ON THE REGULARITY OF SOLUTIONS TO A NONLINEAR ULTRAPARABOLIC EQUATION ARISING IN MATHEMATICAL FINANCE

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1 Differential and Integral Equations Volume 14, Number 6, June 21, Pages ON THE REGULARITY OF SOLUTIONS TO A NONLINEAR ULTRAPARABOLIC EQUATION ARISING IN MATHEMATICAL FINANCE Giovanna Citti, Andrea Pascucci, and Sergio Polidoro Dipartimento di Matematica, Università di Bologna Piazza di Porta S. Donato 5, 4127 Bologna, Italy (Submitted by: G. Da Prato) Abstract. We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance: xx u + u y u t u = f. Using a harmonic analysis technique on Lie groups, we prove that, if the solution u satisfies condition x u in an open set R 3 and f C (), then u C (). 1. Introduction In this paper we study the interior regularity properties of the solutions to the equation in the variables z =(x, y, t) R 3 Lu xx u + u y u t u = f (1.1) satisfying the condition x u(z) z. (1.2) This equation arises in mathematical finance, when studying agents decisions under risk. The problem is the representation of agents preferences over consumption processes. Epstein and Zin in [9] have proposed a utility functional which is the solution of a backward stochastic differential equation. Recently Antonelli, Barucci and Mancino [1] proposed a more sophisticated utility functional that takes into account some aspects of decision making, such as the agents habit formation, which is described as a smoothed average of past consumption and expected utility. In that model the couple Investigation supported by the University of Bologna. topics. Accepted for publication March 2. AMS Subject Classifications: 35K57, 35K65, 35K7. 71 Funds for selected research

2 72 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro of processes utility and habit is described by a system of backward-forward stochastic differential equations. The solution of such a system as a function of consumption and time satisfies the partial differential equation (1.1). Several existence and uniqueness results are known for viscosity solutions of the Cauchy problem associated with equation (1.1), under different hypotheses on the initial data ([23], [1], [1]). However no regularity results are known. Here we are concerned with the regularity of the classical solutions of (1.1) (see Section 3 for the precise definition of classical solution). To this end, condition (1.2) is of crucial importance not only because it is suggested by the model, but also because equation (1.1) could have nonregular solutions if it is suppressed. For example any solution u independent of the variable x satisfies the Burgers equation u y u t u =, which is of hyperbolic type. Our main result is the following. Theorem 1.1. Let be an open subset of R 3 and let f C (). If u is a classical solution of equation (1.1) in and satisfies condition (1.2), then u C (). The operator L in (1.1) can be seen as a degenerate parabolic operator, and it can formally be represented as a sum of squares of nonlinear vector fields. Indeed if we set X = x and Y u = u y t, (1.3) then L can be expressed as Lu = X 2 u + Y u u. (1.4) Condition (1.2) ensures that the vector fields X, Y u and their commutator [X, Y u ]= x u y are linearly independent at every point. This fact suggests a link with the theory of Hörmander s operators. These operators can be written in the form p H = Xi 2 + X (1.5) i=1 where X i, i =,...,p (p N), are linear, smooth vector fields in R N whose generated Lie algebra has maximum rank at every point. It is well known that this last condition, called the Hörmander condition, yields that H is hypoelliptic (see [14]). Under this condition there exists a fundamental solution Γ of the equation (1.5) whose properties have been investigated by [14], [2], [21], [17]. In particular in these papers a control distance d associated to the vector fields and their commutators has been introduced. Moreover

3 nonlinear ultraparabolic equation 73 estimates of Γ and its derivatives are proved in terms of d. Things are particularly easy when the Lie algebra generated by X 1,...,X p is nilpotent and stratified. In this case there exists a nonnegative integer Q, which is called the homogeneous dimension of the space, such that Γ(z,ζ) Cd(z,ζ) Q+2 (1.6) for every z,ζ. Hence a theory of the regularity similar to the classical one has been developed for this type of operator. If we denote by C k,α d the class of functions with derivatives of order k Hölder continuous with respect to the control distance d, then some a priori estimates formally analogous to the classical Schauder ones hold for the solutions of the equation Hu = f (see [13], [11], [2]). Obviously, if ū is a fixed function satisfying (1.2), then the linear operator Lūu = X 2 u + Yūu (1.7) is formally represented as in (1.5), and the associated classes of Höldercontinuous functions will be denoted by Cū k,α. Then we have the following result (see, for example, [2]): Theorem RS1. Let the coefficient ū of Lū be of class C (), and let f Cū k 2,α (), k N, <α<1. If u is a solution of Lūu = f, then u is of class Cū k,α (). This result is optimal if the coefficient ū is of class C, and it can be easily extended with the same technique to less-regular vector fields. In this case, the following holds. Theorem RS2. Assume that ū Cū k+1,α () and f Cū k 2,α (). If u is a solution of Lūu = f, then u is of class Cū k,α (). We stress that this result can not be applied in our nonlinear situation, since the vector fields in (1.3) have only the regularity of the solution; then Theorem RS2 does not provide any gain of regularity. Hence we adapt to this framework a technique introduced by one of the authors in [6] and extended in [7] for studying the regularity of the solutions of another nonlinear operator. Here we are able to prove the following. Theorem 1.2. Assume that R 3 is an open set, ū Cū k 1,α () satisfies (1.2), and f Cū k 2,α (), for any α (, 1). If u is a classical solution of Lūu = f in, then u Cū k,α (), for any α (, 1). This result easily yields Theorem 1.1. The proof of Theorem 1.2 is established in two steps: a freezing method and a regularization procedure.

4 74 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro 1.1. Overview of the freezing method. The freezing method is a wellknown technique, classically used to study the regularity of solutions to linear parabolic operators of the form N a ij xi xj t, (1.8) i,j=1 where z =(x, t) denotes the point in R N R. In this case, the associated frozen operator is simply obtained by evaluating the coefficients at a fixed point : N a ij ( ) xi xj t. i,j=1 This new operator is, up to a linear change of coordinates, the heat operator, and its fundamental solution can be considered as a parametrix of the fundamental solution of the operator in (1.8). An argument much more complicated was used to prove the existence of a fundamental solution for Hörmander-type operators (1.5). Indeed the properties of the operator rely on the vector fields X 1,...,X p and their commutators. If X i are represented by N X i = a ij xj i =1,...,p, j=1 then the constant-coefficient operators N X i,z = a ij ( ) xj j=1 i =1,...,p commute, and the generated Lie algebra is R p with p N. Hence, in general, the operator p X i=1 i,z 2 X,z is not hypoelliptic, and it has not a fundamental solution. Folland and Stein first pointed out that the model operators in this case are operators of the form (1.5) such that the Lie algebra generated by X 1 X p is nilpotent and stratified. Later on Rothschild and Stein introduced an abstract and very general version of the freezing method. The choice of the frozen vector fields X i,z was made in such a way that their generated Lie algebra is nilpotent and stratified, and, at low orders, it has the same structure as Lie(X 1 X p ). With this choice of vector fields, the operator p i=1 X2 i, X,z is hypoelliptic and nilpotent, and its fundamental solution Γ z is a parametrix for the fundamental solution of (1.5). After the existence of the fundamental solution of (1.5) was established, a wide class of Hörmander operators has been studied with the same argument as (1.8): the

