Very weak solutions to hypoelliptic wave equations

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, H\"older, and distributional. For H\"older coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of"very weak solutions"to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique"very weak solution"in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators on $\mathbb{R}^{n}$ or on groups, the results already being new in all these cases.


Introduction
In this paper we are interested in the well-posedness of the following Cauchy problem for general positive hypoelliptic (Rockland operators of homogeneous degree ν) left-invariant differential operators R on general graded Lie group G with the nonnegative propagation speed a = a(t) and with the source term f = f (t) ∈ L 2 (G):    ∂ 2 t u(t) + a(t)Ru(t) = f (t), t ∈ [0, T ], u(0) = u 0 ∈ L 2 (G), ∂ t u(0) = u 1 ∈ L 2 (G).
(1.1) When G = (R n , +) and R = −△ is the positive Laplacian, that is, the equation in (1.1) is the usual wave equation, the well-posedness of the Cauchy problem (1.1) for Hölder functions a = a(t) goes back to Colombini, de Giorgi and Spagnolo [CDS79]. In [CJS87] and [CS82], it was also shown that when G = (R, +) and R = − d 2 dx 2 , the Cauchy problem (1.1) does not have to be well-posed in C ∞ if a ∈ C ∞ is not strictly positive or if it is in the Hölder class a ∈ C α for 0 < α < 1.
We note that following the seminal paper by Rothschild and Stein [RS76], such Rockland operators can be considered as model 'approximations' of general hypoelliptic partial differential operators on manifolds.
Before discussing the obtained results on graded groups, let us briefly recall some necessary facts. The Sobolev space H s R (G) for any s ∈ R is the subspace of S ′ (G) obtained as the completion of the Schwartz space S(G) with respect to the Sobolev norm f H s R (G) := (I + R) s ν f L 2 (G) . (1.2) In the case of stratified Lie groups such spaces and their properties have been extensively analysed by Folland in [Fol75] and on general graded Lie groups they have been investigated in [FR16] and [FR17]. Recall that these spaces do not depend on a particular choice of the Rockland operator R used in the definition (1.2), see [FR16,Theorem 4.4.20]. A brief review of the necessary notions related to graded Lie groups will be given in Section 2. We will also use R-Gevrey (Roumieu) G s R (G) and R-Gevrey (Beurling) type spaces G (1.5) (ii) Let a ∈ C α ([0, T ]) with 0 < α < 1 and a(t) ≥ a 0 > 0. If the initial Cauchy data (u 0 , u 1 ) are in G s R (G) × G s R (G), then there exists the unique solution of the homogeneous Cauchy problem (1.1) (when f ≡ 0) in C 2 ([0, T ], G s R (G)), provided that 1 ≤ s < 1 + α 1 − α ; (iii) Let a ∈ C ℓ ([0, T ]) with ℓ ≥ 2 and a(t) ≥ 0. If the initial Cauchy data (u 0 , u 1 ) are in G s R (G)×G s R (G), then there exists the unique solution of the homogeneous Cauchy problem (1.1) (when f ≡ 0) in C 2 ([0, T ], G s R (G)), provided that , then there exists the unique solution of the homogeneous Cauchy problem (1.1) (when Let H −∞ s and H −∞ (s) be the spaces of linear continuous functionals on G s R and G (s) R , respectively.
