Linear Algebra and its Applications

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1 Linear Algebra and its Applications 466 (15) Nonlinear Analysis 16 (15) 17 Contents lists available at ScienceDirect Contents lists available at ScienceDirect Linear Algebra and its Applications Nonlinear Analysis A proof of Hörmander s Inverse eigenvalue teoremproblem for sublaplacians of Jacobi on matrix Carnot groups wit mixed data Marco Bramanti a, Ying LucaWei Brandolini 1 b, a Dipartimento di Matematica, Department Politecnico of Matematics, di Milano, Nanjing Via Bonardi University of Aeronautics Milano, Italy and Astronautics, b Nanjing 116, PR Cina Dipartimento di Ingegneria gestionale, dell informazione e della produzione, Università degli Studi di Bergamo, Viale Marconi Dalmine B, Italy a r t i c l e i n f o a b s t r a c t a r t i c l e i n f o a b s t r a c t Article istory: In tis paper, te inverse eigenvalue problem of reconstructing Received 16 January 14 a Jacobi matrix from its eigenvalues, its leading principal Article istory: Accepted September We 14 give a self-contained analytical proof of Hörmander s ypoellipticity teorem in Received 1 December 14 submatrix and part of te eigenvalues of its submatrix Available online October te case14 of left invariant sub-laplacians of Carnot groups. Te proof does not rely on Accepted 13 February 15 is considered. Te necessary and sufficient conditions for Submitted by Y. Weipseudodifferential calculus and provides intermediate regularity estimates expressed Communicated by Enzo Mitidieri te existence and uniqueness of te solution are derived. in terms of Sobolev spaces induced by rigt invariant vector fields. MSC: Furtermore, a numerical algoritm and some numerical 15 Elsevier Ltd. All rigts reserved. Keywords: 15A18 examples are given. Hypoelliptic operators 15A57 14 Publised by Elsevier Inc. Hörmander s ypoellipticity teorem Keywords: Jacobi matrix Eigenvalue 1. Introduction Inverse problem Submatrix A linear differential operator L wit smoot coefficients is said to be ypoelliptic in an open set Ω R N if, for every open set Ω Ω, wenever a distribution u D (Ω ) is suc tat Lu C (Ω ) ten u C (Ω ). Te celebrated ypoellipticity teorem of Hörmander [5] provides an almost complete caracterization of second order ypoelliptic operators wit real coefficients. After noting tat every ypoelliptic second order differential operator as necessarily semi-definite principal part, Hörmander proves tat in any open set were te rank of te coefficient matrix is constant, te operator (or its opposite) can be rewritten in te form L = address: weiyingb@gmail.com. 1 Tel.: q j + + c (1.1) were, 1,..., q are real smoot vector fields (tat is, first order differential operators) and c is a smoot function. Motivated by tis preliminary analysis, Hörmander studies operators already written ttp://dx.doi.org/1.116/j.laa in te form (1.1) and proves tat L is ypoelliptic in Ω R N if te vector fields of te Lie algebra / 14 Publised by Elsevier Inc. generated by, 1,..., q span te wole R N at every point x Ω (an assumption wic as been Corresponding autor. addresses: marco.bramanti@polimi.it (M. Bramanti), luca.brandolini@unibg.it (L. Brandolini). ttp://dx.doi.org/1.116/j.na / 15 Elsevier Ltd. All rigts reserved.

2 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) labeled Hörmander s condition ). Conversely, if in an open set U Ω te rank of te Lie algebra is constant and strictly less tan N, ten te operator L is not ypoelliptic in U, ence Hörmander s condition is almost necessary for ypoellipticity. A somewat easier proof of Hörmander s teorem was given by Kon [6], deeply exploiting tecniques of pseudodifferential operators. Te paper by Kon empasizes an intermediate result wic as great independent interest, namely te so-called subelliptic estimates, allowing to control a fractional Sobolev norm H s+ε of u (for some small ε > ) in terms of a norm H s of Lu and some oter less regular norm of u. Altoug trougout te years several autors ave written oter proofs of Hörmander s teorem, it is remarkable tat, apart from te probabilistic approac due to Malliavin [7], all te analytic proofs of tis result substantially go along te line of Kon, based on subelliptic estimates and pseudodifferential operators. After te seminal ypoellipticity result by Hörmander, te teory of Hörmander s operators (1.1) as undergone a tremendous development. In particular, muc more precise estimates for te regularity properties of operators of tis kind ave been obtained: Folland [4] and Rotscild Stein [8] proved, in increasing generality, a priori estimates in L p (1 < p < ) for te igest order derivatives wit respect to te vector fields involved in te equations, tat is estimates on i j u (i, j = 1,,..., q), u, in terms of Lu and u, and analogous iger order estimates. More in detail, Folland [4] studied Hörmander s operators on omogeneous groups, tat is operators of te kind q L = j + were te vector fields are assumed to be left invariant wit respect to a Lie group in R N ( translations ) and omogeneous wit respect to a family of group automorpisms ( dilations ), so tat L be -omogeneous and left invariant (precise definitions will be given later). For tese operators Folland sowed te existence of a global omogeneous fundamental solution, wic is a key tool in proving sarp a priori estimates. Rotscild Stein [8] proved analogous a priori estimates for general Hörmander s operators, also exploiting te fundamental solution built by Folland on groups. It is wortwile noting tat Folland s construction of a omogeneous fundamental solution for Hörmander s operators on omogeneous groups exploits, among oter results, te ypoellipticity of tese operators, a fact wic, instead, is not directly used any longer in te teory developed by Rotscild Stein. Hence we can say tat Hörmander s teorem plays a special role in te context of omogeneous groups. Also, from different points of view one can say tat Hörmander s operators on groups serve as models for general Hörmander s operators. Te corresponding teory, and particularly tat of sublaplacians on stratified omogeneous groups (nowadays called Carnot groups), tat is left invariant -omogeneous Hörmander s operators of te kind q L = as been widely studied in te last decades. Te books [1,1] contain lots of material on tis subject and also references to te original papers. In tis paper we give a new proof of Hörmander s ypoellipticity teorem for sublaplacians in Carnot groups (as far as we know, te first proof in suc simplified context). Te double simplification consisting in assuming an underlying structure of Carnot group and avoiding te presence of te drift term allows to devise a fairly elementary proof. Actually, we completely avoid te use of te teory of pseudodifferential operators. We prove a kind of subelliptic estimates wic are not saped on Euclidean Sobolev spaces of fractional order, but on Sobolev spaces induced by te vector fields temselves, ence more in te spirit of te subsequent developments of te teory of Hörmander s operators. Our proof will consist in two steps. First, in Section 3, we will prove a priori estimates stating tat if u W 1,,loc is a local weak solution to Lu = f in a domain (see Section for precise definition and notation), ten te L norm of any fixed j

