HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Numer 3, Pages S ) Article electronically pulished on Octoer 1, 00 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION ANDREA PASCUCCI Astract. We study the interior regularity properties of the solutions to the degenerate paraolic equation, xu + yu tu = f, x, y, t) R N R R, which arises in mathematical finance and in the theory of diffusion processes. 1. Introduction We consider the degenerate paraolic equation 1.1) L u x + y t )u = f in the variales x, y, t) R N R R, where x denotes the Laplacian operator acting in the variales x =x 1,..., x N ). We aim to prove some new Schauder type interior estimates related to the Hölder classes C k,α naturally associated to L.Our estimates improve the known ones and allow us to study nonlinear equations of the form 1.) x u +,u) y u t u = f,u), recently considered in mathematical finance in [1] and []. We also otain regularity results for the following nonlinear convection-diffusion model proposed y Escoedo, Vazquez and Zuazua in [9]: x u + y gu) t u =0, with particular interest in the case gu) =u u q 1 for q ]1, N+ N+1 [. While we refer to the next section for the precise notation and assumptions on the coefficients and f, we would like to make some preliminary remarks. One of the main features of operator L is the strong degeneracy of its characteristic form due to the lack of diffusion in the y-direction. On the other hand, L can e represented in the form, p 1.3) Xj + X p+1, j=1 where the first-order differential operators vector fields) X j are defined as follows: 1.4) X j = xj,j=1,..., p = N, and X N+1 = y t. A classical result y Hörmander [11] states that if an operator H, intheform1.3), is such that the vector fields X j have smooth coefficients and their commutators, Received y the editors June 7, Mathematics Suject Classification. Primary 35K57, 35K65, 35K70. Investigation supported y the University of Bologna. Funds for selected research topics. 901 c 00 American Mathematical Society

2 90 ANDREA PASCUCCI up to a certain order, span the whole space at every point, then H is hypoelliptic. This means that every weak solution of Hu = f, withf C,issmooth. For instance, if N =1andx, y, t) =x in 1.1), then 1.5) L x = xx + x y t is the linearized prototype of the Kolmogorov operator which, under suitale conditions, descries the proaility density of a physical system with two degrees of freedom cf. [18]). In this case we have X 1 = x, X = x y t, and [X 1,X ] X 1 X X X 1 = y, so that L x is a hypoelliptic operator. More generally, the vector fields in 1.4) verify 1.6) [X j,x N+1 ]= xj ) y, j =1,..., N; therefore, the assumptions of Hörmander s theorem are satisfied if 1.7) C and x x1,..., xn ) 0. Hörmander s result was the starting point of an extensive study of operators H in the form 1.3) with smooth vector fields. A general theory of the regularity analogous to the classical one has een developed oth in Soolev and Hölder spaces y Folland [10], Rothschild and Stein [17], Nagel and Stein [15], and Beals [3]. We also refer to the more recent papers y Krylov [1] and y Lanconelli, Polidoro and the author [13]. The case of operators in the form p a ij X i X j + X p+1 i,j=1 with non-regular coefficients a ij has een considered y Xu [19] and Bramanti and Brandolini [4]. Thanks to the known results cf. [17]), we have the following. Theorem 1.1 Rothschild-Stein). Let u e a classical solution of 1.1) cf. Definition 4.1) in an open suset Ω of R N+. If f C k,α Ω) and the Hörmander condition 1.7) holds, then u C k,α Ω). The regularity assumption on in Theorem 1.1 can e weakened y assuming at least C k+1,α Ω). In this case the proof follows the original one with minor changes and we otain the following: Theorem 1. Rothschild-Stein). Let u e a classical solution of 1.1) in Ω with f C k,α Ω). If C k+1,α Ω) and x 0in Ω, thenu C k,α Ω). In view of the classical Schauder estimates, the previous results do not seem optimal. In particular, we emphasize that they do not allow the treatment of the existence and regularity theory of nonlinear equations. As a matter of fact, the further weaker assumption C k,α is naturally expected. Actually, the techniques used y Rothschild and Stein require the smoothness of the vector fields as an essential hypothesis. On the contrary, here we aim to consider non-regular vector fields. In the recent papers [7], [8] in collaoration with Citti and Polidoro, we considered the nonlinear equation in three variales 1.8) L u u = xx u + u y u t u = f

3 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 903 and we studied the regularity of the solution u y a modification of the classical freezing method. More precisely, we regarded L as a local perturation of a Hörmander s operator on the Heisenerg group. This last operator played the same role as the constant coefficients operators in the classical theory. This technique was introduced y Citti in [5] to study an equation of Levi type. Aiming to adapt those ideas, we immediately realize that, in dimensions higher than three, the Lie algera formally associated to L is not free. This means that the vector fields X j do not satisfy as few linear relations as possile i.e., only those forced y anti-commutativity and the Jacoi identity). As a consequence, the algera that one might naturally associate to L varies from point to point. In order to overcome this prolem and to eliminate the inessential relations among the commutators, we add some extra variales and we lift the operator L to a higher-dimensional space. We recall that a general version of the so-called lifting method for an operator in 1.3) with smooth coefficients, is due to Rothschild and Stein [17]. In our case we make the tentative choice to define the following operator in R N+1 : 1.9) L B = x + y1 + x y x N yn t where x, y, t) =x 1,..., x N,y 1,..., y N,t) denotes the point in R N+1. In order to apply to 1.9) the freezing techniques cited aove, a detailed analysis and careful estimates of the fundamental solutions to the frozen operators are in order. This is done in Section 3 and it is our main proof. Then, we study the regularity properties of L B and finally, we apply our results to the operator L. Weprovethefollowing. Theorem 1.3. Let u e a classical solution of 1.1) in Ω cf. Definition 4.1 and Remark 4.3) and assume the Hörmander type condition 1.10) C 1 Ω) and x 0 in Ω. If, f C k,α Ω), withk and α ]0, 1[, thenu C k,ᾱ Ω) for every ᾱ ]0,α[. We remark that Theorem 1.3 and a ootstrap argument give simple conditions for the interior regularity of solutions to a nonlinear equation of the form 1.). In particular, we refine the results in [7]. Two possile directions for extending Theorem 1.3 come readily to mind. It seems that our technique can e adapted without difficulty to the following, more general, class of ultraparaolic operators in R N+ : a ij xix j + y t i,j=1 where a ij ) is a positive definite matrix with Hölder continuous entries. Secondly, assumption 1.10) could e relaxed y a higher step condition, that is, y requiring that higher-order commutators of the vector fields X j span R N+. In this case, it seems that the proof would e essentially analogous, even if it could ecome consideraly knotty. This paper is organized as follows. In Section we set the notation and we collect some tools for the analysis on nilpotent Lie groups. In Section 3 we provide some estimates of the fundamental solutions of the frozen operators. Section 4 is devoted to the proof of Theorem 1.3.