In particular, in this paper we show the inhomogeneous case of Theorem 1.1 and the case when the initial Cauchy data (u 0 , u 1 ) can be also from the space H −∞ (s) : Theorem 1.2. Let G be a graded Lie group and let R be a positive Rockland operator of homogeneous degree ν. Let T > 0. Then we have ) and the initial Cauchy data (u 0 , u 1 ) are in H with 0 < α < 1 and a(t) ≥ a 0 > 0. Then for initial data and for source term (iii) Let a ∈ C ℓ ([0, T ]) with ℓ ≥ 2 and a(t) ≥ 0. Then for initial data and for source term the Cauchy problem (1.1) has the unique solutions (iv) Let a ∈ C α ([0, T ]) with 0 < α < 2 and a(t) ≥ 0. Then for initial data and for source term Since we are also interested in the case when the time-dependent propagation speed a is less regular than Hölder, let us recall some results in this direction. In [GR15], the authors introduced the notion of "very weak solutions" for a wave-type second order invariant partial differential operator in R n , and proved their existence, uniqueness and consistency with classical or distributional solutions should the latter exist. For similar results in R n , we also refer to [RT17b] for the Landau Hamiltonian, and to [RT17c] for operators with a discrete non-negative spectrum. Thus, the second aim of this paper is to carry out similar investigations for general hypoelliptic operators, namely for positive Rockland operators (1.1), whose spectrum is absolutely continuous. To give some examples, this setting includes: • for G = R n , R may be any positive homogeneous elliptic differential operator with constant coefficients. For example, we can take where a j > 0 and m ∈ N; • for the Heisenberg group G = H n , we can take where a j , b j > 0, m ∈ N, and L = n j=1 (X 2 j + Y 2 j ) is the sub-Laplacian, and • for any stratified Lie group (or homogeneous Carnot group) with vectors X 1 , . . . , X k spanning the first stratum, we can take a j X 2m j , a j > 0, so that in particular, for m = 1, R is a positive sub-Laplacian; • for any graded Lie group G ∼ R n with dilation weights ν 1 , . . . , ν n let us fix the basis X 1 , . . . , X n of the Lie algebra g of G satisfying where D r denote dilations on the Lie algebra. If ν 0 is any common multiple of ν 1 , . . . , ν n , the operator where ω(ε) > 0 (which we will choose later) is such that ω(ε) → 0 as ε → 0, and ψ is a Friedrichs-mollifier, that is, ψ ∈ C ∞ 0 (R), ψ ≥ 0 and ψ = 1.
Let us give the following definition: is said to be C ∞ -moderate if for all K ⋐ R and for all α ∈ N 0 there exist N ∈ N 0 and c > 0 such that sup )moderate if there exists η > 0 and, for all p ∈ N 0 there exists c p > 0 and N p > 0 such that It turns out that if e.g. ω(ε) = ε, then the net (a ε ) ε in (1.7) is C ∞ -moderate. Note that the conditions of moderateness are natural in the sense that regularisations of distributions are moderate, namely by the structure theorems for distributions one can regard (1.8) Thus, by (1.8) we see that while a solution to the Cauchy problems may not exist in the space of distributions E ′ (R), it may still exist (in a certain appropriate sense) in the space on the right hand side of (1.8). The moderateness assumption allows us to recapture the solution as in (1.6) when it exists. However, we note that regularisation with standard Friedrichs mollifiers is not always sufficient, hence the introduction of a family ω(ε) in the above regularisations. Now let us introduce a notion of a "very weak solution" for the Cauchy problem (1.9) Definition 1.4. Let s be a real number.
As usual, a is a nonnegative distribution means that there exists a constant a 0 > 0 such that a ≥ a 0 > 0, while a ≥ a 0 means that a − a 0 ≥ 0, i.e. a − a 0 , ψ ≥ 0 for all nonnegative ψ ∈ C ∞ 0 (R). It can be remarked that it follows then that a is actually a positive measure, although we will not need to make use of this fact in our analysis.
Thus, let us formulate the result of the paper on the existence of very weak solutions of the Cauchy problem (1.9).
Theorem 1.5. (Existence) Let G be a graded Lie group and let R be a positive Rockland operator of homogeneous degree ν. Let T > 0 and s ∈ R.
Then the Cauchy problem (1.9) has a very weak solution of Then the Cauchy problem (1.9) has a very weak solution of H −∞ (s) -type. Now we show that the very weak solution of the Cauchy problem (1.9) is unique in an appropriate sense. For the formulation of the uniqueness statement, we will use the language of Colombeau algebras.
Actually, in this paper, it is sufficient to take K = [0, T ], since the time-dependent distributions can be taken supported in the interval [0, T ].