3 17 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 number of i R-derivatives of u (R i being rigt invariant vector fields) can be controlled in terms of te L norm of u and tose of a (muc larger) number of derivatives of f. Altoug tis kind of estimate is not sarp, it as te same independent interest of Kon s subellipticity estimates, because not only it implies tat if f is smoot also u is smoot, but also expresses a quantitative a priori control on any finite number of derivatives, wic is useful in several situations. Te second part of te proof (Section 4) consists in weakening te assumption on u, from u W 1,,loc to general distribution, wic implies te ypoellipticity of L. Te first part of te proof (Section 3) is nearly self-contained: we will make use of some standard facts from te teory of Carnot groups, collected in Section. In te second part of te proof (Section 4) we also ave to make use of some well-known facts from distribution teory. Our proof will make use bot of te Sobolev spaces induced by te vector fields i (as well as teir rigt-invariant counterparts i R ) and of some seminorms defined in terms of finite differences tat measure te Hölder continuity in L sense along te vector fields of te first layer, and will be introduced and discussed in Section 3. We will sketc te strategy of eac of te two steps of te proof at te beginning of te corresponding section.. Some known facts about Carnot groups In tis section we recall a number of standard definitions and results tat will be useful in te following. For te proofs of tese facts te reader is referred to [4], [1, Capter 1], [1, Capter III, Section 5], and []. A omogeneous group (in R N ) is a Lie group R N, (were te group operation will be tougt as a translation ) endowed wit a one parameter family {D λ } λ> of group automorpisms ( dilations ) wic act tis way: D λ (x 1, x,..., x N ) = (λ α1 x 1, λ α x,..., λ α N x N ) (.1) for suitable integers 1 = α 1 α... α N. We will write = R N,, D λ to denote tis structure. Te number N Q = will be called omogeneous dimension of. Under te cange of coordinates x = D λ (y) te volume element transforms according to i=1 α i dx = λ Q dy (.) wic justifies te name of omogeneous dimension for Q. Note tat we always ave Q N, and Q = N only if te dilations are te Euclidean ones. A omogeneous norm on (also called a gauge on ) is a continuous function : R N [, + ), suc tat, for some constant c >, for every x, y R N, (i) x = x = (ii) D λ (x) = λ x λ > (iii) x 1 c x (iv) x y c ( x + y ). We will always use te symbol to denote a omogeneous norm, and te symbol to denote te Euclidean norm. Concrete ways to define a omogeneous norm on are for instance te following: x = max x k 1 α k k=1,,...,n

4 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) or N 1/Q x = x k Q α k. k=1 It can be proved tat any two omogeneous norms 1, on are equivalent: tere exist two positive constants k 1, k suc tat k 1 x 1 x k x 1 x R N. Also, te following local comparison wit te Euclidean norm olds: for any R > tere exist two positive constants c 1, c (depending on R) suc tat c 1 x x c x 1/Q if x R (or x R). (.3) Altoug in general x y y 1 x, tey are locally equivalent near te origin. More precisely tere exist positive constants c 1 and c and a neigborood of te origin V suc tat for x, y V we ave c 1 x y y 1 x c x y. (.4) To see tis let P y (x) = y 1 x. It follows from te structure of te group operation on a omogeneous group tat Py x (y) is close to te identity in a neigborood of te origin. Since y 1 x = P y (y) + P y x (y) (y x) + o (y x) and P y (y) =, (.4) readily follows. We say tat a smoot function f in R N \ {} is D λ -omogeneous of degree β β-omogeneous ) if R (or simply f (D λ (x)) = λ β f (x) λ >, x R N \ {}. iven any differential operator P wit smoot coefficients on R N, we say tat P is left invariant if for every x, y R N for every smoot function f, were P (L y f) (x) = L y (P f (x)) L y f (x) = f (y x). Analogously one defines te notion of rigt invariant differential operator. Also, P is β-omogeneous (for some β R) if for every smoot function f, λ > and x R N. A vector field is a first order differential operator P (f (D λ (x))) = λ β (P f) (D λ (x)) = N c i (x) xi. i=1 Let g be te Lie algebra of left invariant vector fields over, were te Lie bracket of two vector fields is defined as usual by [, Y ] = Y Y.

5 174 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 Let us denote by 1,,..., N te canonical base of g, tat is for i = 1,,..., N, i is te only left invariant vector field tat agrees wit xi at te origin. Also, 1 R, R,..., N R will denote te rigt invariant vectors fields tat agree wit x1, x,..., xn (and ence wit 1,,..., N ) at te origin. We assume tat for some integer q < N te vector fields 1,,..., q are 1-omogeneous and te Lie algebra generated by tem is g. If s is te maximum lengt of commutators i1, i,..., is 1, is, ij {1,,..., q} required to span g, ten we will say tat g is a stratified Lie algebra of step s, is a Carnot group (or a stratified omogeneous group) and 1,,..., q satisfy Hörmander s condition at step s in R N. Under tese assumptions, let us denote by te canonical sublaplacian on. It will be sometimes useful te compact notation L = f = Rf = q j q j f q j R f. We will make use of te Sobolev spaces W k,p k,p (), W () induced by te systems of vector fields R = { 1,,..., q }, R = 1 R, R,..., q R, respectively. More precisely, given an open subset Ω of R N, we say tat f W 1,p (Ω) (1 p ) if f L p (Ω) and tere exist, in weak sense, j f L p (Ω) for j = 1,,..., q. Inductively, we say tat f W k,p f, j1 j... jk 1 f, belongs to W 1,p k 1,p (Ω) for k =, 3,... if f W (Ω). We set f W k,p (Ω) = f L p (Ω) + k =1 j i=1,,...,q (Ω) and any weak derivative of order k 1 of j1 j... j f Lp (Ω). Te space W k,p R (Ω) as a similar definition. As usual, one can define te Sobolev spaces of functions vanising at te boundary of a domain, W 1,p, (Ω) or W 1,p (Ω) taking te closure of C, R (Ω) in te norm W 1,p 1,p (Ω) or W (Ω), respectively. It can be proved R tat for 1 p < one as W 1,p, R N = W 1,p R N and analogously for R -spaces. We will also use local Sobolev spaces. For example, we will say tat f W k,p,loc (Ω) if for every ϕ C we ave ϕf W k,p (Ω). Te validity of Hörmander s condition at step s implies te following important: (Ω), Proposition.1 (Regularity by Means of Sobolev Spaces). Under te above assumptions we ave: 1. For any 1 p, k=1 W k,p (Ω) C (Ω).

6 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) For any positive integer k and p [1, ] and any Ω Ω tere exists a constant c > suc tat, for every u W ks,p (Ω) we ave u W k,p (Ω ) c u W ks,p (Ω), were W k,p (Ω ) denotes te standard Sobolev space. Let us also recall te definition of control distance induced by te vector fields 1,,..., q in R N. For any δ >, let C x,y (δ) be te class of absolutely continuous mappings ϕ : [, 1] R N wic satisfy ϕ (t) = wit a i : [, 1] R measurable functions, Ten define q a i (t) ( i ) ϕ(t) i=1 a i (t) δ a.e. ϕ () = x, ϕ (1) = y. a.e. d (x, y) = inf {δ > : ϕ C x,y (δ)}, wit te convention inf = +. We also define te d-balls B (x, r) = {y Ω : d (x, y) < r}. A consequence of Hormander s condition is tat actually d (x, y) is finite for every couple of points (Cow Rasevsky connectivity teorem). Te control distance d on a Carnot group as several properties: (i) d is left invariant: and satisfies (ii) d is 1-omogeneous: (iii) te function d (z x, z y) = d (x, y) x, y, z R N (.5) d (x 1 x x k, ) k d (x j, ) ; (.6) d (D λ (x), D λ (y)) = λd (x, y) x, y R N, λ >. x = d (x, ) is a omogeneous norm. More precisely, it also satisfies te stronger properties x 1 = x x y x + y. (.7)