4 904 ANDREA PASCUCCI. Hölder classes and control distances In this section we present some preliminary material and we define the lifted and frozen operators related to L. We egin y defining the Hölder classes related to the vector fields in 1.4). For the reader s convenience, we also give the following standard Definition.1. Let D e a locally Lipschitz continuous vector field on Ω and d a positive) formal degree associated to D. We call u Hölder continuous with exponent α, α ]0,d[, in Ω w.r.t. D and we write u CD α Ω) if, for every compact suset E of Ω, there exists a constant C such that uexp δd)z)) uz) C δ α d, for every z E and suitaly small δ. We refer, for instance, to [16] for the definition and properties of exponential mappings induced y vector fields. We say that u is Lie derivale w.r.t. D in z Ω if the following limit exists: uexp δd)z)) uz) Duz) = lim. δ 0 δ Definition.. Let α ]0, 1[. We set the formal degree of X 1,...,X N in 1.4) equal to one and the formal degree of X N+1 equal to two. We define C α Ω) = N+1 j=1 C α X j Ω). We say that u C 1,α Ω) if X j u C α Ω), j =1,...,N, and u C 1+α X N +1 Ω). Finally, if k N, k, we define y recurrence the class C k,α assuming that C k,α Ω), we say that u C k,α Ω) if Ω) as follows: X j u C k 1,α Ω), j =1,...,N, and X N+1 u C k,α Ω). For greater convenience, when in the sequel we consider the class u C k,α Ω) with k, we always implicitly assume that C k,α Ω). No regularity in the y-direction is seemingly assumed in the definition of C k,α Ω). On the other hand, keeping in mind that [X j,x N+1 ]= xj ) y and Hörmander s condition, it is natural to set the formal degree of the vector field y equal to three and it is possile to prove, y a standard argument ased on the Campell- Hausdorff formula, the following Lemma.3. If k =0, 1, and u C k,α u C k,α Ω), then y u C k 3,α Ω). Ω), thenu C k+α y Ω). If k 3 and By the previous lemma, we have the following inclusion of the space C k,α in the space of Hölder continuous functions in the classical sense: C 3k,α Ω) C k, α 3 Ω). We now lift the original vector fields in 1.4) to R N+1 in such a way that they ecome free. Since we aim to prove a local result, it is not restrictive to suppose that Ω is suitaly small. Then, without loss of generality, y 1.10), we may assume that

5 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 905 x1 0inΩ.Inthesequelwedenoteyz =x, y, t) =x 1,...,x N,y 1,...,y N,t) and ζ =ξ,η,τ) the points in R N+1 and we set Ω 0 =Ω R N 1. We define the lifted vector fields on Ω 0 as.1) D j = xj,j=1,...,n, D N+1 = B, y t, where Bx, y 1,t)=x, y 1,t),x,...,x n ), and y = y1,..., yn ). Thus, the operator L B in 1.9) can e expressed in the form L B = Dj + D N+1. Since x1 0inΩ 0, the commutators D N+ [D 1,D N+1 ]= x1 ) y1, D N+1+j [D j,d N+1 ]= xj ) y1 + yj, j N, j=1 are linearly independent and the system D j ) 1 j N+1 forms a asis of R N+1 at every point of Ω 0. Analogously to Definition., we give the notion of Hölder continuity related to D j ). Definition.4. Let α ]0, 1[. WesettheformaldegreeofD 1,...,D N equal to one and the formal degree of D N+1 equal to two. We define C α B Ω 0)= N+1 j=1 C α D j Ω 0 ). We say that u C 1,α B Ω 0)if D j u CB α Ω 0), j=1,...,n, and u C 1+α D N +1 Ω 0 ). Finally, if k N, k, we define y recurrence the class C k,α B assuming that C k,α Ω), we say that u C k,α B Ω 0)if Ω 0) as follows: D j u C k 1,α B Ω 0 ), j =1,...,N, and D N+1 u C k,α B Ω 0 ). Remark.5. Given a function w = wx, y 1,t) on Ω, we denote again y w its extension to Ω 0 =Ω R N 1, i.e., the function defined y wx, y 1,...,y N,t)= wx, y 1,t). Hence, it is clear that a solution u to 1.1) in Ω is also a solution to L B u = f in Ω 0.Moreover,u C k,α Ω) if and only if u C k,α B Ω 0). We next construct a nilpotent Hörmander operator locally approximating L B and we introduce some distances naturally associated to the vector fields D j in.1). More details aout distances defined y vector fields can e found in [10] and [16]. For fixed z Ω 0, we define the frozen vector fields.) D z j = x j, j =1,...,N, and D z N+1 = z)+ x z), x x) ) y1 + x j yj t. j=

6 906 ANDREA PASCUCCI Since the commutators of D z j and D z N+1 are given y D z N+ [D z 1,D z N+1] = x1 z) y1,.3) D z N+1+j [D z j,d z N+1] = xj z) y1 + yj, j N, and x1 z) 0 y assumption, the Hörmander condition is verified and the operator ).4) L z = D z j + D z N+1 j=1 is hypoelliptic. We call z D z 1,...,D z N+1), the intrinsic gradient related to the system of vector fields defined in.) and.3). For fixed z R N+1, we consider the exponential map Ez z θ) =exp θ, z )z), θ R N+1. It is well known that the map Ez z is a gloal diffeomorphism. Its inverse function θ z z is usually called the canonical change of coordinates and it has the explicit expression.5) θ z z ζ) = Ez z ) 1 ζ) =θ1,...,θ N+1 ), where θ j = ξ j x j, 1 j N, θ N+1 = τ t), θ N+ = 1 [ η 1 y 1 +τ t) z)+ ) x 1 z) ξ 1 + x 1 x 1 ) x1 z).6) ] xj z)η j y j + x j τ t)), j= θ N+1+j = η j y j + τ t)ξ j + x j ), j N. Through the canonical change of coordinates, the vector fields Dj z,d z j corresponding to different points z, z Ω 0, coincide. More precisely, if we set.7) Dj H = θj θ N+1 θ N +1+j, 1 j N, DN+1 H = θn θ j θn +1+j, j=1 D H N+1+j [D H j,d H N+1] = θn +1+j, 1 j N, then, for any smooth function ϕ and z Ω 0, it follows that D z ) j ϕ θ z z =D H j ϕ) θ z z, 1 j N +1. The vector fields in.7) generate a free Lie algera which is isomorphic to the Heisenerg one. Indeed, the vector fields in.7) induce a composition law in R N+1 formally defined y the Campell-Hausdorff formula, or explicitly { θ θ) j = θ j + θ j, if 1 j N +1, θ j + θ j + 1 θj N 1 θn+1 θ ) j N 1 θ N+1, if N + j N +1,

