POINTWISE SCHAUDER ESTIMATES OF PARABOLIC EQUATIONS IN CARNOT GROUPS HEATHER PRICE 6 Abstract. Schauder estimates were a historical stepping stone for establishing uniqueness 1 andsmoothnessofsolutionsforcertainclassesofpartialdifferentialequations. Sincethattime, 0 2 they have remained an essential tool in the field. Roughly speaking, the estimates state that the Hölder continuity of the coefficient functions and inhomogeneous term implies the Hölder n continuity of the solution and its derivatives. This document establishes pointwise Schauder a J estimates for second order “parabolic” equations of the form 3 m1 1 ∂ u(x,t) a (x,t)X X u(x,t)=f(x,t) t ij i j − i,j=1 ] X P where X1,...,Xm1 generate the first layer of the Lie algebra stratificationfor a Carnotgroup. A The Schauder estimates are shown by means of Campanato spaces. These spaces make the . pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. h As a prerequisite device, a version of both the mean value theorem and Taylor inequality are t a established with the parabolic distance incorporated. m [ 1 v 8 1. Introduction 1 4 Schauder estimates are an essential tool in regularity theory for partial differential equations. 3 0 Roughly speaking the Schauder estimates state that given a solution to an inhomogeneous . equation where the coefficients of the operator as well as the inhomogeneous term are both 1 0 Hölder continuous, this regularity transmits through the operator to give Hölder continuity 6 of the derivatives of the solution. These estimates were the key to showing uniqueness and 1 : smoothness of solutions for certain classes of equations [36]. v Juliusz Schauder is credited for the proof in the case of second order linear elliptic equations i X given in [39] and [40]. Though Caccioppoli also had a similar result around the same time, his r a workwas notasdetailed[11]. Höldercontinuity inthemuch simpler case oftheLaplacianisdue to Hopf [30] a few years prior to Schauder’s result. Because of the usefulness of the inequality, a common objective of showing these types of estimates for different types of equations under more general conditions arose. As a result, many methods of proof have emerged. Mentioned here are only a few most relevant to the work of this paper. A more complete discussion on Hölder estimates andregularityofsolutionscanbefoundin[26, Chapter 6]forelliptic equations or [34, Chapter 4] for parabolic equations. One method of deriving Schauder estimates depends on having a representation of a funda- mental solution, explicitly computing derivatives, and relying on methods of singular integrals to get the results. This is demonstrated in [26, Chapter 6] as well as Chapter 1 of this man- uscript. Another method is by means of the Morrey-Campanato classes, which are equivalent to the Hölder spaces and can be seen in [24, Chapter 3] as well as [34, Chapter 4]. A third method is based on a scaling argument and approximation of solutions by Taylor polynomials. It can be found in [41, Section 1.7]. The proof for the Schauder estimates given here has aspects reminiscent of all these methods. 1 2 PRICE The main result of this paper is a generalization of the classic result in two ways. The “pointwise” result requires only Hölder continuity of the coefficients and inhomogeneous term at a single point in order to get the Hölder continuity of the solution and its derivatives at that same point. This pointwise nature can be useful when the source term or coefficients are not well behaved everywhere. It also allows an application to the study of nodal sets [28]. The other generalization is the change from the Euclidean setting to the Carnot group setting where derivatives are given by vector fields which may not necessarily commute. Comparing thesolutionto itsTaylor polynomial, a technique first popularized by Caffarelliin 1998inhiswork onfullynonlinear equations [12], makes possible thepointwise generalizationof the estimate. Caffarelli’s approach was generalized to parabolic equations by Wang in [43], and about a decade later, Han used this same method for proving pointwise Schauder estimates for higher order parabolic and elliptic equations in [28] and [29]. His interest was in the application to nodal sets. These results were extended by Capogna and Han [15] to second order subelliptic linear equations over Carnot groups in 2003. The proof