7 176 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 Trougout te following we will need to use a omogeneous norm satisfying te triangle inequality (.7). Te previous discussion sows tat suc a omogeneous norm always exists. Actually tis particular coice of a omogeneous norm is relevant only witin te proofs, wile te statements of all our results will still old, wit possibly different constants, if we replace tis omogeneous norm wit a different one. Let Exp : g be te exponential map, defined as usual letting Exp () = exp () () were exp (Y ) (x ) = F (1) for F te solution to te Caucy problem F (τ) = Y F (τ) ; F () = x. One as Also recall tat for every smoot function f and every t R exp () (x) = x Exp (). (.8) d (f (x Exp (t))) = f (x Exp (t)). (.9) dt Let us point out a relation between left and rigt invariant operators wic will be very useful in te following. Proposition.. Let L, R be any two differential operators wit smoot coefficients, left and rigt invariant, respectively. Ten L and R commute: for any smoot function f. Proof. We ave LRf = RLf Lf (x) = L y [f (x y)] y= Rf (x) = R z [f (z x)] z= were L, R are constant coefficient differential operators (namely, tey are te variable operators L, R evaluated at te origin). Ten: LRf (x) = L y [Rf (x y)] y= = Ly [Rz [f (z x y)] z= ] y= = L y Rz [f (z x y)] (z,y)=(,) = R z L y [f (z x y)] (z,y)=(,) = RLf (x) since R z, L y obviously commute. We will now introduce on te convolution between test functions and distributions. Trougout te paper we will denote tis operation wit te symbol. Since te group is in general noncommutative also is noncommutative. However, most of its properties are straigtforward adaptations of te same properties tat old for te standard convolution, see e.g. [9, Capter 6]. Hence, we will not give te proofs of te following two propositions. For any given couple of test functions ϕ, ψ C () we define ϕ ψ (x) = ϕ (y) ψ y 1 x dy. It is a simple exercise to ceck tat ϕ ψ C () and tat given ϕ, ψ, ω C () we ave ϕ (ψ ω) = (ϕ ψ) ω.

8 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) Using te left translation operator L x ϕ (y) = ϕ (x y) and te inversion operator qϕ (x) = ϕ x 1 we can also write ϕ ψ (x) = ϕ (y) L q x 1ψ (y) dy so tat given f D () (te space of distributions on ), it is natural to define f ψ (x) = f, L q x 1ψ. (.1) For f D () let also define f q requiring tat for every ϕ C qf, ϕ = f, qϕ. (), One can prove te following: Proposition.3. Let f D () and let ψ, ϕ C (). Ten we ave: (i) f ψ, ϕ = f, ϕ ψ q = qf, ψ qϕ (ii) if P is a left invariant differential operator ten P (f ψ) = f Pψ, (iii) if P is a rigt invariant differential operator ten P (f ψ) = Pf ψ. Remark.4. From (ii) it follows tat f ψ C (). Assume now tat f D () as compact support. In tis case f as a unique extension to a linear functional on C () and terefore (.1) can be used to define te convolution wit functions ψ C (). Also in tis case if P is a left invariant differential operator we ave P (f ψ) = f Pψ. Let now f D () wit compact support and g D (). For every ϕ C () we define f g, ϕ = qf, g qϕ. Note tat g qϕ C () so tat qf, g qϕ is well defined. Te following olds: Proposition.5. Let f, g D () and assume tat f as compact support. We ave te following: (i) If P is a left invariant differential operator ten P (f g) = f Pg.

9 178 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 (ii) If P is a rigt invariant differential operator ten P (f g) = Pf g. 3. Regularity estimates for te sublaplacian in W 1, In order to sow tat wenever Lf is smoot also f is smoot we will prove estimates tat allow to control te Sobolev norm of f in terms of te Sobolev norm of Lf. Our starting point is te following elementary computation. Let f C (), ten q f = ( i f (x)) dx wic gives te estimate f W 1, i=1 = Lf (x) f (x) dx Lf f 1 Lf + f c ( Lf + f ) f C (). (3.1) Assume we knew, instead of (3.1), te following apparently similar estimate f W 1, R c ( Lf + f ) f C (3.) were te Sobolev norm in te LHS is computed using te rigt invariant vector fields. Ten exploiting te fact tat te left invariant operator L commutes wit te rigt invariant operators i R (see Proposition.) we could apply (3.) to te function i R 1 i R... i R k f, obtaining R i1 i R... i R k f W 1, c L R i1 i R... i R k f + R i1 i R... i R k f R = c R i1 i R... i R k Lf + R i1 i R... i R k f ence and iteratively f W k+1, R f W k+1, R c Lf W k, + f W k, R R c Lf W k, + f, R an estimate tat allows to control te derivatives of f in terms of te derivatives of Lf. In oter words, te idea of measuring te degree of smootness of a solution to Lu = f (wit L left invariant) using rigt invariant derivatives apparently trivializes te problem, as if we were andling an elliptic operator wit constant coefficients. However, we do not ave te bound (3.). Neverteless, we will see tat it is possible to control te regularity expressed using rigt invariant vector fields in terms of te left invariant ones, but tis implies a loss of regularity. If s is te step of te Lie algebra of, using one derivative wit respect to te left invariant vector fields, we can only control a rigt invariant regularity of order 1/s. More precisely we will sow tat for small te following bound olds 1/ f ( x) f (x) dx c 1/s f c 1/s ( Lf + f ). (3.3) In te next section we will sow tat iterating te above estimate it is possible to control a full derivative wit respect to te rigt invariant vector fields using s 1 derivatives of Lf. More generally we will sow

10 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) tat it is possible to control k derivatives i R of f using k + s 1 derivatives of Lf and from tis we will deduce tat for a function f tat is locally in W 1,,, wenever Lf is smoot also f is smoot. Before stating precisely our regularity result let us clarify te meaning of Lf for functions tat are locally in W 1,. Definition 3.1. Let f W 1,,loc (). We say tat Lf L loc () if tere exists F L loc () suc tat for every φ C () we ave q j f j φdx = F φdx φ C (), (3.4) and in tis case we will say tat f is a local weak solution to te equation Lf = F. Remark 3.. If f W 1, () and Lf L () ten, by density, (3.4) olds for every φ W 1,, (). Notation 3.3. For a couple of cutoff functions ζ, ζ 1 C (Ω) we will write ζ ζ 1 to say tat ζ 1 1 on sprt ζ. Our regularity result is te following. Teorem 3.4 (Regularity Estimates). Let f W 1,,loc () be a local weak solution to Lf = F L loc (), and, for some open set Ω, let ζ, ζ 1 C (Ω), ζ ζ 1. Ten: (i) For any k = 1,, 3,..., tere exists c > suc tat wenever Lf W k+s 1, R,loc (Ω) ten f W k, R,loc (Ω) and (ii) In particular, if Lf C (Ω) ten f C (Ω). ζf W k, R (RN ) c ζ 1 Lf W k+s 1, R (R N ) + ζ 1f L (R N ). (3.5) Te proof of Teorem 3.4 will be accomplised in Section 3.. In Section 3.1 we develop some properties of finite difference operators tat will be necessary for te proof of te teorem Finite difference operators and Sobolev norm It is well known tat Hörmander condition for te vector fields 1,,..., q implies tat it is possible to connect two points using only integral lines of te vector fields. From an algebraic point of view tis means tat every element of te group can be written as a product of exponentials of te first layer vector fields. Teorem 3.5. Let be a Carnot group. Tere exist absolute constants M N and c > suc tat for every x tere exist real numbers t 1, t,..., t M and indices k 1, k,..., k M {1,,..., q} suc tat and t j c x. x = Exp (t 1 k1 ) Exp (t k ) Exp (t M km )

