded the Sobolev norm W s,2 X R (on a compact set containing the support of ζ 1 ) with the Euclidean Sobolev norm on the same domain; in the second one we have exploited Hörmander's condition.
We want to show that, for F ∈ L 2 (0, T ) , W s 2 ,2 X,loc (G) , Now: where by (2.15) we already know that To apply Lebesgue theorem and conclude the desired result we need to bound g ε with an integrable function independent of ε. Now: and an interative reasoning allows to conclude (2.17) Recalling (2.16) and the fact that we conclude that the right hand side of (2.13) is bounded. Hence the sequence ζu ε is bounded in L 2 (0, T ) , W 1,2 X R (G) , and there exists a subsequence of ζu ε weakly converging in L 2 (0, T ) , W 1,2 X R (G) to some g and in particular weakly converging in L 2 (G T ) to ζu. This is enough to say that ζu ∈ hence (2.11) holds.
Theorem 2.14 (Regularity estimates in x) Let u ∈ W 1,2 (0, T ) , L 2 loc (G) , u (0, ·) = 0, be a weak solution to Lu = F ∈ L 2 (0, T ) , L 2 loc (G) in the sense of (2.10) and let ζ, ζ 1 ∈ C ∞ 0 (G) , ζ ≺ ζ 1 . Then for any k = 1, 2, 3, ..., there exists c = c (k, ζ, ζ 1 , G, ν) > 0 such that whenever ζ 1 F ∈ L 2 (0, T ) , W k+s 2 −1,2 Proof. We will prove (2.18) by induction on k. For k = 1 this is exactly Proposition 2.13. Assume that (2.18) holds up to an integer k and let u ∈ H 0 such that Lu ∈ L 2 (0, T ) , W k+s 2 ,2 X R (G) . By the inductive assumption, . Let X R I be a right invariant differential operator with |I| k, then ζX R I u ∈ L 2 (G T ). We would like to apply Proposition 2.13 to X R I u, but in order to do that we would need to know that X R I u ∈ W 1,2 (0, T ) , L 2 loc (G) with X R I u (0, ·) = 0, which is unclear. Then, let u ε be the mollified version of u as in the proof of Proposition 2.13, so that: which is a smooth function in x, and since X R I φ ε is integrable (although its L 1 (G) norm is not uniformly bounded with respect to ε) we have (see (2.12)) and since ∂ t u ∈ L 2 (0, T ) , L 2 loc (G) , the same is true for ∂ t X R I (u ε ), which equals X R I (∂ t u) ε . Then which also implies at least in weak sense. Actually, noting that L and X R I commute, and L (u ε ) = F ε for a.e. t and x (see (2.14)) for a.e. t. Therefore we can apply Proposition 2.13 to X R I (u ε ) getting Noting that for some I ′ with |I ′ | = |I| − 1, we can proceed iteratively getting, for some different cutoff function ζ 2 ≻ ζ 1 , (2.19) From this bound, which is uniform with respect to ε, reasoning like in the proof of Proposition 2.13 we read that, under the assumption X R I F ∈ L 2 (0, T ) , W s 2 ,2 X R (G) , which is true as soon as F ∈ L 2 (0, T ) , W k+s 2 ,2 X R (G) , we have the uniform boundedness of which implies the weak convergence in L 2 (0, T ) , W 1,2 X R (G) of (a subsequence of) ζX R I (u ε ) to some g. In particular the convergence is in L 2 (G T ) , which implies that for every η ∈ L 2 (0, T ) and φ ∈ C ∞ 0 (G) Pick the cutoff function ζ (x) = 1 on some bounded open set Ω, then for every On the other hand, which implies, for a.e. t and a.e. x ∈ Ω, in the sense of weak derivatives. This means that ζX R I u ∈ L 2 (0, T ) , W 1,2 X R (G) and ζX R I (u ε ) → ζX R I u weakly in L 2 (0, T ) , W 1,2 X R (G) , which also implies, by (2.19), So we are done. Next, we want to derive from the previous result the fact that, for F smooth enough, weak solutions to Lu = F are actually strong solutions. Also, we want to establish Hölder continuity with respect to time of solutions (and their space derivatives): and u is also a strong solution to Lu = F . In particular, for every multiindex I with |I| k we have X R I u ∈ C 0 [0, T ] , L 2 loc (G) and X R I u (0, ·) = 0.