contained in this paper follows the same method. Global and local Schauder estimates have been explored in the group setting as well as the more general case of Hörmander type vector fields. Though this list is not exhaustive, see [5], [7], [9], [10], [27], and [44] for more details. However, the pointwise result contained here seems to be new. Theoutlineofthepaperisasfollows. Section2beginswithbasicdefinitionsrelatedtoCarnot groups before proving the Schauder estimates for the sublaplacian in this setting. Section 3 begins the transition to the parabolic setting. Definitions regarding the product space of a group and the real line are made. The main tools such as group polynomials, Campanato classes, and Sobolev spaces are defined, and several lemmas regarding these items are shown. The proof of the main theorem (stated below) is given in section 4. Before stating the main result, the operator of interest is given by m1 (1.1) H = ∂ a (x,t)X X A t i,j i j − i,j=1 X where the vector fields X ,...,X generate the first layer of the Lie algebra stratification for 1 m1 a Carnot group, and the matrix A = (a ) is Hölder continuous only at the origin. Additionally ij there exist constants 1 < λ Λ < such that ≤ ∞ m1 (1.2) λ ξ 2 a (x,t)ξ ξ Λ ξ 2 for any ξ Rm1. ij i j | | ≤ ≤ | | ∈ i,j=1 X The operator is a non-divergence form similar to the heat equation, but it is not truly parabolic as the title suggests since m may be less than the dimension of the space. 1 Let Q denote the homogeneous dimension of the group G and S2,1 denote the Sobolev space p containing two spatial derivatives and one time derivative. The exact definitions for the Hölder and Campanato classes can be found in Section 3.3. Theorem 1. For Q+2 < p < , let u S2,1( ) and H u = f in with ∞ ∈ p Q1 A Q1 f Lp( ). Assume, for some α (0,1) and some integer d 2, f Cα (0,0) and ∈ Q1 ∈ ≥ ∈ p,d−2 a Cα (0,0). Then u Cα (0,0) and ij ∈ p,d−2 ∈ ∞,d u (0,0) C( u + f (0,0)) k k∞,α,d ≤ k kLp(Q1) k kp,α,d−2 where C = C(G,p,d,α,A) > 0. POINTWISE SCHAUDER ESTIMATES 3 Section 4.1 gives bounds for the heat kernel associated to the constant coefficient equation as well as a few lemmas regarding polynomial expansions of solutions. These are essential for the constant coefficient a-priori estimates shown in Section 4.2. These estimates give specific information about the bounds of the Sobolev norm of solutions, which is then used to give a basic version of the Schauder estimates for the constant coefficient equation as a quick corollary. The corollary is vastly useful because it allows us to transmit information from the polynomials approximating the inhomogeneous term to the polynomial approximating the solution. The freezing technique is then employed to give a-priori estimates for the non-constant coefficient equation, and a weak version of the Schauder estimates is shown. (It is weak in the sense that we still assume some amount of regularity on solutions and inhomogeneous term.) Finally, by comparing the solution to its first order Taylor polynomial and successively applying the a-priori estimates, the final result is obtained. 2. Carnot Groups The setting for the sequel is a special type of Lie group with many structures allowing computations to be done in a fashion similar to the Euclidean setting. Most essential to the proof of the Schauder estimates is the extensive use of the homogeneity of Carnot groups. However, care must be taken in the development of ideas such as scaling and distance. This introduction to Carnot groups aims to make these notions clear and precise while pointing out some of the difficulties of working in these groups, the most obvious of which is the fact that the derivatives do not necessarily commute. Before commencing, let it be known that in this document all vector fields can be written as linear combinators of standard partial derivatives with smooth coefficient functions. That is n X = b (x)∂ where b (x) C∞(Rn). k xk k ∈ k=1 X Definition 2. A Carnot group G of step r 1 is a connected and simply connected nilpotent ≥ Lie group whose Lie algebra g admits a vector space decomposition into r layers. g = V1 V2 Vr ⊕ ⊕···⊕ having the properties that g is graded and generated by V1. Explicitly, [V1,Vj] = Vj+1, j = 1,...,r 1 and [Vj,Vr] = 0, j = 1,...,r. − Let m = dim(Vj) and let X denote a left-invariant basis of Vj where 1 j r and j i,j 1 i m . The