11 18 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 For a proof of tis teorem see e.g. [1, 19..1]. A simple but important consequence of te above teorem is te fact tat it is possible to control te increment of a function using f. We start wit a definition. Definition 3.6 (Finite Difference Operators). For every let us define te operators: f (x) = f (x ) f (x) f (x) = f ( x) f (x). Note tat te operator, wic acts on functions computing te increment of f corresponding to an increment of its variable on te rigt (x x ) is actually a left invariant operator (L y f = L y f); analogously, computes te increment of f corresponding to an increment of its variable on te left and is a rigt invariant operator. Tis duality is a central point in te tecniques wic will be used trougout te paper. Proposition 3.7. Tere exists a constant c = c () suc tat for any and any f W 1, () we ave f c f and for any f W 1, R () we ave f c Rf. Proof. It is enoug to prove te first assertion, since te second one is analogous. Also, it is enoug to prove te assertion for f C (), te case f W 1, () ten follows from te fact tat C () function are dense in W 1, (). wit Let, by Teorem 3.5 we can write = Exp (t 1 k1 ) Exp t M 1 km 1 Exp (tm km ) t i c. Let x and set j = Exp t j kj x = x x j = x j 1 j = x 1 j x M = x. For any f C () using (.9) we obtain f (x ) f (x) = = = M M f (x j ) f (x j 1 ) = f (x j 1 j ) f (x j 1 ) tj M tj M d f xj 1 Exp s kj ds ds kj f x j 1 Exp s kj ds.

12 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) Hence f Since t1 we ave (by Teorem 3.5) k1 f (x Exp (s k1 )) ds + M j= tj kj f x 1 j 1 Exp s kj = f max { t j } kj f x 1 j 1 Exp s kj ds. N kj f C f. kj f (x) dx In order to prove a bound of te kind (3.3) we need to translate te information about a control on te quotient f f in terms of a control on. As already remarked, tis implies a loss of regularity. Te following lemma is an adaptation of te proof of Lemma 4.1 in [5]. Lemma 3.8. Let f L (), U, sprt f U. Tere exists c >, depending on U, suc tat f sup f c sup < 1 1/s < 1 wenever te rigt and side is finite. (Here s is te step of te Lie algebra of.) Proof. Let, 1 and let B = k : k 1/s. For every k we can write f ( x) f (x) dx f ( x) f (x k) dx + f (x k) f (x) dx and since B = c Q/s, integrating over B we obtain f ( x) f (x) dx c Q/s f ( x) f (x k) dx dk B + c Q/s f (x k) f (x) dx dk B = c Q/s f ( x) f (x k) dx dk + c Q/s A + B. B B k f dk We ave B c Q/s c Q/s sup < k 1 sup < k 1 k f k B k dk k f k /s B dk = c /s k f sup < k 1 k. Let us consider A. We make te cange of variable (x, k) = Φ (y, u) = 1 y u, u 1 y 1 y

13 18 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 so tat Since u = y 1 y k 1, A = c Q/s f (y u) f (y) det J Φ (y, u) dydu. Φ 1 ( B) u y 1 y + k 1 c y 1 y 1/s + 1/s. Let F (y, ) = y 1 y. Since F is smoot and F (y, ) =, it follows tat y 1 y c for y ranging in a compact set. Tis appens because x ranges in te compact set U, k 1/s 1, Φ (y, u) is a diffeomorpism, ence also (y, u) range in a bounded set. Hence we can write u c 1/s + 1/s c 1/s. Moreover since te Jacobian J Φ (y, u) is also bounded by a constant depending only on U, A c Q/s f (y u) f (y) dydu c Q/s { u c 1/s } sup < k 1 k f k k f = c sup < k 1 k /s. { u c 1/s } u du Combining Proposition 3.7 wit te above lemma gives f c 1/s f c 1/s ( Lf + f ) (3.6) for every f W 1, () supported in a compact U, and Lf L (), wit c also depending on U. Now we need to recover a full derivative wit respect to rigt invariant vector fields. To do tat we introduce iger order difference operators and adapted seminorms. Definition 3.9. For m = 1,, 3, 4,..., let m m =.... m times =.... m times Ten, for α > and f L () we define te semi-norms m f L () f m,α = sup α : = Exp (t i ) for i = 1,,..., q, t R s.t. < 1 m f R m,α = sup f L () α : = Exp (t i ) for i = 1,,..., q, t R s.t. < 1. We also set for convenience f = f R = f L () f m = f m,m f R m = f R m,m.

14 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) Remark 3.1. From Proposition 3.7 we read in particular tat for any f W 1, seminorms: f 1 c f W 1, () f R 1 c f W 1, R () () or W 1, R (), respectively. It is not difficult to extend tis bound to iger order Teorem Let f L () and let m = 1,,.... Ten: 1. If f W m, () ten m Analogously,. If f W m, () ten R k= m k= f k c f W m, (). (3.7) f R k c f W m, (). (3.8) R Proof. Let us prove (3.7), te proof of (3.8) being similar. Let f W m, () and let = Exp (t j), for some j = 1,,..., q, and < 1. It can easily be proved tat = t. First of all observe tat a simple induction argument sows tat m f (x) = Indeed for m = 1 we ave t t m j f (x Exp (s 1 j ) Exp (s m j )) ds 1 ds m. (3.9) f (x) = f (x Exp (t j )) f (x) = = t t d ds f (x Exp (s j)) ds j f (x Exp (s j )) ds were te last identity follows by (.8) and te definition of exponential of a vector field. Assume now (3.9) olds for some m. Since Exp (s j ) m+1 f (x) = m f (x ) m f (x) t t = m j f (x Exp (s 1 j ) Exp (s m j )) = = = m j f (x Exp (s 1 j ) Exp (s m j )) ds 1 ds m t t t t t t t t and Exp (t j ) commute we ave d m ds j f (x Exp (s j ) Exp (s 1 j ) Exp (s m j )) dsds 1 ds m m j f (x Exp (s 1 j ) Exp (s m j ) Exp (s j )) ds 1 ds m ds d ds m+1 j f (x Exp (s 1 j ) Exp (s m+1 j )) ds 1 ds m+1.