5 nonlinear ultraparabolic equation 75 operators of the form p i,j=1 a ijx i X j +X, where a i,j are not even continuous but the vector fields X i are of class C and satisfy the Hörmander condition for hypoellipticity. Here, for every point, it is convenient to consider as frozen operator p i,j=1 a ij( )X i X j + X, that is an operator with C coefficients. In the same spirit as the results known for elliptic operators, by using the known properties of these operators, many sophisticated results have been obtained under very weak hypotheses on a ij (see, for example, [15], [19], [3], [2]). See also [16] where the regularity properties of this kind of operator have been investigated by a different approach. The few known results about nonlinear operators refer to operators whose nonlinearity depends on C vector fields (see, for example, [25], [4], [24], [26]). Things are different when the vector fields themselves are not smooth, since that operators can not always be considered as simple perturbations of known linear operators. A first regularity result for solutions of a linear equation with continuous vector fields is due to Franchi and Lanconelli [12]. In a more recent study one of the authors introduced a simplification of the freezing method of Rothschild and Stein, for a second-order partial differential operator, based on the notion of intrinsic Taylor expansion of the coefficients [6]. In this paper we use a technique similar to the one in [6]. We consider the linearized operator Lū in(1.7)definedinanopensubsetofr 3. Assuming (1.2), we have that X, Yū, [X, Yū] are linearly independent at every point. We observe that the simplest nilpotent Lie algebra with two generators and of dimension 3 is the Heisenberg algebra. Then, for every point, we associate with X and Yū two frozen vector fields X and Y z of class C and whose generated Lie algebra is the Heisenberg one. This choice ensures that the frozen operator L z = X 2 + Y z is a nilpotent Hörmander-type operator. In particular L z has a fundamental solution Γ z and an associated control distance d z. Unfortunately the distance d z is not equivalent to the distance dū associated with Lū, nor are equivalent two distances d z and d z1 associated with different points and z 1. Since the qualitative behaviour of the solution strictly depends on the control distance, we have to study in detail the properties of these distances. This is done in Section 2 where we also study the properties of the Hölder classes related to these control distances Overview of the regularization procedure. In order to introduce our regularity procedure we consider a solution u in of the linearized equation Lūu = f. Fixing, we represent the function u in terms of

6 76 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro the fundamental solution Γ z of the frozen operator L z : u(z) = Γ z (z,ζ)l z u(ζ)dζ = Γ z (z,ζ)f(ζ)dζ + Γ z (z,ζ)k z (ζ)dζ, (1.9) where K z is a kernel with the behaviour K z (ζ) d q (,ζ), and the exponent q depends on the regularity of ū. In their classical paper [2], Rothschild and Stein choose = z in the representation formula (1.9). Therefore the kernel which appears in the second term of (1.9) becomes Γ z (z,ζ)k z (ζ), and it is less singular then Γ z. Hence it is possible to perform higher-order derivatives with respect to z and to estimate them. On the other hand, Chiarenza, Frasca and Longo [5] noticed for the first time that, even in the parabolic case, it seems to be convenient, when dealing with nonregular coefficients, to keep z different from as long as it is possible. Using their technique in our situation, we can differentiate twice u with respect to z, and we obtain D 2 u(z) = D 2 Γ z (z,ζ)f(ζ)dζ + D 2 Γ z (z,ζ)k z (ζ)dζ. Then, we evaluate the second order derivative of u at : D 2 u( )= D 2 Γ z (,ζ)f(ζ)dζ + D 2 Γ z (,ζ)k z (ζ)dζ. In this way we compute the second derivative of u without differentiating the coefficient of Γ z. The same idea has been used in [7] also for higherorder derivatives, and here we further extend it. Obviously, we can not repeat the preceding arguments for the third derivatives since, for z, the kernel D 3 Γ z (z,ζ)k z (ζ) is not locally integrable. Nevertheless, a rather delicate argument, based on the use of some high-order difference quotients (see Section 3), yields D 3 u( )= DΓ z (,ζ)d 2 f(ζ)dζ + D 3 Γ z (,ζ)k z (ζ)dζ. In this way, we obtain some regularity results for the solutions even though the coefficients of the vector fields and of the fundamental solution of the frozen operator are not regular. 2. Freezing method In this section we describe the freezing method for the linear equation L = X 2 + Yū,

7 nonlinear ultraparabolic equation 77 where X = x,yū = ū y t and ū is a given function satisfying (1.2). In Subsection 2.1, we study the relation between the distances dū, associated with the vector fields X and Yū, and the distance d z associated with the vector fields X and Y z. In Subsection 2.2, we define classes Cū k,α of Hölder-continuous functions with respect to the distance dū, andweprove the existence of a polynomial expansion of Taylor type for functions of class Cū k,α. Finally, in Subsection 2.3, we study the properties of the fundamental solution of the frozen operator Heisenberg group and distances. Here we recall some properties of the Heisenberg algebra and we establish some relations between the control distances corresponding to the linear and to the frozen operators. The Heisenberg algebra is a Lie algebra with two generators, and nilpotent of step two. The simplest representation of the Heisenberg vector fields is X H = θ1 θ 2 2 θ 3 and Y H = θ2 + θ 1 2 θ 3. Clearly [X H,Y H ]= θ3 and all the other commutators vanish. The associated Lie group H 1 is then R 3, endowed with the following composition law: θ θ = ( θ 1 + θ 1,θ 2 +θ 2,θ 3 +θ (θ 1θ 2 θ 2 θ 1) ). A natural dilations group on H 1 is defined by δ H λ (θ) =(λθ 1,λ 2 θ 2,λ 3 θ 3 ), λ >. Since the Jacobian Jδλ H = λ6, the homogeneous dimension of H 1 with respect to (δλ H) λ> is the exponent Q = 6. A norm homogeneous with respect to this dilations group is given by θ H = θ 1 + θ θ The associated distance is obviously defined by d H (θ,θ)= θ 1 θ H.Clearly X H and Y H are respectively δλ H -homogeneous of degree one and two; that is, X H (u δλ H )=λ(x Hu) δλ H, Y H(u δλ H )=λ2 (Y H u) δλ H. Thus, the second-order differential operator L H = XH 2 + Y H has a fundamental solution Γ H which is invariant with respect to the left -translations and δλ H -homogeneous of degree Q +2. We next introduce the canonical coordinates corresponding to the linear operator Lū. IfDis a Lipschitz-continuous vector field, and [, 1] is contained in the domain of the local solution to the Cauchy problem γ (s) =D(γ(s)), γ() = z,

8 78 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro we let exp(d)(z) =γ(1) and call an exponential map the application D exp(d)(z). Since γ(s) =exp(sd)(z), then exp(d)(z) isdefinedfordsufficiently small. If D 1,D 2,D 3 are Lipschitz-continuous vector fields, linearly independent at every point, then the map F z : θ exp(θ D)(z) =exp(θ 1 D 1 +θ 2 D 2 +θ 3 D 3 )(z) is a diffeomorphism of a neighborhood of the origin of R 3 to a neighborhood U z of z. Its inverse function θ D,z = Fz 1 defines the canonical change of variable associated with the vector field D, and center z. When D 1,D 2,D 3 C 1, by the properties of the solutions of the Cauchy problem, the Jacobian matrix JF z of the function F z depends continuously on z. Then, by the local invertibility theorem, the open set U z continuously depends on z. By our assumptions, X, Yū and [X, Yū] =ū x y are linearly independent at every point, but ū x is not Lipschitz continuous, so we cannot define exp (θ (X, Yū, [X, Yū])). We instead consider ū =(X, Yū, y ) and denote by θū,z the associated canonical change of coordinates, defined on U z. This function allows us to introduce a topological structure in a neighborhood of, naturally associated with the vector fields X and Yū. Indeed, by the continuity of U z, there exists r = r( ) > such that the Euclidean ball B(,r) satisfies B(,r) U z, z B(,r). (2.1) Thus, if z,ζ B(,r), then ζ U z, θū,z (ζ) is defined, and we can set dū(z,ζ) = θū,z (ζ) H. (2.2) More explicitly, we have ( 1 ) θū,z (ζ) = ξ x, (τ t),η y+(τ t) ū(γ(s)) ds, (2.3) and 1 dū(z,ζ) = ξ x + τ t 1 2 η y+(τ t) + ū(γ(s)) ds where γ(s) =exp(sθū,z (ζ) ū)(z) with s [, 1]. Remark 2.1. Let andz U z. An integral curve of ū, connecting z and, is γ(s) =exp(sθū,z (z) ū)( ) with s [, 1]. Then, for every s [, 1],wehave dū(γ(s), )= sθū,z (z) H θū,z (z) H = dū(,z). 1 3,