Let us now introduce the Colombeau algebra in the following quotient form: We refer to e.g. [Obe92] for the general analysis of G(R) . Theorem 1.8. (Consistency-1) Let G be a graded Lie group and let R be a positive Rockland operator of homogeneous degree ν. Let T > 0.
. Let u be a very weak solution of H s -type of (1.9). Then for any regularising families a ε and f ε in Definition 1.4, any representative . Let u be a very weak solution of H −∞ (s) -type of (1.9). Then for any regularising families a ε and f ε in Definition Similarly, we can show other consistency "cases" of Theorem 1.8, corresponding to Part (b) of Theorem 1.2 (ii) and Part (b) of Theorem 1.2 (iv): Theorem 1.9. (Consistency-2) Let G be a graded Lie group and let R be a positive Rockland operator of homogeneous degree ν. Let T > 0.
. Let u be a very weak solution of H −∞ (s) -type of (1.9). Then for any regularising families a ε and f ε in Definition 1.4, any representative . Let u be a very weak solution of H −∞ (s) -type of (1.9). Then for any regularising families a ε and f ε in Definition 1.4, any representative (u ε ) ε of u converges in C 2 ([0, T ]; H −∞ (s) ) as ε → 0 to the unique classical solution in C 2 ([0, T ]; H −∞ (s) ) of the Cauchy problem (1.9) given by Part (b) of Theorem 1.2 (iv).
The organisation of the paper is as follows. In Section 2 we briefly recall the necessary concepts of the setting of graded groups. The proof of main results are given in Section 3 for homogeneous Rockland wave equation. Finally, in Section 4 we briefly discuss the differences in the argument in the case of the inhomogeneous Rockland wave equation.

Preliminaries
In this section let us very briefly recall the necessary notation concerning the setting of graded groups. We refer to Folland and Stein [FS82, Chapter 1], or to the recent exposition in [FR16, Chapter 3] for a detailed description of the notions of graded and homogeneous nilpotent Lie groups.
A connected simply connected Lie group G is called a graded Lie group if its Lie algebra g has a vector space decomposition where the g ℓ , ℓ = 1, 2, ..., are vector subspaces of g, all but finitely many equal to {0}, and satisfying We fix a basis {X 1 , . . . , X n } of a Lie algebra g adapted to the gradation. Then, one can obtain points in G through the exponential mapping exp G : g → G as Let A be a diagonalisable linear operator on g with positive eigenvalues. Then, a family of dilations of a Lie algebra g is a family of linear mappings of the form Here, note that D r is a morphism of g, i.e.
where [X, Y ] := XY − Y X is the Lie bracket. Recall that the dilations can be extended through the exponential mapping to G by D r (x) = rx := (r ν 1 x 1 , . . . , r νn x n ), x = (x 1 , . . . , x n ) ∈ G, r > 0, where ν 1 , . . . , ν n are weights of the dilations. The homogeneous dimension of G is denoted by Q := Tr A = ν 1 + · · · + ν n . (2.1) Let G be the unitary dual of G. For a representation π ∈ G, let H ∞ π be the space of smooth vectors. We say that a left-invariant differential operator R on G, which is homogeneous of positive degree, is a Rockland operator, if it satisfies the following Rockland condition: (R) for every representation π ∈ G, except for the trivial representation, the operator π(R) is injective on H ∞ π , i.e. ∀υ ∈ H ∞ π , π(R)υ = 0 ⇒ υ = 0, where π(R) := dπ(R) is the infinitesimal representation of R as of an element of the universal enveloping algebra of G.
For a more detailed discussion of this definition, we refer to [FR16, Definition 1.7.4 and Section 4.1.1], that appeared in the work of Rockland [Roc78]. Alternative characterisations of such operators have been obtained by Rockland [Roc78] and Beals [Bea77], until the resolution in [HN79] by Helffer and Nourrigat of the socalled Rockland conjecture, which characterised operators satisfying condition (R) as left-invariant homogeneous hypoelliptic differential operators on G.