15 184 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 Ten, since = t, by Minkowski s inequality for integrals and te translation invariance of Lebesgue measure on, we obtain t m f L () = t t t j m f (x Exp (s 1 j ) Exp (s m j )) ds L () 1 ds m j m f ds L () 1 ds m t m j m f = L () m f W m, (). Remark 3.1. Te same proof also sows tat m f L () c m f W m, (), (3.1) an inequality tat we will sometimes apply in te following. So far we ave seen tat Sobolev norms control te corresponding seminorms defined by finite difference operators. Let us prove a converse result, for te case m = 1: Proposition Let f L (). 1. If f 1 < ten f W 1, (), wit j f C f 1 for j = 1,,..., q.. If f R 1 < ten f W 1, R (), wit R j f C f R 1 for j = 1,,..., q. Proof. Let φ C () and = Exp (t j ) for some t >, j = 1,,..., q. Ten 1 f (x) φ (x) dx f 1 φ. On te oter and since 1 = Exp ( t j ), we ave, recalling tat = t, 1 f (x) φ (x) dx = 1 f (x) 1φ (x) dx = f (x) φ (x Exp ( t j)) φ (x) dx t ence and for t f (x) φ (x Exp ( t j)) φ (x) t f j φdx f 1 φ dx f 1 φ wic implies tat tere exists, in weak sense, j f L (), wit j f f 1. Te proof of is similar. Let us note an important difference between te seminorms f m for m = 1 and for iger m. Wile for m = 1 te finiteness of f 1 for an L function implies te finiteness of te Sobolev norm W 1,, for iger m te seminorms f m are strictly weaker tan W m,. However, te equivalence in case m = 1 will be enoug for our purposes.

16 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) Regularity estimates We split te proof of Teorem 3.4 into several steps. Let us start rewriting (3.6) in te form f R 1,1/s c ( Lf + f ). (3.11) Recall tat te constant c depends on some compact U containing te support of f. As a first step we will sow, wit an iterative argument, tat tis estimate can be extended to iger order seminorms R m,m/s (Teorem 3.15), wic in turn allows to control R 1,1 by R m,m/s, for m large enoug (see Proposition 3.16). Tis is te estimate tat is needed to implement te idea described at te beginning of Section 3. We start wit some lemmas. Lemma For every m N and we ave were L is te left translation operator m (fg) = m m k k= k f L k m k g (3.1) L f (x) = f ( x). (3.13) Proof. Wen m = 1 it is immediate to see tat (fg) (x) = f (x) L g (x) + f (x) g (x). Te case m > 1 can be obtained by induction. We omit te details. Let us come to te extension of (3.11) to iger order difference operators, also in a localized version. Teorem Let ζ, ζ C () wit ζ ζ. For every m N tere exists c > suc tat if f W 1,,loc () and Lf L loc () ten m 1 ζ f R m,m/s c ζlf R j + ζf, (3.14) wenever te rigt and side is finite. j= Proof. iven two cut-off functions ζ, ζ C () satisfying ζ ζ, for any fixed positive integer n we can always construct intermediate cut-off functions ζ ζ 1 ζ n = ζ. Also, for a fixed positive integer m tere exists ε > suc tat wen < ε we ave m 1 ζ i ζ i+1 for i =, 1,..., n. Te number n will vary in different steps of te proof. We will prove te teorem by induction on m. Let m = 1. Applying (3.11) to ζ f we obtain ζ f R 1,1/s c ( L (ζ f) + ζ f ) q c ζ Lf + i ζ i f + flζ + ζ f i=1 q c ζ Lf + i (ζ 1 f) + ζ 1 f. (3.15) i=1

17 186 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 For te term tat contains te first order derivatives we ave q q i (ζ 1 f) = i (ζ 1 f) i (ζ 1 f) dx = L (ζ 1 f) (ζ 1 f) dx i=1 terefore i=1 = ζ 1 Lf ζ 1 fdx (Lζ 1 ) f ζ 1 dx i=1 q ( i ζ 1 ) ( i f) ζ 1 fdx q ζ 1 Lf ζ 1 f + c ζ f ( i ζ 1 ) ( i (ζ 1 f)) fdx + i=1 q ζ 1 Lf ζ 1 f + c ζ f + i (ζ 1 f) ζ f c ζ 1 Lf + ζ f + ε i=1 q i (ζ 1 f) + c ε ζ f i=1 q i=1 ( i ζ 1 ) f dx q i (ζ 1 f) c ( ζ Lf + ζ f ) (3.16) i=1 wic inserted into (3.15) gives ζ f R 1,1/s c ( ζ Lf + ζ f ). (3.17) Assume now tat (3.14) olds for every m < m and let = Exp (t j ) for some i = 1,..., q suc tat < ε. Since m (ζ f) = m 1 (ζ f) = (ζ 1 g ) wit g = m 1 (ζ f) ζ 1 we ave m (ζ f) 1/s ζ 1 g R 1,1/s c 1/s ( ζ Lg + ζ g ), were we ave applied our teorem for m = 1. Since L and commute we obtain m (ζ f) c 1/s L m 1 (ζ f) + m 1 (ζ f) = c 1/s m 1 L (ζ f) + m 1 (ζ f) = c 1/s m 1 ζ Lf + j ζ j f + flζ + m 1 (ζ f) j c 1/s m 1 (ζ Lf) + j m 1 ( j ζ j f) + m 1 (flζ ) + m 1 (ζ f). (3.18)

18 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) From Lemma 3.14 we obtain m 1 (ζ Lf) c and by (3.1) m 1 As for te second term in (3.18) we ave m 1 ( j ζ j f) c m 1 k= m 1 k= (ζ Lf) c c c k (Lf) L k m k 1 ζ L k m k 1 ζ k (ζ 1 Lf) m 1 k= m k 1 k (ζ 1 Lf) m 1 c m 1 m 1 k= m 1 k= k= ζ 1 Lf R k. (3.19) k ( j ζ ) L k m 1 k j f were we used te fact tat j and m 1 k commute and ζ ζ 1. Using (3.16) and te fact tat L and m 1 k commute we ave j ζ 1 m 1 k ζ f c m 1 k Lf + ζ m 1 k f c m 1 k (ζ 4 Lf) + m 1 k (ζ 3 f) k j ( j ζ ) ζ 1 m 1 k f, (3.) c m 1 k ζ 4 Lf R m 1 k + c (m 1 k)/s ζ 3 f R m 1 k,(m 1 k)/s. Using te inductive assumption for any k =,..., m 1, we obtain j ζ 1 m k 1 m k ζ 3 f R m 1 k,(m 1 k)/s c i= ζ 4 Lf R i + ζ 4 f (ere we implicitly assume tat for k = m 1 te summation is empty). Terefore m k f c m 1 k ζ 4 Lf R m 1 k + c (m 1 k)/s Since k ζ j c k using (3.) we obtain m 1 ( j ζ j f) m 1 c m 1 k= m 1 c m 1 k= m 1 c (m 1)/s m 1 ζ 4 Lf R k + c k= i= m k (m 1 k)/s m ζ 4 Lf R k + c (m 1)/s k= ζ 4 Lf R k + ζ 4f k= ζ 4 Lf R i + ζ 4 f i= ζ 4 Lf R k + ζ 4f ζ 4 Lf R i + ζ 4 f. (3.1).