7 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 907 and the dilations group.8) δ λ θ) =λθ 1,...,λθ N,λ θ N+1,λ 3 θ N+,...,λ 3 θ N+1 ), λ > 0. The space R N+1 endowed with the law and the dilations δ λ is a homogeneous Lie group. The associated Lie algera of the -left-invariant vector fields is the one generated y D H 1,...,DH N+1.WealsoremarkthatDH 1,...,DH N are δ λ-homogeneous of degree one and D H N+1 is δ λ-homogeneous of degree two. Therefore, the Hörmander operator,.9) L H = ) D H j + D H N+1, j=1 has a fundamental solution Γ H which is invariant with respect to the left - translations and it is homogeneous of degree Q + where Q =4N + is the homogeneous dimension of R N+1, ). An explicit expression of Γ H is known see, e.g., [14]); however, here we only use its qualitative properties. A norm homogeneous w.r.t. the dilations in.8) is given y.10) θ H = j=1 ) θ j + θ N+1+j θ N+1 1, and the associated control distance is defined y d H θ, θ) = θ 1 θ H. The following Lie product on R N+1 is naturally induced:.11) z ζ = E z z θ z z z) θ z z ζ)) =x + ξ x, y + η ȳ τ t)j x B z)x x),t+ τ t), where J x B denotes the Jacoian matrix of B w.r.t. the variale x, i.e., the diagonal matrix diag x1, 1,...,1). Correspondingly, we have the dilations group δ z λz) =E z z δλ θ z zz) ), λ > 0, and the associated control distance.1) d z z,ζ) = θ z zζ) H. Lemma.6. There exists a constant c 0, only dependent on Ω 0, such that.13) c 1 0 d zz,ζ) d z z,ζ) c 0 d z z,ζ), for every z,ζ Ω 0,where d z z,ζ) = x ξ + t τ 1 + y1 η 1 +t τ)z) y j η j + t τ)ξ j + x j ) j= 1 3.

8 908 ANDREA PASCUCCI Proof. By.6) and y denoting θ z z ζ) =θ 1,...,θ N+1 ), we have θ N+ = 1 [ η 1 + y 1 + z)τ t) x1 z) j= xj z) η j y j + τ t)ξ ) j + x j ) + 1 ] xj z)τ t)ξ j x j ) + τ t)ξ 1 x 1 ) j= 1 = η 1 + y 1 + z)τ t) xj z) θ N+1+j + 1 ) x1 z) θ jθ N+1 1 θ 1θ N+1. Hence.13) is an immediate consequence of the elementary inequality a.14) a) , a, > 0. j= We stress that the distances d z,d ζ corresponding to different points z,ζ Ω 0 are not, in general, equivalent. Nevertheless, using Lemma.6, it is straightforward to prove the following Lemma.7. There exists a constant c 0, only dependent on Ω 0, such that.15) c 1 0 d zz, z) d z z, z) c 0 d z z, z) and.16) d z z,ζ) c 0 d z z, z)+d z z, ζ)) for every z, z, ζ Ω 0. It is remarkale that the Hölder continuity property related to the vector fields D j can e expressed in terms of the control distances associated to the frozen vector fields. Indeed, we have Lemma.8. Let g e a function on Ω 0 and α ]0, 1[. Suppose that, for every compact suset E of Ω 0, there exists a constant C such that.17) gz) g z) Cd z z, z) α, z, z E. Then g C α B Ω 0). Proof. We have to prove that uexpδd j )z)) uz) cδ α, 1 j N,.18) uexpδd N+1 )z)) uz) cδ α. The first inequality is ovious since d z z,expδd j )z)) = δ, for1 j N. With regard to the second inequality in.18), y assumption.17), it suffices to verify that.19) d z z,expδd N+1 )z)) cδ 1.

9 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 909 Denoting γδ) =expδd N+1 )z), we have δ γδ) =z + 0 = x, y 1 + D N+1 γs))ds Hence, y Lemma.6, we get c 1 0 d zz,γδ)) d z z,γδ)) = δ 1 + δ 0 γs))ds, y + δx,...,y N + δx N,t δ. δ 1 δ 0 γs)) z))ds δ sup γs)) z) s [0,δ] since C 1 Ω) and z ζ d z z,ζ) for every z,ζ in a suitaly compact suset E 0 of Ω) ) 1 δ δ sup d z z,γδ)) E 0 y.14) for ε>0) which yields.19) if ε is suitaly large. δ ε sup E 0 ) 1 ) d zz,γδ)), 3ε The control distances previously introduced also give an estimate of the error in the intrinsic Taylor expansion of a function u C k,α B. To e more precise, as in [7], Theorem.16, the following result can e proved. Proposition.9. Let z Ω 0 and u C k,α B Ω 0). There exists a unique polynomial function P k z u which is a sum of terms δ z λ-homogeneous of degree less than or equal to k and verifies.0) uz) =P k z uz)+od z z,z) k+α ), as z z. For instance, in the case k =0, 1, we have P 0 z uz) =u z) and P 1 z uz) =u z)+ x u z),x x. Hence, the frozen vector fields defined in.) are otained y considering the firstorder intrinsic) Taylor expansion of the coefficients of the original vector fields in.1). In particular, we have D N+1 D z N+1 = P 1 z ) y 1. We end this section y stating a technical lemma, which will e used in the proof of Theorem 1.3. For fixed z, z Ω 0 and a constant M 1, we define the set.1) Ω M z, z) ={ζ Ω 0 Md z z, z) d z z,ζ)}.