9 nonlinear ultraparabolic equation 79 Since θū,z (ζ) is only a local diffeomorphism, it does not introduce a group structure on R 3. In order to overcome this problem, we define a new vector field in the following way: for every fixed, we define the frozen operator of Yū as follows: Y z = Yū(z )+ū x( )(x x ), where we have denoted, as usual, u x ( )= x u( ).Obviously X, Y z and [X, Y z ]=ū x ( ) y are of class C and all the commutators of higher order are null. Now we set z =(X, Y z, [X, Y z ]) and L z = X 2 + Y z. The map θ exp(θ z )(z) the canonical coordinates and of center z. As a consequence of the Campbell- Hausdorff formula we have is a global diffeomorphism. We denote by θ () z associated with z X ( u θ () ) =(XH u) θ (), Y z ( u θ ( ) ) =(YH u) θ (), ( ū x ( ) y u θ ( ) ) z =( θ3 u) θ (). (2.4) ( (z Then, as a direct consequence, we have L z u θ )) =(LH u) θ ().The diffeomorphism θ () naturally induces a Lie group structure with dilations on R 3. Indeed, we define the composition law z ζ =(θ () ) 1 (θ () (z) θ () (ζ)), (2.5) the dilations δ () λ (z) =(θ () ) 1 (δλ H (θ() (z))), λ >, and the function d z defined by d z (z,ζ) = ( θ () (z) ) 1 θ ( ) (ζ) H, which is a quasi-distance, in the sense that there exists a positive constant C = C( ) such that d z (,ζ) C(d z (,η)+d z (η, ζ)), z,η,ζ R 3. (2.6) Then G z =(R 3, ) is the Lie group associated with the Lie algebra L z = Lie(X, Y z ), generated by X and Y z, and it is isomorphic to H 1. The quasidistance we have introduced can be represented as d z (z,ζ) = θ () z (ζ) H, z,ζ R 3, (2.7) and, more explicitly, θ () z (ζ) = (2.8) ( 1 ( (ū(z ξ x, (τ t), η y +(τ t) )+ū x ( ) ξ+x 2x ))), ū x ( ) 2 d z (z,ζ) = (2.9)

10 71 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro ξ x + τ t ( (ū(z η y +(τ t) )+ū x ( ) ξ+x 2x )) 1 3. ū x ( ) 2 We next describe some relations between d z and dū. Remark 2.2. In the sequel, we shall also use the distance defined by d z (z,ζ) = θ () z (ζ) H, θ( ) z (ζ)=(ξ x, (τ t),η y+(τ t)ū( )). (2.1) This distance is equivalent to d z, in the sense that there exists a positive constant C, which depends only on ū x ( ), such that 1 C d z (,ζ) d z (,ζ) C d z (,ζ), ζ R 3. Here and in the sequel, C will denote a constant which will not always be the same. The proof of the above statement relies only on the following elementary inequality, (ab) a b 1 2, a, b >, (2.11) and on the explicit expression of d z provided in (2.9). Lemma 2.3. Let U z be a neighborhood of such that (θ 1,θ 2,θ 3 ) θū,z (ζ) is defined for every ζ U z.if( θ 1, θ 2, θ 3 ) θ () (ζ)is defined as in (2.1), then we have θ 3 θ 3 C τ t dū(,ζ), ζ U z. (2.12) Proof. Since ū is a locally Lipschitz-continuous function (in the Euclidean sense), we get ū(γ(s)) ū( ) C 1 γ(s) C 2 dū(,γ(s)), where γ(s) = exp(sθū,z (ζ) ū)( ), for s [, 1], and C 1, C 2 depend only on U z. Thus the assertion follows from expressions (2.3) and (2.1): θ 3 θ 3 τ t 1 (by Remark 2.1) C 2 τ t dū(,ζ). ū(γ(s)) ū( ) ds C 2 τ t 1 dū(,γ(s)) ds Proposition 2.4. For every z 1, there exists a compact neighborhood K of z 1, and a positive constant C = C(K), such that i) C 1 d z (,ζ) dū(,ζ) Cd z (,ζ), ii) d z (,z) C(d z (,ζ)+d ζ (ζ,z)), iii) dū(,z) C(dū(,ζ)+dū(ζ,z)), for every z,,ζ K.

11 nonlinear ultraparabolic equation 711 Proof. We first remark that there exists r = r(z 1 ) > such that K B(z 1,r) U z U z U ζ. (i) If θū,z (ζ) and θ () (ζ) are the functions defined in (2.3) and (2.1), they have the first two components in common. Then, using Lemma 2.3, we get d z (,ζ)= θ () (ζ) H θū,z (ζ) H + θ 3 θ dū(,ζ)+c τ t 2 3 dū(,ζ) 1 3 C1 dū(,ζ). On the other hand, again from Lemma 2.3 and (2.11), we get θ 3 θ 3 C τ t dū(,ζ) C ( 2( τ t ) δ 3 (δd ū(,ζ)) 3) for any positive δ. Therefore dū(,ζ) d z (,ζ)+ θ 3 θ ( (2C δ 1 ) ) dz (,ζ)+δ ( C 3 )1 3 dū(,ζ). By choosing a suitably small δ> and by Remark 2.2, we get the thesis. (ii) We observe that y y +ū( )(t t ) 1 3 y η+ū(ζ)(t τ) η y +ū( )(τ t ) (t τ)(ū(z ) ū(ζ)) 1 3 d ζ (ζ,z)+ d z (,ζ)+( t τ dū(,ζ)) 1 3, since ū is Lipschitz continuous. The last term can be estimated by (2.11) and (i), as follows: (t τ)dū(,ζ) t τ 1 2 +C 1 dū(,ζ) 2 3 d ζ (ζ,z)+c d z (,ζ). By the definition of d z (,z), this last inequality and Remark 2.2 yield the assertion. (iii) It is a direct consequence of (i) and (ii). Remark 2.5. Since we are proving a local result, from now on we shall always work in a compact set K, satisfying the assumptions of Proposition 2.4.

12 712 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro 2.2. Hölder-continuous functions and Taylor polynomials. Definition 2.6. Let, <α<1 and let D be a locally Lipschitz continuous vector field on. We say that u CD α () if there exists a positive constant C such that u(exp(hd)( )) u( ) C h α, (2.13) for every suitably small h. We say that u CD α (), if (2.13) holds uniformly on compact subsets of. Definition 2.7. Let anddbe a Lipschitz-continuous vector field in. We say that there exists the Lie derivative of u with respect to D in, if the following limit exists: u(exp(hd)( )) u( ) Du( ) lim. h h We denote by Cū() (or Cū()) the set of continuous functions in. If u Cū(), and there exists Xu Cū(), we say that u Cū(). 1 If k 2, Xu Cū k 1 () and Yūu Cū k 2 (), then we say that u Cū(). k Remark 2.8. We remark that if u C 1 () and D can be expressed as D = d 1 x + d 2 y + d 3 t, with d 1,d 2,d 3 Lipschitz-continuous functions, then there exists the Lie derivative of u and it can be expressed as Du( )=d 1 x u( )+d 2 y u( )+d 3 t u( ),. We next define the spaces of Hölder-continuous functions related to the linear operator Lū. Definition 2.9. Let ū be a C 1 function satisfying (1.2), and let <α<1. We say that u Cu ᾱ () if u CX α () and u C α Yū(). 2 We say that u Cū 1,α () if Xu Cu() ᾱ and u C 1+α 2 (). Yū We say that u Cū 2,α () if Xu Cū 1,α () and Yūu Cu ᾱ (). Let k 3 and suppose that ū Cū k 2,α (). We say that u Cū k,α () if Xu Cū k 1,α () and Yūu Cū k 2,α (). For the sake of simplicity, in the sequel, when we write u Cū k,α (), for k 3, we will assume implicitly that ū Cū k 2,α (). Remark 2.1. For every fixed z, we agree to work only in a compact neighborhood K of z 1 satisfying the assumptions of Proposition 2.4. Thus dū(,z) is defined for every,z K. We will see that, as a simple consequence of Theorem 2.16, that the class Cu ᾱ () is defined in such a way that,