homogeneous dimension of G is defined as Q = r km . For≤simpl≤icity, we wi≤ll se≤t Xj= X . We call X the horizontal vector fields and cakll=t1heirkspan, denoted HG, i i,1 i { } P the horizontal bundle. We call X the vertical vector fields and refer to their span, i,j 2≤j≤r denoted VG, as the vertical bun{dle. }Then g = HG VG. In fact, the Lie algebra spans the whole tangent space of the group (TG = g). ⊕ Because of the stratification of the Lie algebra, there is a natural dilation on g. If X = r X where X Vk, then the dilation can be defined by δ (X) = r skX . It is k=1 k k ∈ s k=1 k worth noting here that while dilation mappings are defined on the Lie algebra, the mapping Pexp δ exp−1 gives the dilation on the group G. However, the same notationP, δ , will be used s s ◦ ◦ for both maps. Recall the definition of the exponential of a vector field, X. Fix a point p G. Let γ(t) be ∈ a curve such that γ(0) = p and dγ(t) = X . (The existence of such a curve is guaranteed dt γ(t) 4 PRICE by the theorem of existence and uniqueness of systems of ordinary differential equations.) The exponential map is defined as exp (X) = γ(1), or more generally, exp (tX) = γ(t). p p For Lie groups, p is taken as the identity element, and the exponential map provides a means of relating the Lie algebra to the group itself. exp : g G. → And in the special case of Carnot groups, the exponential map is an analytic diffeomorphism, and the Baker-Campbell-Hausdorff formula holds for all X and Y in g. For a proof see [17, Theorem 1.2.1]. The Baker-Campbell-Hausdorff formula (BCH) gives a more complete picture of how the exponential map relates the algebra to the group. Take two vector fields in the Lie algebra, X and Y. The BCH is given by explicitly solving for the vector field Z in the equation exp(Z) = exp(X) exp(Y). · Z = log(exp(X) exp(Y)) · 1 1 1 (2.1) = X +Y + [X,Y]+ [X,[X,Y]] [Y,[X,Y]]+ 2 12 − 12 ··· The formula continues with higher order commutators. For nilpotent Lie groups, it is clear the summation will eventually terminate. Definition 3. Let X ,...,X be a basis for a nilpotent Lie algebra g, and consider a map 1 n { } Ψ : Rn G → Ψ(s ,...,s ) = exp(s X +...+s X ). 1 n 1 1 n n The coordinates given by the map Ψ are called exponential coordinates or canonical coordinates of the first kind. Canonical coordinates of the second kind are defined similarly by taking Ψ(s ,...,s ) = exp(s X ) exp(s X ). 1 n 1 1 n n ··· The foundation has now been laid to give a few examples of Carnot groups. Example 4. The Heisenberg group, Hn, is a step 2 Carnot group whose underlying manifold is R2n+1. Taking x,x′ R2n and t,t′ R, the group operation is given by ∈ ∈ n (x,t) (x′,t′) = (x+x′,t+t′ +2 (x′x x x′ )). · i n+i − i n+i i=1 X The vector fields below form a left-invariant basis for the Lie algebra, h = V1 V2. ⊕ 1 1 X = ∂ x ∂ , X = ∂ + x ∂ for i = 1,...,n i xi − 2 n+i t i+n xn+i 2 i t and T = ∂ t The horizontal bundle is given by V1 = span X ,...,X , and the vertical bundle is then 1 2n { } V2 = span T leading to a homogeneous dimension of 2n+2. { } Example 5. The Engel group, K , is an example of a step 3 Carnot group with a homogeneous 3 dimension of 7. See [17] and [18] for more information. This group has an underlying manifold of R4, and the group operation is given by x x′ = (x +x′, x +x′, x +x′ +A , x +x′ +A ) · 1 1 2 2 3 3 3 4 4 4 POINTWISE SCHAUDER ESTIMATES 5 where 1 A = (x x′ x x′) 3 2 1 2 − 2 1 and 1 1 A = (x x′ x x′)+ (x2x′ x x′(x +x′)+x (x′)2). 4 2 1 3 − 3 1 12 1 2 − 1 1 2 2 2 1 The Lie algebra can be graded as g = V1 V2 V3 by letting V1 = span X ,X , V2 = 1 2 ⊕ ⊕ { } span X , and V3 = span X . Using the Baker-Campbell-Hausdorff formula, expressions for 3 4 { } { } the vector fields can be found. x x x x 2 3 1 2 X = ∂ ∂ + ∂ 1 x1 − 2 x3 − 2 12 x4 (cid:16) (cid:17) x x2 X = ∂ + 1∂ + 1∂ 2 x2 2 x3 12 x4 x 1 X = ∂ + ∂ 3 x3 2 x4 X = ∂ 4 x4 Notice that [X ,X ] = X and [X ,X ] = X and all other commutators are trivial. 1 2 3 1 3 4 For demonstrative purposes, return to the first Heisenberg group, H1. This group is of great interest and widely studied not only because of its appearance in applications but also because there are only two 3 dimensional simply connected nilpotent Lie groups: H1 and R3. It is simple to check the left-invariance of the vector fields for H1. Let f(x,y,t) be a left translation by (a,b,c). f(x,y,t) = (x+a,y +b,t+c+1/2(ay bx)) − The differential is 1 0 0 (2.2) df = 0 1 0 .   