10 910 ANDREA PASCUCCI We remark that we can choose M sufficiently large so that.) c M ) 1 d z z,ζ) d z z, ζ) c M d z z,ζ), ζ Ω M z,z), for some constant c M. Indeed, y Lemma.7, we have c0 ) d z z,ζ) c 0 d z z, z)+d z z, ζ)) c 0 c 0 d z z, z)+d z z, ζ)) c 0 M +1 d z z, ζ), and, on the other hand, ) d z z, ζ) d z z,ζ) c 0 d z z,z)+d z z,ζ)) c 0 M + d zz,ζ), so that 1 1 ) d z z,ζ) d z z,ζ). c 0 M Lemma.10. Let u C k,α B Ω 0) and k 4. For every compact suset E of Ω 0, there exists a constant c = ce) such that.3) P k z uζ) P k z uζ) cd z z, z) α d z z,ζ) k, P k z uζ) P 1 z uζ)) P k z uζ) P 1 z uζ)) cd z z, z) α d z z,ζ) k,.4) for z, z E and ζ Ω M z, z). Proof. The proof is a direct and tiresome computation. We only show.3) for k =1. Wehave P 1 z uζ) P 1 z uζ) = uz) P 1 z uz)+ ξ x, x uz) u z)) since u C 1,α B Ω 0)andy.)) cd z z,z) α d z z, ζ). 3. Parametrices The proof of Theorem 1.3 is ased on a representation formula for classical solutions to 1.1) in terms of the fundamental solution Γ z of the frozen operator L z in.4). In this section, we provide some crucial estimates of Γ z,with z Ω 0 =Ω R N 1 cf. Proposition 3.1). Most of the results of this section are rather technical. We denote y Γ z z,ζ) resp. Γ H θ)) the fundamental solution of L z resp. of L H in.9)), evaluated in z resp. in θ) and with pole in ζ resp. in the origin). We note that 3.1) Γ z z,ζ) =Γ z ζ 1 1 z,0) = x1 z) Γ H θ z z ζ 1 z) ), where the product isdefinedin.11). We introduce some auxiliary notation. We denote the identity y D 0 = D z 0 = D0 H and for every multi-index σ =σ 1,...,σ m ) {0, 1,...,N +1} m,weset D z σ = D z σ D z 1 σ m, Dσ H = Dσ H 1 Dσ H m. We call the weight of σ the numer 3.) σ = m σ 1 +m σ +3m σ 3

11 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 911 where m σ 1,mσ,mσ 3, respectively, are the cardinalities of the sets {σ j σ 1 σ j N}, {σ j σ σ j = N +1}, {σ j σ N + σ j N +1}. Analogously, if C σ,α Ω), we put D σ = D σ1 D σm. For greater convenience, whenever we consider a derivative D σ with σ, in the sequel we agree to assume C σ,α Ω). Moreover, to avoid any amiguity, when we have a function F which depends on several variales, we systematically write Dz)F instead of DFz). Then, for instance, Dz)Γ z,ζ) denotes the D-derivative of Γ z,ζ) evaluated at the point z. The following estimate of Γ z and its derivatives is well known: for every z Ω 0 and multi-index σ {0, 1,...,N +1} m there exists a positive constant c, such that 3.3) D z σz)γ z,ζ) cd z z,ζ) Q+ σ, z,ζ Ω 0, z ζ. Moreover, the constant c in 3.3) depends continuously on z. In the proof of Theorem 1.3, we shall also need to compare the fundamental solutions of frozen operators corresponding to different points of Ω 0. The main result of this section is the following. Proposition 3.1. Let σ {1,...,N +1} m. c = cσ, Ω 0 ) such that There exists a positive constant 3.4) D σ z)γ z,ζ) D σ z)γ z,ζ) cd z z, ζ) Q+ σ α d z z, z) α, and, if σ 3, 3.5) D σ z)γ z,ζ) D σ z)γ z,ζ) cd z z, ζ) Q+ σ α d z z, z) α, for every z, z Ω 0 and ζ Ω M z,z) cf..1)). The proof of Proposition 3.1 is ased on two lemmas. The first one gives an expression of D j in terms of the frozen vector fields in.). Lemma 3.. Let z, z Ω 0 and σ {1,...,N +1} m and C σ,α Ω), if σ ). For every smooth function ϕ, we have 3.6) D σ ϕz) = Λ µ, z z)r z z)) αµ D z µ ϕz) µ {1,...,N+} m where µ σ + α µ, α µ m σ,and 3.7) R z z) = z) P 1 z z) x1 z) Λ µ, z z) = c i µ) i = z) z) x z),x x, x1 z) ν J µ,i D ν z)r z ) βν. In 3.7), J µ,i is a suitale suset of {1,...,N +1} m 1, ν σ and β ν m. Moreover, we have 3.8) Λ µ, z z)r z z)) αµ Λ µ, z z)r z z)) αµ cd z z, z) αµ+α

12 91 ANDREA PASCUCCI for some positive constant c = cσ, Ω 0 ). Proof. We proceed y induction on σ. If σ =1,, the assertion is trivial since D σ = D z σ if mσ = 0 cf. 3.)), and we have D N+1 = D z N+1 + R z D z N+. We next consider σ. If j =1,...,N, y induction, we have D j D σ ϕz) = D j z)λ µ, z R z ) αµ ) D z µ ϕz) µ {1,...,N+} m Analogously, we have D N+1 D σ ϕz) = µ {1,...,N+} m +Λ µ, z z)r z z)) αµ D z j D z µϕz)). D N+1 z)λ µ, z R z ) αµ ) D z µ ϕz) ) +Λ µ, z z)r z z)) αµ D z N+1D z µϕz)+λ µ, z z)r z z)) αµ+1 D z N+D z µϕz). Then the thesis is a straightforward verification. In particular, 3.8) is a consequence of the fact that, y assumption, C σ,α Ω) and R z z) =0. The proof of 3.5) in Proposition 3.1 is rather delicate since we need to estimate the fundamental solutions of frozen operators related to different points z, z. Here we use the canonical change of coordinates.5) and we investigate the properties of the fundamental solution Γ H. The next lemma provides us with some asic estimates. Lemma 3.3. Let C 1,α Ω). There exists a positive constant c = cω), such that 3.9) d H θ z z ζ),θzζ) z ) cd z z,ζ) 1 α 3 d z z, z) α 3, for every z, z Ω 0 and ζ Ω M z,z). Proof. Let us denote θzζ)) z θ z zζ) =θ. Keeping in mind formulas.5) and.6), we get N ) d H θ z z ζ),θz z ζ)) = x x + t t 1 + θ j 1 3. If j N, wehave θ N+1+j = η j ȳ j + τ t)x j + x j ) + ξ j x j )τ t) therefore, = ξ j x j )τ t) y j ȳ j + t t)x j + x j ) j=n+ η j y j + τ t)ξ ) j + x j ) ) +ξ j x j )τ t) ξ j x j )τ t); 3.11) θ N+1+j 1 3 d z z, z)+d z z, ζ) 3 d z z, z) dz z,ζ) 3 d z z, z) 1 3 since ζ Ω M z, z) andy.)) cd z z, ζ) 1 α 3 d z z,z) α 3.