13 nonlinear ultraparabolic equation 713 for every u C ᾱ u (), u(z) u( ) Cdū(,z) α, z, K. Remark In the sequel we will use the following simple result: if <α<1andk 1, then Cū k,α () Cū k 1,β (), β (, 1). We next prove some regularity results in the direction y for a function belonging to the spaces Cū k,α (). Proposition Let ū be a C 1 -function satisfying (1.2) and let <α<1. i) If u Cu ᾱ (), then u C α 3 y (); ii) If u Cū 1,α (), then u C 1+α 3 y (); iii) If u Cū 2,α (), then u C 2+α 3 y (); iv) If u Cū k,α (), withk 3, then there exists y u and it belongs to (). C k 3,α ū Remark We explicitly note that Cū 3k,α () C k, α 3 () C k, α 3 ū (), for every <α<1andk 1, where C k,β () is the space of functions with derivatives up to order k that are β-hölder-continuous functions in the Euclidean sense. Thus, if u Cū k,α () for every k N, then u C (). Proof of Proposition We start with a preliminary remark. If ū C (), the proof is a consequence of the Campbell-Hausdorff formula. The assertion could be deduced from the general theory also if ū Cū 4,α (). Since ū is only of class C 1, we proceed by direct computation. Let be a point in and, for a suitably small δ>, let z 1 =exp(δx)( ),z 2 =exp ( δ 2 Yū) (z1 ), z 3 =exp( δx)(z 2 ),z 4 =exp ( δ 2 Yū) (z3 ).We claim that z 4 =exp ( (δ 3 +o(δ 3 ) ))ū x y (z ), as δ. (2.14) We denote by (x j,y j,t j )=z j,forj=,1,...,4. Let γ j :[,1] bethe integral path connecting the point z j 1 to z j,j=1,2,3,4. It is easy to see that γ 1 (s) =(x +δs, y,t ), then z 1 =(x +δ, y,t ), s ) γ 2 (s)= (x +δ, y + δ 2 ū(γ 2 (τ)) dτ, t δ 2 s, 1 γ 3 (s) = (x +δ(1 s),y +δ 2 ū(γ 2 (τ)) dτ, t δ 2), (2.15)

14 714 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro γ 4 (s) = (x,y +δ 2( 1 ū(γ 2 (τ)) dτ s ) ) ū(γ 4 (τ)) dτ,t +δ 2 (s 1) ; then it is clear that x 4 = x and t 4 = t. In order to estimate y 4 y,we observe that ū(γ 2 (τ)) ū(γ 4 (τ)) = ū( z(τ)) (γ 2 (τ) γ 4 (τ)) (2.16) ( 1 τ ) = δū x ( z(τ)) + δ 2 ū y ( z(τ)) ū(γ 2 (σ)) dσ + ū(γ 4 (σ)) dσ +(1 2τ)δ 2 ū t ( z(τ)) = δū x ( )(1+o(1)), as δ, since ū x is a continuous function. As a consequence of (2.15), we get 1 y 4 y = δ 2 (ū(γ 2 (τ)) ū(γ 4 (τ))) dτ = δ 3 ū x ( )(1 + o(1)); (2.17) as δ, then (2.14) is proved. Moreover, since y 4 depends continuously on δ and u x ( ), the function δ y 4 is surjective in a neighborhood of y. Hence, for every β sufficiently small, there exists a δ = δ(β) such that the point (x,y +β,t ) can be written as z 4 in (2.15) (with δ = δ(β)). We stress that (2.17) also yields δ(β) 3 1 as β. (2.18) β ū x ( ) After these preliminary considerations, we conclude the proof as follows. (i) Let β be chosen as above. Since u Cu ᾱ (), we have then, since ū x ( ), u( ) u(z 1 ) Cδ α, u(z 1 ) u(z 2 ) Cδ α, u(z 2 ) u(z 3 ) Cδ α, u(z 3 ) u(z 4 ) Cδ α ; τ (2.19) u( ) u(z 4 ) 4Cδ α =4C ( β ) α 3 as β, ū x ( )(1 + o(1)) and this proves (i). (ii) We consider the functions γ j :[,1] R, defined in (2.15) for j = 1,...,4 and we apply the Taylor expansion of first order to u γ j. Since, by hypothesis, u is of class C 1,a as a function of the first variable x, wehave u( ) u(z 1 )=δu x (z 1 )+O(δ α+1 ), (2.2) u(z 2 ) u(z 3 )= δu x (z 2 )+O(δ α+1 ).

15 nonlinear ultraparabolic equation 715 Since u C 1+α 2 (), Yū u(z 1 ) u(z 2 ) Cδ α+1, u(z 3 ) u(z 4 ) Cδ α+1. Then, by using (2.2), (2.18) and the fact that u x Cu ᾱ (), we conclude that u( ) u(z 4 )=O(δ α+1 )=O(β α+1 3 )asβ. (iii) In this case we use Taylor polynomials of order 2. As δ tends to zero, we have u( ) u(z 1 )=δu x (z 1 )+ δ2 2 u xx(z 1 )+O(δ 2+α ), u(z 2 ) u(z 3 )= δu x (z 2 ) δ2 2 u xx(z 2 )+O ( δ 2+α), u(z 1 ) u(z 2 )=δ 2 Yūu(z 2 )+O(δ α+2 ), u(z 3 ) u(z 4 )= δ 2 Yūu(z 3 )+O(δ α+2 ). Hence we obtain u( ) u(z 4 )=δ(u x (z 1 ) u x (z 2 )) + δ2 2 (u xx(z 1 ) u xx (z 2 )) +δ 2 (Yūu(z 2 ) Yūu(z 3 )) + O ( δ 2+α) (since u x Cū 1,α () and u xx,yū Cu()) ᾱ = O ( δ 2+α), as δ, and this yields (iii). (iv) We first consider the problem for k = 3. By using a Taylor polynomial of higher order, we get, as δ tends to zero, u( ) u(z 1 )=δu x (z 1 )+ δ2 2 u xx(z 1 )+ δ3 6 u xxx(z 1 )+O ( δ 3+α), u(z 1 ) u(z 2 )= δ 2 Yūu(z 2 )+O ( δ 3+α), u(z 2 ) u(z 3 )= δu x (z 2 ) δ2 2 u xx(z 2 ) δ3 6 u xxx(z 2 )+O ( δ 3+α), (2.21) u(z 3 ) u(z 4 )=δ 2 Yūu(z 3 )+O ( δ 3+α). Then u( ) u(z 4 )=δ(u x (z 1 ) u x (z 2 )) + δ2 2 (u xx(z 1 ) u xx (z 2 )) (2.22) + δ 2 (Yūu(z 2 ) Yūu(z 3 )) + δ3 6 (u xxx(z 1 ) u xxx (z 2 )) + O ( δ 3+α). Since u x Cū 2,α (), Yūu x Cu ᾱ () and dū(,z 2 ) δ,wehave u x (z 1 ) u x (z 2 )= δ 2 Yūu x (z 2 )+O ( δ 2+α) = δ 2 Yūu x ( )+O ( δ 2+α).

16 716 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro Since u xx C 1+α 2 ū () and u xxx Cu ᾱ (), we have u xx (z 1 ) u xx (z 2 )=O ( δ 1+α), u xxx (z 1 ) u xxx (z 2 )=O(δ α ). Moreover, by the fact that Yūu C 1,α ū (), x Yūu C ᾱ u () and dū(,z 2 ) δ,weget Yūu(z 2 ) Yūu(z 3 )=δ x Yūu(z 2 )+O ( δ 1+α) = δ x Yūu( )+O ( δ 1+α). Inserting in (2.22), we finally obtain u( ) u(z 4 )=δ 3 ( x Yūu( ) Yūu x ( )) + O(δ 3+α ) or,inotherwords, u(x,y +β,t ) u(x,y,t ) = δ(β)3 [ ] X, Yū u(z )+O(δ(β) α ) β β 1 [ ] X, Yū u(z ), as β. ū x ( ) (2.23) This proves the existence of y u. The regularity follows from the fact that u y (z) = 1 ū x ( ) ( xyūu(z) Yūu x (z)), z. The proof in the case k>3is immediate: the existence of y u has been proved, while its regularity directly follows from the above identity. We introduce the Taylor polynomials related to the spaces Cū k,α () above considered. Definition Let =(x,y,t ),k Nand let ū be a C 1 function such that (1.2) holds. We denote by Pz k any function of the form Pz k (x, y, t) = c i,j,m (x x ) i (t t ) j (y y +(t t )ū( )) m (2.24) i+2j+3m k where i, j, m N {} and c i,j,m are real constants. We say that Pz k is a polynomial of initial point and δ () λ -degree k. Remark The functions x x,t t and y y +(t t )ū( )are δ () λ -homogeneous of degree 1, 2 and 3, respectively, since they are θ 1, θ 2 and ū x ( )(θ 3 + θ 1 θ 2 ), where (θ 1,θ 2,θ 3 ) θ () (z)definedin(2.8). We next state the main result of this subsection.