In this paper we will deal with the Rockland differential operators which are positive in the sense of operators.
We also refer to [FR16,Chapter 4] for an extensive presentation concerning Rockland operators and their properties, as well as for the consistent development of the corresponding theory of Sobolev spaces. In [CR17] the corresponding Besov spaces on graded Lie groups and their properties are investigated. Spectral properties of the infinitesimal representations of Rockland operators have been analysed in [tER97]. For the pseudo-differential calculus on graded Lie groups, we refer to [FR14] and [FR16].
Let π be a representation of G on the separable Hilbert space H π . We say that a vector v ∈ H π is a smooth or of type C ∞ if the function π be the space of all smooth vectors of a representation π. Let π be a strongly continuous representation of G on a Hilbert space H π . For every X ∈ g and v ∈ H ∞ π we denote Then dπ is a representation of g on H ∞ π (see e.g. [FR16, Proposition 1.7.3]), that is, the infinitesimal representation associated to π. By abuse of notation, we will often write π instead of dπ, therefore, we write π(X) instead of dπ(X) for any X ∈ g.
By the Poincaré-Birkhoff-Witt theorem, any left-invariant differential operator T on G can be written in a unique way as a finite sum which allows us to look at T as an element of the universal enveloping algebra U(g) of g, where all but finitely many of the coefficients c α ∈ C are zero and X α = X 1 · · · X |α| , with X j ∈ g. Therefore, the family of infinitesimal representations {π(T ), π ∈ G} yields a field of operators that turns to be the symbol associated with T . Let π ∈ G and let R be a positive Rockland operator of homogeneous degree ν > 0. Then, from (2.2) we obtain the following infinitesimal representation of R associated to π, where π(X) α = π(X α ) = π(X α 1 1 · · · X αn n ) and [α] = ν 1 α 1 + · · · + ν n α n is the homogeneous degree of the multiindex α, with X j being homogeneous of degree ν j .
Recall that R and π(R) are densely defined on D(G) ⊂ L 2 (G) and H ∞ π ⊂ H π , respectively (see e.g. [FR16, Proposition 4.1.15]). Let us denote the self-adjoint extension of R on L 2 (G) by R 2 and keep the same notation π(R) for the self-adjoint extensions on H π of the infinitesimal representations.
By the spectral theorem for unbounded operators [RS80, Theorem VIII.6], we write where E and E π are the spectral measures corresponding to R 2 and π(R).
Moreover, for any f ∈ L 2 (G) one has F (φ(R)f )(π) = φ(π(R)) f (π), (2.3) for any measurable bounded function φ on the real line R (see e.g. [FR16, Corollary 4.1.16]). Note that the infinitesimal representations π(R) of a positive Rockland operator R are also positive, because of the relations between their spectral measures.
In [HJL85] Hulanicki, Jenkins and Ludwig showed that the spectrum of π(R), with π ∈ G\{1}, is discrete and lies in (0, ∞), which allows us to choose an orthonormal basis for H π such that the infinite matrix associated to the self-adjoint operator π(R) has the following form where π ∈ G\{1} and π j ∈ R >0 . Now, since we will also deal with the Fourier transform on G, let us briefly recall it.
As usual we identify irreducible unitary representations with their equivalence classes. For f ∈ L 1 (G) and π ∈ G, the group Fourier transform of f at π is defined by with integration against the biinvariant Haar measure on G. It implies a linear mapping f (π) from the Hilbert space H π to itself that can be represented by an infinite matrix once we choose a basis for H π . Consequently, we have F G (Rf )(π) = π(R) f (π).
In the sequel, when we write f (π) m,k , we will be using the same basis in the representation space H π as the one giving (2.4).

Proofs of main results
In this section we prove main results of the paper when f ≡ 0, and in the case when f ≡ 0 we refer to Section 4 for the differences in the argument in this case.
First, we need to prove the following result: respectively.
We prove Theorem 1.2 (i) in Section 4. Now let us prove Part (b) of Theorem 1.2 (ii), (iii) and (iv).