19 188 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 As to te last two terms in (3.18) we can bound m 1 (flζ ) + m 1 (ζ f) (m 1)/s flζ R m 1,(m 1)/s + ζ f R m 1,(m 1)/s. (3.) Inserting (3.19), (3.1) and (3.) in (3.18) we get m 1 m (ζ f) c m/s ζ 4 Lf R k + ζ 4f + flζ R m 1,(m 1)/s + ζ f R m 1,(m 1)/s. k= By te inductive assumption, we obtain so tat m flζ R m 1,(m 1)/s + ζ f R m 1,(m 1)/s c i= ζ Lf R j + ζ f m 1 m (ζ f) c m/s ζ 4 Lf i + ζ 4 f. Recall now tat we cose = Exp (t i ) wit < ε. To obtain (3.14) it is enoug to observe tat wen > ε one as te trivial estimate m (ζ f) c ζ f c ζ 4 f ε. Te previous teorem allows to control te regularity of a function f, measured using difference operators, by means of te regularity of Lf. Unfortunately we cannot apply directly Proposition 3.13 to bound R i f since it requires an estimate for te first order difference of f wile (3.14) contains iger order differences. A result of M.A. Marcaud allows to bound te first order difference operator using iger order one (see e.g. [3, Capter, Teorem 8.1]). In te following proposition we adapt tis classical result to our setting. Proposition 3.16 (Marcaud Inequality on Carnot roups). Let f L () and assume tere exist A >, an integer m > 1 and 1 < α < suc tat for every 1 we ave m f A α. (3.3) Ten tere exists c >, independent of f, suc tat for 1 f c (A + f ). Proof. For every integer k 1 let and observe tat Q k (x) is a polynomial satisfying Q k (x) = 1 k (x + 1) k x 1 (x 1) k = (x 1) k+1 Q k (x) + k x 1 k. (3.4) For every let L denote te left translation operator (3.13). Ten clearly f (x) = (L I) f. Also note tat L L = L, were from now on we will write n =. n times

20 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) Replacing in Eq. (3.4) te variable x wit te translation operator L gives (L I) k = k (L I) k + Q k (L ) (L I) k+1 (note tat all te operators involved commute). Since L f = f we ave Q k (L ) g M k g were M k only depends on te coefficients of te polynomial Q k (x). It follows tat k f k k f + Q k (L ) k+1 f k k f + M k k+1 f. Applying n times te above inequality we get We will now prove tat k f M k n j= jk k+1 j f + k(n+1) k n+1 f. (3.5) k ck (A + f ) α for 1 < k m, f c k (A + f ) for k = 1, (3.6) were m is te integer appearing in (3.3). Clearly (3.3) gives te above bound wen k = m. We now assume tat (3.6) olds for k + 1 and we will prove it for k (assuming k 1). For 1, using (3.5) we obtain k n f M k jk (A + f ) j α + k(n+1) k n+1 f j= M k (A + f ) n jk jα α + k(n+1) k f j= were in te last inequality we used te fact tat n = n. n times Coosing n N in suc a way tat n n+1 gives k n f M k (A + f ) α j(α k) + k f. Since 1 < α <, if k te sum is bounded independently of n so tat k f c k (A + f ) α. j= If k = 1 we obtain f M 1 (A + f ) α n j= j(α 1) + f M 1 (A + f ) α (α 1)(n+1) + f M 1 (A + f ) α c 1 α + f = c 1 (A + f ). Corollary Let f L () and assume tat for ε (, 1) and some integer m > 1 te seminorm f R m,1+ε is finite. Ten f R 1 f c R m,1+ε + f.

21 19 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 Proof. Since f R m,1+ε is finite we can write m f f R m,1+ε 1+ε and by te previous lemma wit α = 1 + ε and A = f R m,ε, we obtain f c f R m,1+ε + f and te tesis follows. Before proceeding wit te proof of Teorem 3.4 we still need te following. Proposition Let f L loc () and assume tat in te sense of distributions Lf = F wit F L loc (), tat is for every φ C (), f (x) Lφ (x) dx = F (x) φ (x) dx. If F W s, R,loc olds: (Ω) ten f W 1, R,loc (Ω) and for every ζ, ζ 1 C (Ω) wit ζ ζ 1 te following estimate ζf W 1, R (RN ) c ζ 1 F W s, R (RN ) + ζ 1f L (R N ). (3.7) Remark As te reader will notice, te proof of te above proposition requires te knowledge of Teorem 3.4 for k = 1. In turn, te above proposition is required in te proof of te inductive step of Teorem 3.4 (but not in te proof of te case k = 1), so tat our argument is not circular. Proof. Let φ C () suc tat φ, Define, for any ε >, φ (x) = for x 1 and φ (x) dx = 1. and φ ε (x) = ε Q φ (D ε 1x) f ε (x) = (φ ε f) (x) = φ ε (y) f y 1 x dy. Since by Proposition.3 and Remark.4 f ε is smoot, we can apply to f ε te estimate of Teorem 3.4 for k = 1: ζf ε W 1, R () c ζ 1 L (f ε ) W s, R () + ζ 1f ε L (). (3.8) By known properties of mollifiers we ave ζ 1 f ε L ζ 1 f L. Also, by Proposition.5, observe tat L (f ε ) = L (φ ε f) = φ ε Lf = (Lf) ε. Unfortunately te left convolution wit φ ε provides convergence in W k, k, but not in W and we are forced to te following roug R estimate, ζ 1 L (f ε ) ζ 1 Lf W s, R () c ζ 1L (f ε ) ζ 1 Lf W s, () (3.9)

22 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) were we used te fact tat since at every point te vector fields 1 R, R,..., q R and teir commutators up to te step s span R N it is possible to represent every derivative xi by means of a linear combination of rigt invariant differential operators of te kind j R 1 j R j R k wit k s. Since Lf W s,,loc () te RHS in (3.9) vanises as ε +. In particular te RHS in (3.8) is bounded, ence te sequence ζf ε is bounded in W 1, (Ω). Since W 1, (Ω) is a separable Hilbert space by Banac Alaoglu teorem tere exists R R a subsequence of ζf ε converging weakly in W 1, (Ω). Tis sows tat ζf W 1, (Ω). Moreover te weak R R convergence in W 1, R (Ω) implies ence (3.7) olds. ζf W 1, R (RN ) lim inf ζf ε W 1, R (RN ) c ζ 1 Lf W s, R (RN ) + ζ 1f L (R N ) Proof of Teorem 3.4. We will prove (i) by induction on k. We start wit k = 1. Let f W 1,,loc () suc tat Lf = F W s, (Ω). By Corollary 3.17 and Teorem 3.15 we ave R,loc s ζf R 1 ζf c R s+1,1+1/s + ζf c ζ 1 Lf R j + ζ 1f were ζ, ζ 1 C (Ω) suc tat ζ ζ 1. Also, by Proposition 3.13 and Teorem 3.11 ζf W 1, R (RN ) c ζf R 1 + ζf s c ζ 1 Lf R j + ζ 1f c ζ 1 Lf W s, R (RN ) + ζ 1f wic is te case k = 1. j= Assume now tat (3.5) olds up to te integer k and let f W 1,,loc () suc tat Lf W k+s, (Ω). By R,loc te inductive assumption we know tat f W k, R,loc (). Let R I be any rigt invariant differential operator wit I k. Ten I Rf L loc (Ω) and since R I and L commute we ave, in te sense of distributions j= L R I f = R I Lf W s, R,loc (Ω). It follows tat we can apply Proposition 3.18 to I Rf and conclude tat R I f W 1, (Ω) and satisfies R,loc te estimate ζ I R f c W 1, ζ R (RN ) 1 I R Lf + W s, ζ R (RN ) 1 I R f L. (R N ) Tis means tat f W k+1, R,loc (Ω) and, introducing oter cutoff functions suc tat ζ ζ ζ 1 ζ ζ 3 and exploiting te inductive assumption, ζ f W k+1, R (RN ) = ζ f W k, R (RN ) + I =k R I (ζ f) W 1, c ζ 1f W k, R (RN ) + I =k c ζ Lf W s+k 1, R (R N ) + c R (RN ) ζ1 R I f W 1, I =k ζ 3 Lf W s+k, R (RN ) + ζ 3f L (R N ) R (RN ) ζ I R Lf W s, R (RN ) + ζ I R f L (R N )