13 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 913 Next we prove the inequality 3.1) θ N+ 1 3 cd z z,ζ) 1 α 3 d z z,z) α 3, which y 3.10) suffices to conclude the proof of the lemma. Noting that η j y j + x j τ t) =η j y j + τ t)ξ j + x j ) τ t)ξ j x j ), we have θ N+ = 1 t t)ξ 1 x 1 + ξ 1 x 1 )+ η ȳ +τ t) z) η y +τ t)z) x1 z) x1 z) [ xj z) x1 z) η j ȳ j + x j τ t)) ] x j z) x1 z) η j y j + x j τ t)) j= 3.13) = 1 t t)ξ 1 x 1 + ξ 1 x 1 )+A 1 + A + A 3 + A 4, where A 1 = y ȳ +t t) z) τ t)z) z)) + x1 z) x1 z) + x 1 z) x1 z) η y +τ t)z)), x1 z) x1 z) so that, since C 1,α Ω) C 1 Ω), we have ) A c d z z, z)+d z z,ζ) 3 d z z, z) d z z,z) α 3 dz z,ζ) 3.14) since ζ Ω M z, z) andy.)) cd z z,ζ) 1 α 3 d z z, z) α 3. Also, xj z) A = j= x1 z) ) x j z) η j ȳ j + x j τ t)) x1 z) so that, since C 1,α Ω), we have 3.15) A 1 3 cd z z, z) α 3 d z z,ζ). Moreover, xj z) A 3 = η j y j + τ t)x j + x j ) η j ȳ j + τ )) t)ξ j + x j ) x1 z) j= which can e estimated as efore: 3.16) A cd z z,ζ) 1 α 3 d z z, z) α 3. Finally, A 4 = 1 xj z) x1 z) τ t)ξ j x j ) τ t)ξ j x j )) = 1 j= j= xj z) x1 z) t t)ξ j x j )+τ t)x j x j ))

14 914 ANDREA PASCUCCI so that 3.17) A cd z z,z) 3 d z z,ζ) dz z,ζ) 3 d z z,z) 1 3 since ζ Ω M z, z) andy.)) cd z z,ζ) 1 α 3 d z z, z) α 3. Plugging inequalities 3.14), 3.15), 3.16) and 3.17) ack into 3.13), we otain 3.1). This concludes the proof. Remark 3.4. Let C 1,α Ω). If z, z Ω 0, ζ Ω M z,z) andθ R N+1 are such that d H θ z z ζ),θ ) d H θ z z ζ),θzζ) z ), then there exists a positive constant c = cω 0 ) such that 3.18) θ H c θ z zζ) H. Indeed, we have d z z,ζ) = θ z zζ) H c d H 0,θ)+d H θ z z ζ),θ )) d H 0,θ)+d H θ z z ζ),θz z ζ))) y 3.9) and since ζ Ω M z,z)) c d H 0,θ)+M α 3 d z z, ζ) ). Thus, if M is suitaly large, we get 3.18). We are now in position to prove Proposition 3.1. Proof of Proposition 3.1. We first prove estimate 3.4) y using Lemma 3.. We have D σ z) D σ z)) Γ z,ζ) A 1 + A, where A 1 = Λ µ, z z)r z z)) αµ Λ µ, z z)r z z)) αµ D z µ z)γ z,ζ) µ {1,...,N+} m and A = µ {1,...,N+} m Λ µ, z z)r z z)) αµ D z µ z) D z µ z)) Γ z,ζ). Hence, y 3.3) and 3.8), we have A 1 c d z z, z) αµ+α d z z,ζ) Q+ µ µ {1,...,N+} m since ζ Ω M z, z) and, y Lemma 3.), µ σ + α µ ) cd z z,z) α d z z,ζ) Q+ σ.

15 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 915 On the other hand, y the mean value theorem, for some z such that d z z, z) d z z,z), we have D z µz) D z µ z) ) Γ z,ζ) = θ z z z), z D z µ z)γ z,ζ) since ζ Ω M z, z)) 3 c d z z,z) j d z z, ζ) Q+ µ j j=1 cd z z,ζ) Q+ µ α d z z, z) α. As efore, we conclude that A cd z z,ζ) Q+ σ α d z z, z) α. Next we prove 3.5). By 3.1) and Lemma 3., we have D σ z)γ z,ζ) D σ z)γ z,ζ) Λµ,z z) x1 z) DH µ θzζ)) z Λ µ, z z) x1 z) DH µ θ z z ζ) )) Γ H A 1 + A where µ σ A 1 = µ σ A = µ σ Λ µ,z z) x1 z) Λ µ, z z) x1 z) D H µ Γ H θzζ)) z, Λ µ, z z) x1 z) Dµ H Γ H θz z ζ)) DH µ Γ H θ z z ζ) ). Since the function z Λ µ,zz) x1 z) Cα BΩ 0 ), we get A 1 cd z z, z) α d z z,ζ) Q+ σ. On the other hand, y the mean value theorem, there exists θ such that d H θ z z ζ),θ ) d H θ z z ζ),θz z ζ)), and D H µ Γ H θz z ζ)) DH µ Γ H θ z z ζ) ) = θ z z ζ)) θ z z ζ), HDµ H Γ Hθ) 3 c d H θ z z ζ),θzζ) z ) j Q+ µ j θ H j=1 cd z z,z) α d z z,ζ) Q+ σ α, where the last inequality follows from Lemma 3.3 and Remark 3.4 note that the assumption C 1,α Ω) of Lemma 3.3 is fulfilled ecause σ 3). The proof is completed. We end this section y descriing the fine ehaviour of the vector fields D z j through the right translations w.r.t. the law in.11).