17 nonlinear ultraparabolic equation 717 Theorem Let, <α<1,k N {} and assume that u Cū k,α (). Then there exists a polynomial function Pz k u,ofδ () λ -degree k, such that u(z) =P k u(z)+o(dū(,z) k+α ) as z. (2.25) Proof. We prove our result by a classical argument that relies on Proposition We observe that the regularity assumption is given in terms of the geometry corresponding to θū,z while we obtain polynomial functions that are homogeneous with respect to δ () λ. The assertion is obvious if k =. We first prove (2.25) for k =1,2. We proceed essentially as in the proof of Proposition We consider a point =(x,y,t ) and a compact neighborhood K of such that every z =(x, y, t) K can be connected to as follows. Let γ 1 (s) =exp(s(x x )X)( ) and z 1 = γ 1 (1)=(x, y,t ) γ 2 (s)=exp( s(t t )Yū)(z 1 ) and z 2 = γ 2 (1)=(x, y 2,t) γ 3 (s)=exp(s(y y 2 ) y )(z 2 ). Clearly x x dū(,z), t t dū(,z) 2. (2.26) Let us now estimate y y 2.Wehave y y 2 =dū(z,z 2 ) 3 C 2 (dū(z, )+dū(,z 1 )+dū(z 1,z 2 )) 3 C 2 ( dū(z, )+ x x + t t 1 2) 3 Cdū(,z) 3. (2.27) Let us give a detailed proof of (2.25) for k = 1. The case k = 2 can be treated analogously. u(z) u( )=u(z 1 ) u( )+u(z 2 ) u(z 1 )+u(z) u(z 2 )= (since u C 1,α () as a function of its first variable, u C 1+α 2 () and Yū u C 2+α 3 y ()) = u( )+(x x )u x ( )+O(dū(,z) 1+α )+O( t t 1+α 2 )+O( y y2 2+α 3 )= (by (2.26) and (2.27)) = u( )+(x x )u x ( )+O(dū(,z) 1+α ), as z. For k 3 we simply argue by induction. We recall that, by our convention, we assume ū Cū k 2,α (). If γ is the Euclidean segment connecting z and

18 718 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro,wehave u(z) u( )=(x x ) 1 +(y y +ū( )(t t )) +(t t ) 1 x u( γ(s)) ds (t t ) 1 1 y u( γ(s)) ds (ū( γ(s)) ū( )) y u( γ(s)) ds. Yūu( γ(s)) ds (2.28) By the inductive hypothesis, u x has a Taylor expansion of the form (2.24) (2.25) of δ () λ -degree k 1. Thus we have 1 1 (x x ) x u( γ(s)) ds =(x x ) c i,j,m (x x ) i (t t ) j i+2j+3m k 1 1 (y y +(t t )ū( )) m s i+j+m ds +(x x ) = + O i+2j+3m k 1 O(dū(, γ(s)) k 1+α )ds c i,j,m i + j + m +1 (x x ) i+1 (t t ) j (y y +(t t )ū( )) m ( dū (,z) k+α), as z. In the same way we can handle the second and the third term in the righthand side of (2.28), since, by our assumption, Yūu Cū k 2,α () and y u Cū k 3,α (). The last term can be estimated as follows. Let c i,j,m (respectively ĉ i,j,m ) denote the coefficients of Pz k 2 ū (respectively Pz k 3 u y ). Then we have 1 (t t ) (ū( γ(s)) ū( )) y u( γ(s)) ds 1 ( ) =(t t ) Pz k 2 ū( γ(s)) ū( )+O(dū(, γ(s)) k 2+α ) ( ) Pz k 3 u y ( γ(s)) + O(dū(, γ(s)) k 3+α ) ds (since Pz k 2 ū ( γ(s)) ū( )=O(dū(,z)), as z ) 1 = c i,j,m ĉ i1,j 1,m 1 (x x ) i+i 1 (t t ) j+j 1+1 <i+2j+3m k 2, i 1 +j 1 +m 1 k 3 (y y +(t t )ū( )) m+m 1 s i+i 1+j+j 1 +m+m 1 ds + O(dū(,z) k+α ).

19 nonlinear ultraparabolic equation Parametrix. In this subsection, we provide some results about the fundamental solution of the frozen operator L z. As we previously noticed, the second-order differential operator L H = XH 2 + Y H has a fundamental solution Γ H, which is invariant with respect to the left -translations and δλ H -homogeneous of degree Q+2. Hence a fundamental solution of L z is given by Γ z (z,ζ) = 1 (( ) ) ū x ( ) Γ H θ () (ζ) θ () (z) (2.29) anditisδ () λ -homogeneous of degree Q + 2. We remark that in [22], Kolmogorov wrote explicitly the fundamental solution of the operator xx + x y t, which is, up to a canonical change of coordinates, the fundamental solution of L H or L z. However here we don t make use of that explicit formula, but we use only its local behavior. For the sake of convenience, here and in the sequel we systematically use the following notation: D 1 = X, D () 1 = X, D1 H = X H, (2.3) D 2 = Yū, D () 2 = Y z, D2 H = Y H. Besides we denote D 3 = y,d () 3 =ū x ( ) y,d3 H = θ 3.We also denote the identity by D = D () = D H. For every multi-index σ =(σ 1,...,σ m ), with σ r {,1,2,3},1 r m N,weset D σ =D σ1 D σm, D () σ = D () σ 1 D () σ m, Dσ H = Dσ H 1 Dσ H m. (2.31) We call height of σ the natural number m σ = σ r. (2.32) r=1 We remark that D () σ (resp. Dσ H )isaδ () λ (respectively δλ H)-homogeneous operator of degree σ. Since Γ z depends on many variables, the notation D(z 1 )Γ z (,ζ) shall denote the D-derivative of Γ z (z,ζ) with respect to the variable z, evaluated at the point z 1. If ϕ C, a simple relation holds between the derivatives D σϕ and D () σ ϕ.

20 72 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro Lemma If ū Cū k,α (), and σ {,1,2} k+1 with σ k+1, then for every function ϕ C () the derivative D σϕ can be represented as D σ ϕ(z) = (ū Pz 1 C ū) kϱ (z) ϱ D µ ū(z)d (ū ϱ I x ( )) hϱ ϱ (z) ϕ(z), σ µ J ϱ where I σ and J ϱ are suitable subsets of {, 1, 2, 3} k+1, C ϱ are nonnegative constants, and h ϱ and k ϱ are nonnegative integers such that ϱ σ +k ϱ, k ϱ h ϱ, µ k ϱ 3 σ, µ I ϱ, ϱ I σ. If J ϱ is empty, we set D σ ū =1. µ J ϱ Proof. Since the function ϕ is of class C (), by Remark 2.8 its Lie derivatives can be represented in terms of the standard partial derivatives, and it is not necessary to use the exponential function. If σ 2, the assertion follows directly from the definition. Indeed, if σ {(1), (1, 1)}, then D σ = D () σ, while, if σ = (2), then D σ = D () σ + (ū P1 ū) D () 3. ū x ( ) The general assertion follows by induction on σ. Analogously it is not difficult to prove the following. Lemma If ū C 1,α ū (), σ {,1,2} k+1 with σ k+1, and, z 1, then for every function ϕ C () D (z 1) σ ϕ(z) = D () σ ϕ(z)+ C ϱ (ū Pz 1 1 ū) kϱ ( )(x x ) hϱ (ū x ( ) ū x (z 1 )) jϱ D () ϱ ϕ(z), ϱ J σ where J σ is a suitable family of subsets of {, 1, 2, 3} k+1, C ϱ are nonnegative constants, and j ϱ and k ϱ are nonnegative integers such that ϱ σ +k ϱ +h ϱ h ϱ j ϱ, ϱ I σ. Proof. If σ {(1), (1, 1)} the assertion is obvious. If σ = (2), then D (z 1) σ ϕ(z) D () σ ϕ(z) = ( Pz 1 1 ū(z) Pz 1 ū(z) ) y ϕ(z) = ( ū( ) Pz 1 1 ū( ) (x x )(ū x (z 1 ) ū x ( )) ) y ϕ(z).