Let u ∈ W 1,2 (0, T ) , L 2 loc (G) , u (0, ·) = 0, be a weak solution to Lu = F ∈ L 2 (0, T ) , W h,2 X R ,loc (G) . By Theorem 2.14, if ζ 1 F ∈ L 2 (0, T ) , W k+s 2 −1,2 then ζu ∈ L 2 (0, T ) , W k,2 X R (G) . In particular, if h ≥ 2s + s 2 − 1 then u ∈ L 2 (0, T ) , W 2,2 X,loc (G) and this implies that u is actually a strong solution to the equation Lu = F , so that for a.e. t and a.e. x we have (2.20) This identity allows to transfer further x-regularity of both F and u to u t : if, for some k = 1, 2, 3, ..., we know that h ≥ k + 2s + s 2 − 1, then by Theorem 2.14 u ∈ L 2 (0, T ) , W k+2s,2 X R ,loc (G) , so that X i X j u ∈ L 2 (0, T ) , W k,2 X R ,loc (G) , hence by (2.20) u t ∈ L 2 (0, T ) , W k,2 X R ,loc (G) and u ∈ W 1,2 (0, T ) , W k,2 X R ,loc (G) . This implies that for |I| k, X R I u ∈ C 0 [0, T ] , L 2 loc (G) . Moreover we can write, for every t 1 , t 2 ∈ [0, T ] and a.e. x ∈ G, Letting t 1 = 0 in (2.21) we get an identity which can also be differentiated with respect to X R I , giving which implies X R I u (0, ·) = 0. This completes the proof of (i). Next, multiplying both sides of (2.22) for ζ ∈ C ∞ 0 (G) and taking L 2 (G)-norms we get, recalling that X R I commutes with L: By Theorem 2.14 this implies that for some h large enough and any cutoff function ζ 1 such that ζ ≺ ζ 1 . On the other hand, letting v (x) = u (t 2 , x) − u (t 2 , x) and noting that every cartesian derivative ∂ α x v (x) can be bounded, uniformly on a compact set of G by a suitable linear combination of X R I v, we arrive to a bound for some integer h 1 > h. And since also the sup of |∂ α x u (t 2 , ·) − ∂ α x u (t 1 , ·)| can be bounded, by Sobolev embeddings, by suitable L 2 norms of higher order derivatives, we also have a control This ends the proof of (ii). The previous result also shows that Then the equality We end this section with an easy example showing that the regularity properties of the solution cannot be improved for bounded measurable coefficients a ij (t).

Example 2.16 Let us consider the uniformly parabolic operator
Then U solves the problem x), so that, as soon as α > 1 2 , Moreover, Since α > 1 2 can be chosen as close to 1/2 as we want, this shows that the regularity with respect to t expressed by Theorem 2.15 cannot be improved. Also, note that the Hölder continuity w.r.t. t cannot be improved to Lipschitz continuity just remaing far off t = 0: if we multiply the above U (t, x) for |t − t 0 | α we get a similar example exhibiting a α-Hölder continuity w.r.t. t near t = t 0 .

Regularization of distributional solutions
In this section we want to extend our smoothness result, established in Theorem 2.15 (iii) for functions in W 1,2 (0, T ) , L 2 loc (G) , to more general distributions. First of all, we need to make precise the distributional notions that we will use.
The proof of a regularity result for distributional solutions usually begins identifying the given distribution, locally, with some derivative of a continuous function, in view of the classical result about the local structure of distributions. For distributions in the class L 2 ((0, T ) , D ′ (Ω)) we could not find in the literature any reference for a similar result. So we will explicitly assume that our distribution could be seen, on a fixed domain compactly contained in Ω, as a space derivative of a suitable function: ) for some open set Ω ⊆ G. We will say that u satisfies the x-finite order assumption on Ω if: there exists a function h ∈ L 2 (0, T ) , L 1 loc (Ω) and a multiindex α such that If u ∈ W 1,2 ((0, T ) , D ′ (Ω)), we will say that u satisfies the x-finite order assumption on Ω if (3.1) holds with h ∈ W 1,2 (0, T ) , L 1 loc (Ω) . Note that if u ∈ W 1,2 ((0, T ) , D ′ (Ω)) satisfies the x-finite order assumption on Ω ′ , then h ∈ C 0 [0, T ] , L 1 loc (Ω) . In particular, saying that u (0, ·) = 0 means that h (0, ·) = 0 a.e. in Ω.
The aim of this section is to prove that: For some bounded domain Ω ⊂ G, let u be a distributional solution to Lu = F in Ω T with F ∈ L 2 ((0, T ) , D ′ (Ω)). Assume that u satisfies the x-finite order assumption (see Definition 3.2) and u (0, ·) = 0 in Ω. Then, for every domain Ω ′ ⋐ Ω, if In order to prove Theorem 3.3 we will adapt the technique used in [3, §4] for sublaplacians.
Let us consider the second order differential operator built using the whole canonical base of right invariant vector fields. This is a right-invariant (but no longer homogeneous) uniformly elliptic operator in G, which at the origin coincides with the standard Laplacian. The fundamental solution of the Laplacian can be proved to be a parametrix for L R : and such that in the sense of distributions (Here δ is the Dirac mass as a distribution in R N ).