16 916 ANDREA PASCUCCI Lemma 3.5. Let C 1,α Ω). For every smooth function ϕ and multi-index µ {1,...,N +} m, we have 3.19) D z µ z)ϕ ζ 1 )= P ν, z ζ)d z ν z ζ 1 )ϕ, ν {1,...,N+1} m where P ν, z is a polynomial δ z λ -homogeneous of degree k ν,with ν µ + k ν, whose coefficients are functions of z of class CB αω 0). Proof. Since z ζ 1 =x ξ,y η + J x B z)τx x) τ t)ξ),t τ), we have, for j N, D z 1z)ϕ ζ 1 )=D z 1z ζ 1 )ϕ + τd z N+z ζ 1 )ϕ, ) D z j z)ϕ ζ 1)=D z j z ζ 1 )ϕ + τ D z N+1+j 1 D z N+ z ζ 1 )ϕ, x1 z) D z N+1z)ϕ ζ 1 )= D z N+1 + z ζ 1 )ϕ, j= ξ j D z N+1+j D z N+ z)ϕ ζ 1)=D z N+ z ζ 1 )ϕ. The thesis easily follows y induction on µ. 4. Hölder regularity In this section we prove Theorem 1.3. We first give the definition of a classical solution of 1.1). Definition 4.1. A classical solution of 1.1) is a function u C 1 Ω), with secondorder derivatives xix j u CΩ), 1 i, j N, verifying 1.1). For greater convenience, since in the following proof we deal with several estimates, we shall denote y c a constant which will not e always the same. Proof of Theorem 1.3. We proceed y induction on k. Let us remark that, if k 7, the proof is standard and the thesis follows y differentiating the equation. More precisely, if, f C k,α Ω), with 0 <α<1andk 7, then y the inductive hypothesis, u C k 1,ᾱ Ω) for every ᾱ ]0,α[. Susequently, y differentiating equation 1.1) w.r.t. the variale y and y denoting, as usual, y u = u y,weget L u y = f y y u y C k 5,ᾱ Ω), ᾱ ]0,α[. Thus, we deduce that u y C k 3,ᾱ Ω). Next we differentiate equation 1.1) w.r.t. the variale x j, j =1,...,N 1, and we get L u xj = f xj xj u y C k 3,ᾱ Ω), ᾱ ]0,α[. Therefore, we deduce that u xj C k 1,ᾱ Ω). Finally, we differentiate equation 1.1) w.r.t. to X N+1 = y t : L X N+1 u)=x N+1 f + x, x u y + u y x C k 4,ᾱ Ω), ᾱ ]0,α[. Hence, X N+1 u C k,ᾱ Ω) and this proves that u C k,ᾱ Ω) for every ᾱ ]0,α[. We next consider 3 k 6. We set the prolem in dimension N +1 and we prove that u C k,ᾱ B Ω 0). Then, y Remark.5, we infer that u C k,ᾱ Ω). We split

17 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 917 the proof into two steps: existence of the derivatives and Hölder continuity. Since the case k = is easier, we shall only sketch its proof separately at the end. The case 3 k 6: existence of the derivatives. Let, f C k,α Ω), with 0 <α<1and3 k 6. By the inductive hypothesis, u C k 1,ᾱ Ω) for every ᾱ ]0,α[. Since we aim to prove a local result, we only prove the existence of the derivatives of order k of u in a domain E 0,whereE 0 is a compact suset of Ω 0. To this end, we represent the solution u in terms of the fundamental solution Γ z, z E 0. More precisely, we consider a cut-off function ϕ C0 Ω 0) such that ϕ =1 in a neighorhood of E 0. We remark that it is not restrictive to assume that 4.1) supp ϕ) Ω M z,z), z, z E 0, where M is large so that.) holds for every z, z E 0.Wehave uϕ)z) = Γ z z,ζ)l z uϕ)ζ)dζ 4.) Ω 0 = ) Γ z z,ζ) ϕ f P 1 z )u y1 + x u, x ϕ + ul z ϕ ) dζ. Ω 0 Consequently, the solution u can e expressed in the form uz) =I 1, z z) I, z z), z E 0 where I j, z z) = Γ z z,ζ)u j, z ζ)dζ, j =1,, Ω 0 with U 1, z C0 Ω 0 )and 4.3) U, z ζ) cd z z,ζ) k +ᾱ, z E 0,ζ Ω 0, ᾱ ]0,α[. Indeed, it suffices to put U 1, z = ϕ P z k f + P z k 4 u y1 P k 4.4) z P 1 z )) 4.5) + P z k x u, x ϕ +L z ϕ) P z k u, U, z = ϕ u y1 P k z ) + P z k + x u P z k x u, x ϕ + L z ϕ u P z k u ) P 1 z ) u y1 P z k 4 ) u y1 + f P k z f ) where we agree that P h z u 0ifh < 0. Hence, 4.3) is a consequence of the regularity assumptions on u,, f and of the estimate P z k ζ) P 1 z ζ) =O d z z, ζ) ), as ζ z, which can e easily deduced from the homogeneity property of the Taylor polynomial P h z. Note that I 1, z is a smooth function on E 0. Indeed, y the change of variales ζ = ζ 1 z, wehave 4.6) I 1, z z) = Γ z ζ,0)u 1, z z ζ 1 )d ζ and U 1, z C0 Ω 0). On the other hand, the function 4.7) J σ z) = D σ z)γ z,ζ)u, z ζ)dζ, z E 0, Ω 0

18 918 ANDREA PASCUCCI is well defined for σ = k. Indeed, y Lemma 3., we have 4.8) D σ z)γ z,ζ) = Λ z)d z µ, z µ z)γ z,ζ) µ σ cd z z,ζ) Q+ σ ; therefore, y 4.3), the integral in the right-hand side of 4.7) converges. We claim that, for every multi-index σ {1,...,N+1} m,with σ k, wehave 4.9) D σ u z) =D σ z)i 1, z J σ z), z E 0. The proof of 4.9) is ased on the use of some suitale high-order difference quotients related to the system D j, j =1,...,N +1. For z Ωandδ R \{0} sufficiently small, we set j) u, z, δ) = uexpδd j)z)) uz), 1 j N, δ N+1) u, z, δ) = uexpδ D N+1 )z)) uz) δ. Also, if σ {1,...,N +1} m, y recurrence we define σ u, z, δ) = σm) σ1,...,σ m 1)u,,δ),z,δ). The following result can e proved as in [6], Remark 4.. Lemma 4.. Let u C k 1,α Ω) and σ {1,...,N +1} m, σ = k. Ifthereexists lim σu,,δ)=v δ 0 uniformly on compact susets of Ω, then there exists D σ u = v. We are now in a position to prove 4.9). Since I 1, z is a smooth function, y the mean value theorem, we have σ I 1, z, z, δ) =D σ z δ )I 1 z for some z δ such that 4.10) d z z,z δ ) c 1 δ, where the constant c 1 depends only on σ andontheconstantc 0 in.16) for instance, c 1 = mc m 1 0 is fine). Thus, σ I 1, z, z, δ) convergestod σ z)i 1, z as δ tends to zero uniformly in z E 0. Therefore, y Lemma 4., in order to prove 4.9), it suffices to show that 4.11) lim σ I, z, z, δ) =J σ z) δ 0 uniformly on E 0. Let us consider a cut-off function χ C0 R, [0, 1]) such that χs) =1fors and χ vanishes for s 1. We set ) d z z,ζ) 4.1) I, z,δ z) = Γ z z,ζ)χ U, z ζ)dζ, z E 0, c 1 Mδ Ω 0 where M,c 1 are the constants in 4.1), 4.10) respectively. Note that z I, z,δ z), d z z,z) c 1 δ,