Proof of Part (b) of Theorem 1.2 (ii), (iii) and (iv). Since the way of deriving Parts (b) of Theorem 1.2 (ii), (iii), (iv) from Parts (ii), (iii), (iv) of Theorem 1.1, respectively, is similar, let us show it only for Part (b) of Theorem 1.2 (iii), which will be useful in investigating the weak solution of (1.1). Recall the characterisation of H −∞ (s) . Since u 0 , u 1 ∈ H −∞ (s) and by Lemma 3.1 we see that there exist positive constants A 1 and C 1 such that for all m, k ∈ N. By the proof of [RT17a, Case 3 of Theorem 1.1, Page 20], we know that there exist positive constants C and K such that for 1 ≤ s < σ = 1 + ℓ/2 and some K ′ > 0 small enough, and all m, k ∈ N, where π m are strictly positive real numbers from (2.4).
Putting (3.1) into (3.2) we obtain that for all t ∈ [0, T ] there exist positive constants C 2 and A 2 such that which implies that there exist positive constants A and C > 0 such that This completes the proof of Part (b) of Theorem 1.2 (iii). Similarly, Parts (ii) and (iv) of Theorem 1.1 imply Part (b) of Theorem 1.2 (ii) and Part (b) of Theorem 1.2 (iv), respectively.
Proof of Theorem 1.5. (i) Assume that the coefficient a = a(t) is a distribution with compact support contained in [0, T ]. Then, we note that the formulation of (1.9) might be impossible in the distributional sense due to issues related to the product of distributions. Therefore, we replace (1.9) with a regularised equation. Namely, if we regularise the coefficient a by a convolution with a mollifier in C ∞ 0 (R), then we get nets of smooth functions as coefficients. For this, we take ψ ∈ C ∞ 0 (R), ψ ≥ 0 with ψ = 1, and ω(ε) > 0 such that ω(ε) → 0 as ε → 0 to be chosen later. Then, we define ψ ωε and a ε by and a ε (t) := (a * ψ ω(ε) )(t) for all t ∈ [0, T ], respectively. Using these representations of ψ ωε and a ε and identifying the measure a(t) with its density, we get where we have used that a(t) is a positive distribution with compact support (hence a Radon measure) and ψ ∈ C ∞ 0 (R), supp ψ ⊂ K, ψ ≥ 0 in above. Here, note that a 0 does not depend on ε.

(3.18)
From the definition of the very weak solution u, we also know that there exists a representative (u ε ) ε of u such that for suitable embeddings of a(t).
where n ε (t) = (a ε (t) − a(t))R u(t) ∈ C([0, T ]; H s R ) and converges to 0 in this space as ε → 0. By virtue of (3.19) and (3.20) we note that u − u ε solves the following Cauchy Then, similarly as in the proof of Theorem 1.7, we reduce above to a system and apply the group Fourier transform to get the following energy estimate since the coefficient a ε (t) is regular enough. Then, noting |( V − V ε )(0, π) m,k | = 0 and n ε → 0 in C([0, T ]; H s R ) and continuing to discussing as in Theorem 1.7 we arrive at |( V − V ε )(t, π) m,k | ≤ c(ω(ε)) α for some positive constants c and α, which concludes that . Furthermore, since any other representative of u will differ from (u ε ) ε by a C ∞ ([0, T ]; H s R ) -negligible net, the limit is the same for any representative of u. (ii) The Part (ii) can be proven as Part (i) with slight modifications. This completes the proof of Theorem 1.8.
Similarly, Theorem 1.1 (ii), (iii) and (iv) imply Parts (a) of Theorem 1.2 (ii), (iii) and (iv), respectively. Then, Parts (b) of Theorem 1.2 (ii), (iii) and (iv) can be proved as in homogeneous cases using Parts (a) of Theorem 1.2 (ii), (iii) and (iv). In the same way as in the proof of homogeneous cases of Theorems 1.5, 1.7, 1.8 and 1.9, their inhomogeneous cases can be proven with slight modifications.