23 19 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 so we are done. Part (ii) of te teorem follows from te fact tat W k, (Ω ) C (Ω ). R k=1 4. Hypoellipticity of te sublaplacian In tis section we will prove te ypoellipticity result extending Teorem 3.4 from functions in W 1, generic distributions. Teorem 4.1 (Hypoellipticity of te Sublaplacian). Let Ω R N be an open set and let f D (Ω) suc tat Lf C (Ω), ten f C (Ω). Tat is L is ypoelliptic in R N. To prove tis extension we will use a regularizing operator tat commutes wit L, namely te convolution wit a parametrix (tat is an approximate fundamental solution) of a rigt invariant elliptic operator. Let us consider te second order differential operator N L R = R j built using te wole canonical base of rigt invariant vector fields. From te structure of te vector fields i R we can read tat at te origin te principal part of L R coincides wit te classical Laplacian. Using te fundamental solution of te Laplacian we will construct a parametrix for L R tat we will name γ (Proposition 4.). We will ten study te operator T f = γ f, and we will sow tat if f is a distribution, ten for K large enoug T K f W 1,,loc (Proposition 4.4 and Corollary 4.5). Since T and L commute (Proposition 4.6), ten if Lf is smoot also LT K f = T K Lf is smoot and terefore by Teorem 3.4 we see tat T K f is smoot. Now, if γ were te fundamental solution of L R ten just applying K times L R to T K f we would obtain tat f is smoot. Te fact tat γ is only an approximate fundamental solution introduces a minor difficulty tat is addressed in Lemma 4.8. Let us start wit te construction of γ. Proposition 4. (Parametrix of a Rigt Invariant Elliptic Operator). Let V be a neigborood of te origin. Tere exist γ C ( \ {}) and ω C ( \ {}), bot supported in V, satisfying and suc tat in te sense of distributions c γ (x) x N (4.1) c xi γ (x) x N 1 i = 1,,..., N (4.) c ω (x) x N (4.3) L R γ = δ + ω. Proof. Let W V be a neigborood of te origin, suc tat also W W V and let η C (W ) identically 1 in a neigborood of te origin. Let γ (x) = c N 1 x N be te fundamental solution of te standard Laplacian and set γ 1 = γη. An easy computation based on te structure of te vector fields i R sows tat we can write to

24 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) L R = + i,j b ij x i x j + i c i (4.4) x i were b ij (x) and c i (x) are polynomials and b ij () =, see [1, Corollary ]. For every test function ϕ C R N we ave L R γ 1, ϕ = γ 1 (x) L R ϕ (x) dx = γ 1 (x) ϕ (x) dx + γ 1 (x) b ij (x) ϕ (x) + ϕ (x) c i (x) dx x i,j i x j x i i = ηγ (x) ϕ (x) dx + (γ 1 b ij ) (x) (γ 1 c i ) (x) ϕ (x) dx. x i x j x i i i,j Observe tat te integration by parts is justified by te fact tat b ij vanises at te origin so tat γ 1 b ij can be differentiated twice witout losing summability. A simple computation sows tat ηγ ϕ = γ (ηϕ) div (γϕ η) + γϕ η + ϕ γ η and terefore Since γϕ η C ηγ ϕdx = (γ (ηϕ) div (γϕ η) + γϕ η + ϕ γ η) dx = ϕ () div (γϕ η) dx + ϕ (γ η + γ η) dx. () te first integral vanises in te last expression, so tat L R γ 1 = δ + ω 1 (4.5) were ω 1 = (γ η + γ η) + i,j x i x j (γ 1 b ij ) i x i (γ 1 c i ). Observe tat for small x we ave c ω 1 (x) x N 1 (4.6) and tat ω 1 is supported in W. Unfortunately ω 1 is still a bit too big. We need to add a term to γ 1 to improve te approximation. Let It is immediate to ceck tat and terefore γ = γ 1 + γ 1 ω 1. (4.7) L R (γ 1 ω 1 ) = L R γ 1 ω 1 = ( δ + ω 1 ) ω 1 = ω 1 + ω 1 ω 1 L R γ = L R γ 1 + L R (γ 1 ω 1 ) = δ + ω

25 194 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 were ω = ω 1 ω 1. We are left to ceck tat ω satisfies te properties stated in te proposition. Since ω 1 is supported in W it is clear tat ω is supported in W W V. It remains to sow tat ω C ( \ {}) and tat ω (x) c x N. Let us fix x and a neigborood B r (x ) suc tat B r (x ). Let ψ C () be a cut-off function wic is identically 1 in a neigborood of te origin and wit support disjoint from B r (x ). We can also assume tat x w 1 and w 1 x are bounded away from zero wen w sptrψ and x B r (x ). For any x B r (x ) we write ω (x) = ω 1 x w 1 ω 1 (w) ψ (w) dw + ω 1 x w 1 ω 1 (w) ψ x w 1 dw + ω 1 x w 1 ω 1 (w) 1 ψ (w) ψ x w 1 dw = I 1 (x) + I (x) + I 3 (x). Let us sow tat te tree terms are smoot. Let P R be any rigt invariant differential operator, since ω x w 1 is not singular on te support of ψ, P R I 1 (x) = R P R ω 1 x w 1 ω 1 (w) ψ (w) dw so tat I 1 (x) is smoot. Similarly, if P is any left invariant differential operator we can write PI (x) = ω 1 (w) Pω 1 w 1 x ψ (w) dw and I (x) is smoot. Finally observe tat in I 3 tere are no singular terms, we can easily differentiate under te integral sign and terefore also I 3 is smoot. As to te growt of ω, using (4.6) and (.4) we easily obtain ω (x) x w 1 1 N w 1 N dw V c x w 1 N w 1 N dw c x N. V 1 Finally, let us prove (4.1) and (4.). Recall tat γ = γ 1 + γ 1 ω 1 wit γ 1 (x) = c N η (x). Since γ x N 1 and ω 1 are compactly supported, also γ is compactly supported and Moreover, since, γ (x) c x N + c c x N + c c x N. W W 1 1 x w 1 N w 1 1 x w 1 N w xi γ 1 (x) c N 1 x N 1 N dw N dw