19 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 919 is a smooth function. Indeed, if d z z,z) c 1 δ, then y.16), the argument of the cut-off function χ in 4.1) satisfies d z z,ζ) c 1 Mδ 1 c 1 δ d z z,z)+d z z,ζ)) 1, ζ, d z z,ζ) c 1 δ, so that χ vanishes in a neighorhood of the pole of Γ z z,ζ). We claim that 4.13) sup d z z,z) c 1δ I, z,δ z) I, z z) cδ k+ᾱ, 4.14) sup d z z,z) c 1δ D σ z)i, z,δ J σ z) cδᾱ, for some positive constant c which depends continuously on z. Taking the claim for granted, we immediately conclude the proof of 4.11) and consequently of 4.9)): σ I, z, z,δ) J σ z) y the mean value theorem, for some z δ such that d z z, z δ ) c 1 δ) σ I, z, z, δ) σ I, z,δ, z, δ) + D σ z δ )I, z,δ J σ z) y 4.13) and 4.14)) cδᾱ. We are left with the proof of the claim. We egin y proving 4.13). We have I, z z) I, z,δ z) Γ z z,ζ) U, z ζ) dζ d z z,ζ) 4c 1Mδ y 3.3) and 4.3)) cδ k +ᾱ d z z,ζ) Q+ dζ d z z,ζ) 4c 1Mδ since, y assumption, d z z,ζ) c 0 d z z,z)+d z z,ζ)) c 0 c 1 δ + d z z, ζ))) cδ k +ᾱ d z z,ζ) Q+ dζ cδ k+ᾱ. d zz,ζ) c 0c 11+4M)δ Next, we prove 4.14). We have D σ z)i, z,δ J σ z) A 1 z, z)+a z, z), where )) d z z,ζ) A 1 z, z) = D σ z)γ z,ζ) 1 χ U, z ζ) dζ c 1 Mδ and Ω 0 Ω 0 ) d z z,ζ) A z, z) = D σ z) D σ z)) Γ z,ζ) χ U, z ζ) dζ. c 1 Mδ Proceeding as in the proof of 4.13) and using 4.8), we otain A 1 z,z) cδᾱ.

20 90 ANDREA PASCUCCI On the other hand, we remark that, if d z z,z) c 1 δ,then )) d z z, ) supp χ Ω M. c 1 Mδ Hence, y estimate 3.4) of Proposition 3.1 and 4.3), we get A z,z) c d z z,z)ᾱd z z,ζ) Q+ᾱ dζ cδᾱ. Ω M This concludes the proof of 4.14). The case 3 k 6: Hölder continuity of the derivatives. Let α ]0, 1[ and 3 k 6. In the previous step, we have proved that, if, f C k,α Ω), and u C k 1,ᾱ Ω), for any ᾱ ]0,α[, then u is D σ -differentiale for any σ of weight σ k and the representation formula 4.9) of D σ u holds. In this step, we aim to prove that 4.15) D σ u CᾱBE 0 ), σ {1,...,N +1} m, σ = k, and ᾱ ]0,α[, 4.16) D σ u C 1+ᾱ D N +1 E 0 ), σ, σ = k 1, and ᾱ ]0,α[, where E 0 is any domain fixed as in the preceding step. As a consequence of 4.15) and 4.16), we deduce that u C k,ᾱ Ω). We prove 4.15) y means of Lemma.8 and we consider separately the terms D σ z)i 1, z and J σ z) in formula 4.9), with z E 0 and σ = k. Letuseginwith J σ and prove that 4.17) J σ z) J σ z) cd z z, z)ᾱ, z,z E 0, ᾱ ]0,α[, for some constant c = ce 0, ᾱ, k). By a standard decomposition, we otain J σ z) J σ z) =A 1 z, z)+a z, z)+a 3 z, z), where A 1 z, z) = D σ z)γ z,ζ)u,z ζ) D σ z)γ z,ζ)u, z ζ)) dζ, A z, z) = A 3 z, z) = Ω 0\Ω M z,z) Ω M z,z) Ω M z,z) D σ z)γ z,ζ) D σ z)γ z,ζ)) U, z ζ)dζ, D σ z)γ z,ζ)u,z ζ) U, z ζ)) dζ. By 4.3) and 4.8), it is straightforward to prove that A 1 z, z) cd z z,z)ᾱ, z, z E 0. Next, we estimate A z, z) y using 3.5) of Proposition 3.1. We deduce that, for every ᾱ ]0, ᾱ[, there exists a constant c = ce 0, ᾱ), such that A z, z) c d z z, ζ) Q d z z,z)ᾱ dζ cd z z,z) ᾱ, z,z E 0. Ω M z,z) Analogously, we have 4.18) A 3 z, z) cd z z,z) ᾱ, z, z E 0.