21 nonlinear ultraparabolic equation 721 Now, let us suppose that the assertion is true for every multi-index of height less than or equal to k 1. We choose σ =(σ 1,σ ), of height σ = k. We assume for simplicity that σ 1 = 1, since the proof is similar if σ 1 =2. Then D (z 1) σ ϕ= ( D () 1 =D () σ + ϱ J σ D () σ ϕ + ϕ+ ϱ J σ ϱ J σ ) C ϱ (ū Pz 1 1 ū) kϱ ( )(x x ) hϱ (ū x ( ) ū x (z 1 )) jϱ D () ϱ ϕ h ϱ C ϱ (ū Pz 1 1 ū) kϱ ( )(x x ) hϱ 1 (ū x ( ) ū x (z 1 )) jϱ D () ϱ ϕ C ϱ (ū P 1 z 1 ū) kϱ ( )(x x ) hϱ (ū x ( ) ū x (z 1 )) jϱ D () 1 D () ϱ ϕ. Remark It is well known that for every compact set K and for every multi-index σ, there exists a positive constant C such that D () σ (z)γ z (,ζ) Cd z (z,ζ) Q+2 σ, z,,ζ K, z ζ. By Lemma 2.17, we also have D σ (z)γ z (,ζ) Cd z (z,ζ) Q+2 σ. Fixing an open subset of R 3, a positive constant M and two points,z, we set M = {ζ :d z (,ζ) Md z (,z)}. (2.33) This set is defined in such a way that the function z Γ z (z,ζ) is smooth, if ζ belongs to M. Indeed, if M is sufficiently large, then, by (2.6), d z (z,ζ) 1 C d z (,ζ) d z (,z) ( 1 C 1 ) dz (,ζ), (2.34) M for every ζ M. Proposition 2.2. Let k N. Let ū Cū k 1,α () and let K be a compact subset of. There is a positive constant C, such that, for every multi-index σ, σ = k, we have (D σ (z) D σ ( )) Γ z (,ζ) ( C d z (,z)d z (,ζ) Q+1 k + d z (,z) α d z (,ζ) Q+2 k), for every z, K, ζ M defined in (2.33) for suitable M>. (2.35)

22 722 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro Proof. We apply Lemma 2.17, and denote Ĩσ = {ϱ : k ϱ =}. Then, we have (D σ (z) D σ ( ))Γ z (,ζ) = (ū Pz 1 C ū) kϱ (z) ϱ D µ ū(z)d (ū x ( )) hϱ ϱ (z) (z)γ z (,ζ) ϱ I µ J σ\ĩσ ϱ + ( ) C ϱ D µ ū(z) D µ ū( ) D () ϱ ( )Γ z (,ζ) ϱ Ĩσ µ J ϱ µ J ϱ + ( ) C ϱ D µ ū(z) D () ϱ (z) D () ϱ ( ) Γ z (,ζ). ϱ Ĩσ µ J ϱ Now we estimate each term separately. By simplicity let us call them S 1, S 2, S 3 respectively. We first note that if with M> C,ζ M and z is such that d z (, z) d z (,z), then we have d z ( z,ζ) 1 C d z (,ζ) d z (, z) 1 Cd z (,ζ) d z (,z) (by definition of M ) ( 1 C 1 M )d (,ζ). (2.36) We first consider S 1 S 1 C d z (,z) kϱ(1+α) d z (z,ζ) Q+2 ϱ ϱ σ +k ϱ (using (2.33), (2.34), the fact that ϱ σ +k ϱ and that K is bounded) Cd z (,z) α d z (,ζ) Q+2 σ. Analogously we can estimate S 2. Indeed, since D µ ū C α for every µ such that µ kand ϱ σ,weget S 2 Cd z (,z) α d z (,ζ) Q+2 σ. Finally S 3 ( ) C ϱ D µ ū(z) D () ϱ (z) D () ϱ ( ) Γ z (,ζ)= ϱ σ µ I ϱ (by the mean value theorem, for some z such that d z (, z) d z (,z)) = C ϱ θ () (z), ( z D () ϱ )( z)γ z (,ζ) ϱ σ

23 nonlinear ultraparabolic equation 723 C ϱ ϱ σ r=1 3 d z (,z) r d z ( z,ζ) Q r+2 ϱ (by (2.36), and the definition of M ) Cd z (,z)d z (,ζ) Q+1 σ. The main results of this section are contained in the following statement. Proposition Let k N, ū Cū k 1,α () and K be a compact subset of. There exist two positive constants C and M, such that D σ (z)γ z (,ζ) D σ ( )Γ z (,ζ) ( C d z (,z)d z (,ζ) Q+1 σ + d z (,z) α d z (,ζ) Q+2 σ ) (2.37), for every multi-index σ, σ = k, and for every z, K, ζ M defined in (2.33). The proof of the above statement relies on the following. Lemma Let be a bounded open set, ū Cū 1,α (), Ka compact subset of and M>. Then there exists a positive constant M such that (θ () (ζ) ( θ z (z) (ζ))) 3 M (d z (,ζ) 3 d z (,z) α +d z (,ζ) 2 d z (,z)), (2.38) for every z, K and for every ζ M (the notation ( ) 3 in the left-hand side denotes the third component of the considered vector). Moreover, if M is sufficiently big, there exist two constants M 1,M 2 >1 such that, for every z, K and for every ζ M, we have M 1 θ () (ζ) ( θ z (z) (ζ)) H θ () (ζ) H (2.39) and, for every θ R 3 such that d H (θ () (ζ),θ) d H (θ z (z) (ζ),θ () (ζ)), we have d z (,ζ) M 2 θ H. (2.4) Proof. A straightforward computation gives θ () (ζ) ( θ z (z) (ζ)) = ( x x,t t, 1 2 (x x )(2τ t t ) + 1 ū x ( ) (η y +ū( )(τ t )) 1 ū x (z) (η y +ū(z)(τ t))). If we denote by θ 3 the last component of θ () θ x x ( τ t + τ t ) (ζ) ( θ z (z) (ζ)), we have

24 724 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro + ( 1 ū x (z) 1 ) (η y +ū( )(τ t )) ū x ( ) + 1 ū x (z) (y y +ū( )(τ t ) ū(z)(τ t)) ( C 1 x x ( τ t + τ t ) + ūx(z) ū x ( ) η y +ū( )(τ t ) ū x (z)ū x ( ) + y y +ū( )(t t ) + t τ ū(z) ū( ) ū x ( )(x x ) + ū x ( ) t τ x x ) ( C 2 dz (,z)(d z (,ζ) 2 +d z (z,ζ) 2 )+d z (,z) α d z (,ζ) 3 +d z (,z) 3 +d z (z,ζ) 2 d z (,z) 1+α + d z (z,ζ) 2 d z (,z) ), since ū x C ᾱ u (). By using the inequality d z (z,ζ) C (d z (,z)+d z (,ζ)) we then obtain ( θ 3 C 3 dz (,ζ) 2 d z (,z)) + d z (,ζ) 3 d z (,z) α +d z (,z) 3 +d z (,ζ) 3+α + d z (,ζ) 2 d z (,z) 1+α). Since,z K and ζ M, from the above inequality we immediately deduce (2.38). Moreover, using the fact that is bounded and that ζ M, we get θ () (ζ) ( θ z (z) (ζ)) H d z (,z)+ θ d (,ζ) (d M +M1 3 z (,ζ) 3 M α + d (,ζ) 3 )1 3 ; M then, by choosing M sufficiently great, we also obtain (2.39). Finally, if d H (θ () (ζ),θ) d H (θ z (z) (ζ),θ () (ζ)), from (2.39) we obtain θ H d H (θ () (ζ), ) d H (θ () (ζ),θ) d H (θ () (ζ),) d H (θ z (z) (ζ),θ () (ζ)) ( 1 1 ) dh (θ () z M (ζ), ) 1 d z (,ζ). 1 M 2 This proves (2.4) and concludes the proof of the lemma.