Let us now consider three open sets in G, Ω ′ ⋐ Ω ′′ ⋐ Ω and let V be a neighborhood of the origin such that V −1 • Ω ′ ⊂ Ω ′′ . Define γ as in Proposition 3.4, with γ supported in V . The convolution with γ defines a regularizing operator that acts on functions u ∈ L 1 loc (G T ) as follows. For every x ∈ Ω ′ and t ∈ [0, T ] we set The subscript V in T V recalls that the definition of the operator depends on the choice of the neighborhood V used to define γ. Note that Namely, for x ∈ Ω ′ and x • y −1 ∈ sprt γ, the point y = x • y −1 −1 • x ranges in V −1 • Ω ′ ⊂ Ω ′′ , hence introducing characteristic functions, which by Young's inequality gives, at least for a.e. t, and hence Also, T V acts on distributions u ∈ L 2 ((0, T ) , D ′ (Ω)) as follows. For every ϕ ∈ D (Ω ′ ) we set Observe that the assumption on V implies that T * V ϕ is a test function in Ω ′ . Namely, for x ∈ sprt ϕ and x • y −1 ∈ sprt γ the point y ranges in Ω ′′ ⋐ Ω. The function T * V ϕ is smooth, as one can see writing and computing left invariant derivatives Therefore the pairing (3.6) is well defined. Also, from the previous identity we easily read that if (just by definition of L 2 ((0, T ) , D ′ (Ω))), so that Next, we need to prove the regularizing properties of T V . The following result is an adaptation of [3,Prop. 4.4.].

Proposition 3.5 (Regularizing properties of T V )
Let Ω ′ ⋐ Ω ′′ ⋐ Ω. There exists a neighborhood V of the origin such that the operator T V defined in (3.6) has the following properties.
(1) Let u ∈ D ′ ((0, T ) × Ω) such that u = ∂ α ∂x α g, for some g ∈ L 2 (0, T ) , L 1 loc (Ω) and multiindex α. Then T V u ∈ L 2 ((0, T ) , D ′ (Ω ′ )) and Remark 3.6 Throughout the next proof, and also in other deductions in the following, all the stated equalities of the kind A (t) = B (t) hold for a.e. t. Rigorously speaking, we should write chains of equalities of the kind and then deduce that A (t) = B (t) a.e.
Proof. This proof is an adaptation of the proof of [3,Prop. 4.4]. ( We can write where by (3.3) h k (z) are locally integrable functions, smooth outside the pole, and Z k are polynomials (these polynomials also depend on the index i corresponding to ∂ yi , but for simplicity we suppress this unimportant index). Hence Now, for suitable polynomials a β,k , hence we can write follows from (3.5) and Young's inequality since, by (3.2), γ ∈ L r (G) for 1 r < N N −2 . Taking L 2 (0, T ) norms we get (2).
(3) We know that any derivative ∂ xi γ (i = 1, 2, ..., N ) is integrable and supported in V , hence each function X R i γ (i = 1, 2, ..., N ) is a linear combination with polynomial coefficients of integrable functions, compactly supported in V , so that X R i γ ∈ L 1 (G). Also, for a.e. t ∈ [0, T ], uχ Ω ′′ (t, ·) ∈ L 2 (G) hence by Young's inequality and . This holds for i = 1, 2, ..., N (not just for the first q derivatives). Now, let us recall that the left invariant vector fields X i (i = 1, 2, ..., N ) can be written as linear combinations with polynomial coefficients of the right invariant vector fields X R i . Hence by the boundedness of Ω ′ we also have (4). Let u ∈ C ∞ Ω . From we read that for x ∈ Ω ′ and any left invariant differential operator P we can write showing that PT V u (t, ·) ∈ C ∞ (Ω ′ ). Moreover, so that, for every k = 0, 1, 2, ...
and also Corollary 3.7 Let Ω ′ ⋐ Ω ⋐ G. For every distribution u ∈ D ′ ((0, T ) × Ω) such that u = ∂ α x g for some multindex α and g ∈ L 2 (0, T ) , L 1 loc (Ω) there exist a neighborhood of the origin V and an integer K such that ( The proof follows exactly that of [3,Corollary 4.5].

Proposition 3.8
Let Ω ′ ⋐ Ω and V small enough so that V • Ω ′ ⋐ Ω. Then for any distribution u ∈ L 2 ((0, T ) , D ′ (Ω)) and every left invariant operator P on G we have Also, if u ∈ W 1,2 ((0, T ) , D ′ (Ω)) then The previous proposition can be obviously iterated writing for any fixed positive integer K, provided V is chosen small enough to have • Ω ′ ⋐ Ω.