26 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) and we ave xi (γ 1 ω 1 ) (x) = xi γ (x) = xi γ 1 (x) + xi (γ 1 ω 1 ) (x) = xi γ1 x w 1 ω 1 (w) dw N ( xk γ 1 ) x w 1 xi x w 1 k ω1 (w) dw k=1 were xi x w 1 k is a polynomial. Terefore near te pole; ence ( xk γ 1 ) x w 1 c x w 1 N 1 c x w N 1 xi (γ 1 ω 1 ) (x) c W 1 1 c x w N 1 dw N 1 w x N. Let us now consider tree open sets in, Ω Ω Ω and let V be a neigborood of te origin suc tat V 1 Ω Ω. Define γ as in Proposition 4., wit γ supported in V. Te convolution wit γ defines a regularizing operator tat acts on functions f L 1 loc (Ω) as follows. For every x Ω we set T f (x) = γ f (x) = γ x y 1 f (y) dy. (4.8) Note tat T : L 1 (Ω ) L 1 (Ω ). Namely, for x Ω and x y 1 sprt γ, te point y = x y 1 1 x ranges in V 1 Ω Ω, ence introducing caracteristic functions, (χ Ω T f) (x) = (γχ V ) x y 1 (fχ Ω ) (y) dy, or wic by Young s inequality gives χ Ω T f = γχ V fχ Ω (4.9) T f L 1 (Ω ) γ L 1 (V ) f L 1 (Ω ). Also, T acts on distributions f D (Ω) as follows. For every ϕ C (Ω ) we set T f, ϕ = f, T ϕ (4.1) were We claim tat T ϕ (y) = γ x y 1 ϕ (x) dx. T : D (Ω) D (Ω ).

27 196 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 Observe tat te assumption on V implies tat T ϕ is a test function in Ω. Namely, for x sprt ϕ and x y 1 sprt γ te point y ranges in Ω Ω. By Remark.4, T ϕ is smoot and terefore te pairing (4.1) is well defined. Remark 4.3. Te definition of te operator T depends on te coice of te neigborood V used to define γ. In te following, in order to stress tis dependence, we will also write T V. Proposition 4.4 (Regularizing Properties of T V ). Let Ω Ω Ω. Tere exists a neigborood V of te origin suc tat te operator T V defined in (4.1) as te following properties. (1) Let f D (Ω) suc tat f = D α g, for some g L 1 loc (Ω) and multiindex α. Ten T V f D (Ω ) and T V f = β α 1 D β A β in Ω for suitable A β L 1 loc (Ω ). () Let f L p loc (Ω) for some 1 p < N, ten T V f L p (Ω ) for every 1 p > 1 p N and (3) Let f L loc (Ω) ten T V f W 1, (Ω ). (4) Let f C (Ω) ten T V f C (Ω ). T V f L p (Ω ) c f L p (Ω ). Proof. (1) Let f = yi D α g for some α wit α = α 1. Ten, for ϕ C (Ω ) T V f, ϕ = yi D α g (y), γ x y 1 ϕ (x) dx = D α g (y), yi γ x y 1 ϕ (x) dx. We can write yi γ x y 1 N = ( k γ) x y 1 yi x y 1 N = k x y 1 Z k (x, y) k=1 were by (4.) k (z) are locally integrable functions, smoot outside te pole, and Z k are polynomials. Hence N T V f, ϕ = D α g (y), k x y 1 Z k (x, y) ϕ (x) dx = D α g (y), k=1 k=1 k=1 N k (w) Z k (w y, y) ϕ (w y) dw since te function y N k=1 k (w) Z k (w y, y) ϕ (w y) dw belongs to C (Ω) N = g (y), k (w) ( 1) α D α y [Z k (w y, y) ϕ (w y)] dw. Now, k=1 ( 1) α D α y [Z k (w y, y) ϕ (w y)] = β α D β ϕ (w y) a β,k (w, y)

28 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) for suitable polynomials a β,k, ence T V f, ϕ = g (y) Next, observe tat = = g (y) β α b β (x) = N k (w) k=1 k=1 β α N k x y 1 D β ϕ (x) D β ϕ (w y) a β,k (w, y) dw dy β α D β ϕ (x) a β,k x y 1, y dx dy N g (y) k x y 1 a β,k x y 1, y dy dx. k=1 N g (y) k x y 1 a β,k x y 1, y dy k=1 belongs to L 1 loc (Ω ), since g L 1 loc (Ω), k L 1 and is compactly supported in V, a β,k are polynomials. Hence, letting A β (x) = ( 1) β b β (x) we can write T V f, ϕ = ( 1) β A β (x) D β ϕ (x) dx = β α β α 1 D β A β, ϕ wit A β L 1 loc (Ω ), so tat T V f as te desired structure. () follows from (4.9) and Young s inequality since, by (4.1), γ L r () for 1 r < N N. (3) We know tat any derivative xi γ (i = 1,,..., N) is integrable and supported in V, ence eac function i R γ (i = 1,,..., N) is a linear combination wit polynomial coefficients of integrable functions, compactly supported in V, so tat i Rγ L1 (). Also, fχ Ω L () ence by Young s inequality tat is T V f L (Ω ), and T V f = γ (fχ Ω ) L (), χ Ω R i T V f = R i γ (fχ Ω ) L (), tat is R i T V f L (Ω ). Tis olds for i = 1,,..., N (not just for te first q derivatives). Now, let us recall tat te left invariant vector fields i (i = 1,,..., N) can be written as linear combinations wit polynomial coefficients of te rigt invariant vector fields R i. Hence by te boundedness of Ω we also ave i T V f L (Ω ) for i = 1,,..., N in particular T V f W 1, (Ω ). (4) Let f C (Ω). From T V f (x) = γ x y 1 f (y) dy = γ (w) f w 1 x dw

29 198 M. Bramanti, L. Brandolini / Nonlinear Analysis 16 (15) 17 we read tat for x Ω and any left invariant differential operator P we can write PT V f (x) = γ (w) Pf w 1 x dw, sowing tat PT V f C (Ω ). Corollary 4.5. Let Ω Ω. For every distribution f D (Ω) tere exist a neigborood of te origin V and an integer K suc tat (T V ) K f W 1, (Ω ). Proof. For fixed Ω Ω and positive integer K to be cosen later, tere exists a neigborood V of te origin suc tat Let V V V Ω Ω. K times Ω j = V V V Ω j times Ω = Ω for j = 1,,..., K so tat Ω K Ω. If f D (Ω), ten te restriction of f to Ω K coincides wit a finite sum of derivatives of locally integrable (actually, continuous) functions: f = M D αj g j for suitable g j L 1 (Ω K ) and multiindices α j, see e.g [11, Corollary 3, p. 63]. Applying point (1) in Proposition 4.4 k 1 times, wit k 1 large enoug we find tat T k1 V (f) L1 (Ω K k1 ). Applying point () in Proposition 4.4 k times (wit k large enoug) provides Finally, using point (3) we obtain wic is te desired result, for K = k 1 + k +. T k1+k V (f) L (Ω K k1 1 k ). T k1+1+k+1 V (f) W 1, (Ω K k 1 1 k 1), Proposition 4.6. Let Ω Ω and V small enoug so tat V Ω Ω. Ten for any distribution f D (Ω) and every left invariant operator P we ave PT V f = T V Pf in D (Ω ). Remark 4.7. Te previous proposition can be obviously iterated writing PT K V f = T K V Pf in D (Ω ) for any fixed positive integer K, provided V is cosen small enoug to ave V V V Ω Ω. K times