25 nonlinear ultraparabolic equation 725 Proof of Proposition Let us first prove that, for every multi-index σ, σ, we have Dσ H Γ H ( θ z (z) (ζ)) Dσ H Γ H ( θ () (ζ)) ( C d z (,z) α d z (,ζ) Q+2 σ + d z (,z)d z (,ζ) Q+1 σ ) (2.41), for some positive constant C. Indeed,we have DσΓ H H ( θ z (z) (ζ)) Dσ H Γ H ( θ () (ζ)) = (by the mean value theorem and denoting H =(X H,Y H, θ3 )) = θ () (ζ) ( θ z (z) (ζ)), H Dσ H Γ H (θ) (where d H (θ () (ζ),θ) d H (θ z (z) (ζ),θ () (ζ))) x x X H Dσ H Γ H (θ) + t t Y H Dσ H Γ H (θ) + (θ () (by (2.4) and (2.38)) (ζ) ( θ z (z) (ζ))) 3 θ3 Dσ H Γ H (θ) C(d z (,z)d z (,ζ) Q+1 σ + d z (,z) α d z (,ζ) Q+2 σ ). Assertion (2.41) is then proved. Now we can apply Lemma 2.17, again denoting Ĩσ = {ϱ : k ϱ =}.Wehave D σ ( )Γ z (,ζ) D σ (z)γ z (,ζ) = ( C ϱ D µ ū( )D ū x ( ) hϱ ϱ (z) ( )Γ z (,ζ) ϱ Ĩσ µ I ϱ C ϱ D µ ū(z)d (z) ū x (z) hϱ ϱ (z)γ z (,ζ)) µ I ϱ 1 1 C ϱ D µ ū( ) D µ ū(z) ū x ( ) hϱ ū ϱ Ĩσ µ I x (z) hϱ ϱ µ I ϱ + C ϱ D µ ū(z) D ϱ (z) (z)γ z (,ζ) D () ϱ ( )Γ z (,ζ) ϱ Ĩσ µ I ϱ D () ϱ ( )Γ z (,ζ) (by (2.29)) = ϱ Ĩσ C ϱ 1 ū x ( ) hϱ 1 D µ ū( ) ū µ I x (z) hϱ ϱ µ I ϱ D µ ū(z) D () ϱ ( )Γ z (,ζ)

26 726 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro + 1 C ϱ D µ ū(z) ū x ( ) (DH ϱ Γ H )( θ () (ζ)) 1 ū x (z) (DH ϱ Γ H )( θ z (z) (ζ)) ϱ Ĩσ µ I ϱ (by 2.41) C (d z (,z)d z (,ζ) Q ϱ +1 + d z (,z) α d z (,ζ) Q ϱ +2) ϱ Ĩσ (since ϱ σ, ϱ I ϱ ) ( C d z (,z)d z (,ζ) Q σ +1 + d z (,z) α d z (,ζ) Q σ +2). Proposition Let ϕ C (). Then D () σ (z) ( ϕ( ζ 1 ) ) = p ϱ (ζ)(d () ϱ ϕ)(z ζ 1 ) ϱ J σ where denotes the law group (2.5) in G z and p ϱ is a polynomial of δλ H- degree k ϱ in the first two components of ζ such that ϱ k ϱ + σ, ϱ J σ. Proof. If we prove the assertion on the Heisenberg group, it will be proved in any group (G z, ), by the canonical change of variables. Let us start with σ =1. X H (θ)ϕ( θ 1 )=( θ1 θ 2 2 θ 3 )ϕ ( θ 1 θ 1,θ 2 θ 2,θ 3 θ (θ 1 θ 2 θ 2 θ1 ) ) = ( 1 ϕ θ 2 2 3ϕ θ 2 2 3ϕ ) (θ θ 1 )= ( 1 ϕ θ 2 θ 2 3 ϕ θ 2 3 ϕ ) (θ θ 1 ) 2 =(X H ϕ)(θ θ 1 ) θ 2 3 ϕ(θ θ 1 ). This proves the claim since the δλ H-degree of θ 2 is two. An analogous direct computation shows that Y H (θ)(ϕ( θ 1 ))=(Y H ϕ)(θ θ 1 )+ θ 1 3 ϕ(θ θ 1 ). The general assertion follows by iterating the previous arguments. In the sequel we shall need the following results. Remark If u Cū k,α (), the coefficients c i,j,m in Pz k u depend on the derivatives of u of order less than or equal to k. Hence c i,j,m Cu ᾱ (). If K is a compact subset of and σ is a multi-index, then there exists C> such that D σ Pz k u(ζ) D σ Pz k u(ζ) Cd z (,z) α, and (Pz k u(ζ) Pz 1 u(ζ)) (Pz k u(ζ) Pz 1 u(ζ)) Cd z (,z) α d z (,ζ) 2,

27 nonlinear ultraparabolic equation 727 for every z,,ζ K. Remark Let u Cū k,α (), with k 5. Then Pz k u(ζ) Pz k u(ζ) Cd z (,z) α d z (,ζ) k, for every z, andζ M, where M is defined in (2.33). 3. Regularization results In this section we prove Theorems 1.1 and 1.2. We consider the linear equation in R 3, Lūu = u xx +ūu y u t = f. (3.1) We say that a function u C 1 () is a classical solution of (3.1), if there exists xx u C() and equation (1.1) is satisfied at every point of. In order to study the regularity of u, we first represent it in terms of the fundamental solution Γ z : (uϕ)(z) = Γ z (z,ζ)l z (uϕ)(ζ) dζ (3.2) for any C () function ϕ. Then we set, as usual, U ε (z, )= Γ z (z,ζ)χ z,ε(z,ζ)l z (uϕ)(ζ) dζ (3.3) where χ z,ε(z, ) is a cut-off function, vanishing in a neighborhood of the pole of Γ z (z, ). As we pointed out in the introduction we can not use the standard theory, based on uniform convergence of U ε (z, ) and its derivatives to u and its derivatives. Instead we use a different technique, introduced in [6] and [7] and based on a weak definition of local uniform convergence and on the representation of higher-order derivatives as limits of suitable different quotients. We represent the functions u and U ε in (3.2) and (3.3) as the sum of two terms u = I 1 (, )+I 2 (, ) U ε =I 1,ε (, )+I 2,ε (, ), where I 1 (, )isc,i 1,ε uniformly converges to I 1, while I 2,ε converges to I 2 in the sense of the following definition. Definition 3.1. Let (F ε ) be a family of continuous functions on, let f : R,letα (, 1) and k N. We say that F ε (z, ) f ( ), as ε, locally uniformly of order k + α if for every compact set K there exists C>such that F ε (z, ) f( ) Cε k+α, z, K, d z (z, ) ε.

28 728 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro We next state an existence result for the derivatives D σ u (introduced in formula (2.31)). Lemma 3.2. Let σ 1and let u C σ 1,α ū (), a function which can be represented as u(z) =I 1 (z, )+I 2 (z, ), where I 1 is smooth as a function of z in, and the function z D σ (z)i 1 (, ) is continuous in z uniformly in. Assume that there exists a family I 2,ε of smooth functions and a continuous function I2 σ such that I 2,ε(z, ) I 2 (, ), as ε, locally uniformly of order σ + α, and D σ I 2,ε (z, ) I2 σ (), as ε, locally uniformly of order α. Then D σ u( ) exists and, for every in, D σ u( )=D σ ( )I 1 (, )+I2(z σ ). The proof is postponed to Subsection 3.1. In Subsection 3.2, we prove Theorem 1.2 by using Lemma 3.2. Finally, in Subsection 3.3, we conclude the proof of Theorem Derivatives and difference quotients. The main ideas of the proof of Lemma 3.2 are already contained in [6] and [7], but the lemma is not stated explicitly; hence we give here the proof. It is based on the following definition: Definition 3.3. If g : R, for every z andh Rsufficiently small, we define (i) (z)g(h) = g(exp(hi D (i) )(z)) g(z) h i, i =1,2. For every multi-index σ =(σ 1,...,σ m ) {1,2} m, we define by recurrence σ (z)g(h) = (σ1 )(z)( (σ2,...,σ m)g(h))(h). Remark 3.4. If g C σ ū () then, by the mean value theorem, we have σ (z)g(h) =D σ g(z h ), for a suitable z h such that dū(z h,z) h. Hence there exists lim σ(z)g(h) =D σ g(z), h uniformly on the compact sets. As in [7], Remark 4.2, the following result holds. Lemma 3.5. Let σ 1and let g C σ 1 ū (). If there exists lim σg(h) =w h uniformly on the compact subsets of, then there exists D σ g = w.