21 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 91 Indeed, 4.18) is an immediate consequence of 4.8) and of the following estimate: there exists a positive constant c = ce 0 ) such that 4.19) U,z ζ) U, z ζ) cd z z,z)ᾱd z z,ζ) k, z, z E 0,ζ Ω M z, z). Let us prove 4.19). We have U,z U, z = B 1 z, z)+ϕb z, z)+b 3 z, z), where B 1 z, z) =ϕ u y1 P k z + P k z B z, z) = P k z P k z x u P k x u, x ϕ, P 1 z ) u y1 P k 4 z z ) + P z k f Pz k f )) B 3 z, z) =L z ϕ u Pz k u ) L z ϕ u P z k u ). By.3) of Lemma.10, we get at once ) u y1 P k z P 1 z ) u y1 P z k 4 ) u y1, B 1 z, z)ζ) cd z z,z)ᾱd z z,ζ) k, z, z E 0,ζ Ω M z, z). We next remark that B z, z) = P k z + [ P k z P 1 z ) P k 4 z u y1 P k 4 z u y1 ) Pz 1 ) P z k P 1 z )] u y1 P z k 4 ) u y1. Therefore, y.3) and.4) of Lemma.10, we infer B z, z)ζ) cd z z,z)ᾱd z z,ζ) k, z, z E 0,ζ Ω M z, z). Finally, we oserve that B 3 z, z) =L z ϕ L z ϕ) u P k z = P 1 z P 1 z ) y1 ϕ u P k z u ) + L z ϕ P k z u P k z u ) u ) + L z ϕ P z k u Pz k u ). Thus, applying again Lemma.10, we estalish 4.19) and, consequently, 4.17). We conclude the proof of 4.15) y showing that 4.0) D σ z)i 1,z D σ z)i 1, z cd z z,z) α, z,z E 0, for some constant c = ce 0,k). We denote ζ = ξ, η, τ) andy 4.1) z R z) ζ 1 z) z ζ 1 =x ξ,y η + J x B z)τx x) τ t)ξ),t τ),

22 9 ANDREA PASCUCCI the right translation w.r.t. the law in.11). By 4.6), for every z E 0 and multi-index σ {1,...,N +1} m of weight σ = k, wehave 4.) D σ z)i 1,z = = µ k Ω 0 Γ z ζ,0)d σ z)u 1,z R z) ζ 1 ))d ζ Λ µ,z z) y Lemma 3.) Γ z ζ,0)d z µ z)u 1,zR z) ζ 1 ))d ζ y Lemma 3.5) = Λ µ,z z) Γ z ζ,0)p ν,z ζ)d νz z ζ 1 )U 1,z d ζ µ k ν {1,...,N+1} m y the change of variales ζ = ζ 1 z) = Γ z z,ζ)v z ζ)dζ, where V z ζ) = We oserve that µ k ν {1,...,N+1} m Λ µ,z z)p ν,z ζ 1 z)d z ν ζ)u 1,z. 4.3) V z ζ) V z ζ) cd z z,z) α, z, z E 0, ζ Ω 0, for some constant c. Indeed, 4.3) follows from 3.8), the fact the function z P ν,z is of class CB α y Lemma 3.5, and the estimate D z 4.4) ν Pz k uζ) D z σp k z uζ) cd z z,z) α. At this point, the proof of 4.0) is analogous to the one of 4.17). Indeed it suffices to oserve that D σ z)i 1,z D σ z)i 1, z = A 1 z, z)+a z, z)+a 3 z, z), where A 1 z, z) = Γ z z,ζ)v z ζ) Γ z z,ζ)v z ζ)) dζ, A z, z) = A 3 z, z) = Ω 0\Ω M z,z) Ω M z,z) Ω M z,z) Γ z z,ζ) Γ z z, ζ)) V z ζ)dζ, Γ z z,ζ)v z ζ) V z ζ)) dζ, and to conclude as efore, y using Proposition 3.1. Thus, 4.15) is proved. We omit the proof of 4.16) since it is essentially analogous.

23 HÖLDER REGULARITY FOR A KOLMOGOROV EQUATION 93 The case k =. Let, f C α Ω), with 0 <α<1. Since, y assumption 1.10), C 1 Ω), we have L B uz) L z uz) = P 1 z ) y1 uz) =O d z z, z)), as z z. In this case, a much simpler choice of the frozen operators provides an approximation of L of the same order. Indeed, for convenience, let us denote y z =x, y, t) a point of Ω. For fixed z Ω, we define L z) = x + z)+x 1 x 1 ) y t. Then, L z) is a Hörmander operator which, in the case N = 1, up to a straightforward change of variales, coincides with the Kolmogorov operator in 1.5). Moreover, we have L uz) L z) uz) =z) z) x 1 x 1 )) y uz) ) = O d z) z,z), as z z, where d z) denotes the control distance associated to L z). Given a cut-off function ϕ, werepresentthesolutionu in terms of the fundamental solution Γ z) of L z) : uϕ)z) = Γ z) z,ζ)l z) uϕ)ζ)dζ. Ω Since L z) uϕ) C 0 Ω), it is standard to prove that u C 1,ᾱ Ω), for every ᾱ ]0, 1[. Moreover, it is not difficult to adapt the previous arguments and to show that u C,ᾱ Ω), for every ᾱ ]0,α[. Remark 4.3. Theorem 1.3 holds true if we assume that u is a locally Lipschitz continuous, strong solution to 1.1) instead of a classical solution. We recall that u is a strong solution to 1.1) if it has weak derivatives and equation 1.1) is satisfied almost everywhere. In order to justify our claim, it suffices to remark that the proof of Theorem 1.3 is ased only on the representation formula 4.) and on the oundedness of the first-order derivatives of the solution. References [1] Antonelli, F., Barucci, E., and Mancino, M. E. A Comparison result for FBSDE with Applications to Decisions Theory. Math. Methods Oper. Res. 001, 54 3), [] Antonelli, F. and Pascucci, A. On the viscosity solutions of a stochastic differential utility prolem. To appear in J. Differential Equations. [3] Beals, R. L p and Hölder estimates for pseudodifferential operators: sufficient conditions. Ann. Inst. Fourier 1979, 9 3), MR 81c:47049 [4] Bramanti, M. and Brandolini, L. L p estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Trans. Amer. Math. Soc. 000, 35 ), [5] Citti, G. C regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci., Serie IV 1996, 3, MR 98:3507 [6] Citti, G. and Montanari, A. C regularity of solutions of an equation of Levi s type in R n+1. Ann. Mat. Pura Appl.4) 001, 180 1), MR 00f:35049 [7] Citti, G., Pascucci, A., and Polidoro, S. On the regularity of solutions to a nonlinear ultraparaolic equation arising in mathematical finance. Diff. Integral Equations 001, 14 6), MR 00f:35118 [8] Citti, G., Pascucci, A., and Polidoro, S. Regularity properties of viscosity solutions of a non-hörmander degenerate equation. J. Math. Pures Appl. 001, 80 9),

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