29 nonlinear ultraparabolic equation 729 Proof of Lemma 3.2 Since z I 1 (z, ) is smooth, by Remark 3.4, we have σ (h)i 1 (, )( ) D σ ( )I 1 (, ), as h, locally uniformly on compact sets. On the other side σ ( )I 2 (, )(h) I2 σ ( ) σ ( )I 2 (, )(h) σ ( )I 2,ε (, )(h) + σ ( )I 2,ε (, )(h) I2 σ ( ) (by the hypotheses on the local uniform convergence of order k + α and Remark 3.4) C 1 ε α + D σ (z h )I 2,ε (, ) I2(z σ ) Cε α. Then σ ( )u(h) D σ ( )I 1 (, )+I2(z σ ), as h, uniformly on the compact sets. By Lemma 3.5, we infer that D σ ( )u = D σ ( )I 1 (, )+I2(z σ ) Linear operators with Cū k,α coefficients. The aim of this subsection is the proof of Theorem 1.2. Let K be fixed according to Remark 2.5 and let K 1 be any compact set K 1 int(k). We study the regularity of u in K 1. We fix a function ϕ C (int(k)) such that ϕ 1 in a neighborhood of K 1. It is nonrestrictive to assume that, if z, K 1, then Md z (,z) d z (K 1,supp( ϕ)), Md z (z, ) d z (K 1,supp( ϕ)), (3.4) where M is the constant of Lemma Remark 3.6. With this choice of function ϕ and compact set K 1,wehave C 1 d z (z,ζ) d z (,ζ) C 2 d z (z,ζ), ζ supp( ϕ),z, K 1 (3.5) where C 1,C 2 are positive constants depending only on ū and K. In particular, d z (,z) Cd z (z,ζ), ζ supp( ϕ), z, K. (3.6) Proof. By (3.4) and Proposition 2.4 (ii), we have d z (,ζ) C(d z (,z)+d z (z,ζ)) C( 1 M d (,ζ)+d z (z,ζ)), ζ supp( ϕ); thus, if M is sufficiently large, we get d z (,ζ) Cd z (z,ζ), (3.7) for every ζ supp( ϕ). Exchanging the role of and z in (3.7), we get (3.5).

30 73 Giovanna Citti, Andrea Pascucci, and Sergio Polidoro Proposition 3.7. Let us assume that the coefficient ū in equation (3.1) is of class Cū k 1,α (), 2 k 6, and that f is of class Cū k 2,α (). Let u Cū k 1,α () be a classical solution of (3.1). Then, for every z, K 1, u = uϕ can be represented as u(z) =uϕ(z) = Γ z (z,ζ)n 1 (ζ, )dζ (3.8) + Γ z (z,ζ)n 2,k (ζ, )dζ + Γ z (z,ζ)n 3,k (ζ, )dζ. where N i,k (, ) is supported in the support of ϕ, and (i) supp(n 1 (, )) supp( ϕ) and N 1 (ζ, ) N 1 (ζ,z) Cd z (,z) α d z (,ζ); (3.9) (ii) N 2,k (, ) is of class C and for every multi-index σ D σ (ζ)n 2,k (, ) D σ (ζ)n 2,k (,z) Cd z (,z) α, ζ K; (iii) there exists a constant C, dependent only on the choice of ϕ and K 1, such that for every ζ K and z, K 1 N 3,k (ζ, ) Cdz k 2+α (,ζ), (3.1) N 3,k (ζ, ) N 3,k (ζ,z) Cd z (,z) α d z (,ζ) k 2. (3.11) Proof. By definition of the fundamental solution, we have uϕ(z) = Γ z (z,ζ)l z (uϕ)(ζ) dζ = Γ z (z,ζ)(ul z ϕ +2XuXϕ) dζ + = + Γ z (z,ζ)lūu(ζ)ϕ(ζ)dζ + Γ z (z,ζ)(l z Lū)u(ζ)ϕ(ζ) dζ Γ z (z,ζ)(ul z ϕ +2XuXϕ) dζ (3.12) Γ z (z,ζ)f(ζ)ϕ(ζ)dζ Γ z (z,ζ)(ū P 1 ū)(ζ) η u(ζ)ϕ(ζ) dζ. In order to use a uniform notation, in the sequel we will set P k u =fork a negative integer. Then, for every k 2, we have (ū Pz 1 ū)(ζ) η u(ζ) =(Pz k 1 +(Pz k 1 ū(ζ) Pz 1 ū(ζ))pz k 4 ū(ζ) Pz 1 ū(ζ))(u η (ζ) Pz k 4 u η (ζ)) u η (ζ)+(ū Pz k 1 ū)(ζ) η u(ζ).

31 nonlinear ultraparabolic equation 731 Then (3.8) is satisfied by inserting these expressions in formula (3.12) and by choosing the kernels N 1, N 2,k,andN 3,k as follows: N 1 (ζ, )=u(ζ)l z ϕ(ζ)+2xu(ζ)xϕ(ζ), N 2,k (ζ, )=P k 2 N 3,k (ζ, )=(f(ζ) P k 2 +(P k 1 f(ζ)ϕ(ζ)+(p k 1 ū(ζ) Pz 1 ū(ζ))pz k 4 u η (ζ)ϕ(ζ), f(ζ))ϕ(ζ)+(ū(ζ) Pz k 1 ū(ζ))u η (ζ)ϕ(ζ) ū(ζ) Pz 1 ū(ζ))(u η (ζ) Pz k 4 u η (ζ))ϕ(ζ). Let us prove (i). The support of N 1 (, ) is clearly a subset of supp( ϕ). Formula (3.9) can be proved as follows: N 1 (ζ, ) N 1 (ζ,z) = u(ζ)(l z ϕ(ζ) L z ϕ(ζ)) = u(ζ) P 1 z ū(ζ) Pz 1 ū(ζ) η ϕ(ζ) (by Remark 2.25) C ( d z (,z) 1+α + d z (,z) α d z (,ζ) ) Cd z (,z) α d z (,ζ). Condition (ii) easily follows from the definition of N 2,k and Remark Let us prove (3.1) for k =2: N 3,2 (ζ, )=(f(ζ) f( ))ϕ(ζ)+ ( ū(ζ) P 1 ū(ζ) ) u η (ζ)ϕ(ζ) Cd z (,ζ) α. We observe that, for every k 3, Pz k 1 ū(ζ) Pz 1 ū(ζ) =O ( d z (,ζ) 2), as ζ. Hence, we get N 3,3 (ζ, )= ( f(ζ) Pz 1 f(ζ) ) ϕ(ζ)+ ( ū(ζ) Pz 2 ū(ζ) ) u η (ζ)ϕ(ζ) +O ( d z (,ζ) 2) =O ( d z (,ζ) 1+α), as ζ. This proves (3.1) for k = 3. We can proceed analogously for k 4. Let us prove (3.11): N 3,k (ζ, ) N 3,k (ζ,z) P k 2 + P k 1 + (P k 1 + P k 1 ū(ζ) P 1 ū(ζ) P k 4 ū(ζ) P 1 ū(ζ)) (P k 1 z ū(ζ) Pz k 1 ū(ζ) u η (ζ) ϕ(ζ) (by Remark 2.25 and Remark 2.24) f(ζ) Pz k 2 f(ζ) ϕ(ζ) u η (ζ) Pz k 4 u η (ζ) ϕ(ζ) ū(ζ) Pz 1 ū(ζ)) u η (ζ) Pz k 4 u η (ζ) ϕ(ζ) Cd z (,z) α d z (,ζ) k 2.