b/2 a/2 1 −   1 Consider first the vector field X = ∂ (y/2)∂ = 0 . If we are to first take the X 1 x t 1 −   y/2 − derivative at a point p = (x,y,t) and then apply theleft translation in the tangent space, we get df X = ∂ +( b/2 y/2)∂ . On the other hand, if we left translate the point and then find 1 x t · − − the derivative at the translated point, we get X f(x,y,t) = ∂ +( b/2 y/2)∂ . Therefore 1 x t ◦ − − f X = X f showing X is left-invariant. The same can be done for the other vector fields. ∗ 1 1 ◦ A nice derivation of the vector fields is given in [14]. For H1, the underlying manifold is R3, but X and X do not span all of the tangent space. 1 2 The commutator, T = ∂ , recovers the missing direction. This group is said to satisfy Hör- t mander’s condition. In fact, Carnot groups in general satisfy Hörmander’s condition, meaning that the basis of vector fields along with all of their commutators up to some finite step will span the entire tangent space. This property is essential when considering Carnot groups as sub-Riemannian manifoldsbecause itensures thatany two pointsinthegroupcanbeconnected by a path that lies entirely in the span of V1. This type of path is referred to as a horizontal path. To be more precise, stated below is the fundamental theorem. Theorem 6. (Chow’s Theorem) If a smooth distribution satisfies Hörmander’s condition at some point p, then any point q which is sufficiently close to p can be joined to p by a horizontal curve. 6 PRICE Because the L2 energy required to travel only along directions in vertical bundle is infinite, the horizontal vector fields give the so called “admissible" directions. Getting from one point to another requires traveling along a horizontal curve, and Chow’s theorem alleviates any concern that we might not be able to find an appropriate path. We will always be able to get there by moving along horizontal curves so long as the vector fields satisfy Hörmander’s finite rank condition. Carnot groups can also be viewed as sub-Riemannian manifolds, and it is always possible to define g a Riemannian metric with respect to which the Vj are mutually orthogonal. A curve γ : [0,1] G is horizontal if the tangent vector γ′(t) lies in V1 for all t. The Carnot- → Carathéodory distance (CC-distance) can now be defined. Definition 7. Let p,q G. ∈ 1 m1 1/2 d (p,q) = inf γ′(t),X 2 dt , cc i|γ(t) g Z0 i=1 ! X(cid:10) (cid:11) where the infimum is taken over all horizontal curves γ such that γ(0) = p, γ(1) = q and , h· ·ig denotes the left invariant inner product on V1 determined by the metric g. Chow’s Theorem gives the existence of the horizontal curve, γ. Another consequence of his work is that the CC-distance is finite for connected groups. The CC-distance is not a true distance in that it lacks the triangle inequality. However it does satisfy the quasi-triangle inequality meaning there exists a positive constant, A, depending on the group G such that d (x,y) A(d (x)+d (y)). cc cc cc ≤ We will also make use of the distance defined by the gauge norm. Let x be the coordinates i,k for a point x G, then ∈ r m1 (2.3) x 2r! = x 2r!/k. i,k | | | | k=1 i=1 XX For x,y G, we then let d(x,y) = xy−1 as defined above. ∈ | | The gauge distance is equivalent to the CC-distance but has the advantage of being a Lips- chitz function. We say they are equivalent due to the fact that there exists a constant, a(G), dependent on the group such that a−1d (x,y) d(x,y) ad (x,y). cc cc ≤ ≤ For a proof, as well as other properties of these metrics see [35]. 3. The Parabolic Setting 3.1. Parabolic Distance, Balls, and Cylinders. Throughout this paper, the relevant space is G R, where x is reserved as a space variable in G and t is thought of as a time variable in R. × This is also a Carnot group where time derivatives appear in the first layer of the stratification. However, homogeneity considerations of the operator would dictate the time derivative should have weight 2, and a different dilation mapping would be needed rather the natural one given previously. An alternative viewpoint is the one given by Rothschild and Stein. In [37], they referred to this situation (where the algebra is generated by a vector field in the second layer, X , in addition to the vector fields X ,...,X in the first layer) as a type II stratified group. o 1 m1 POINTWISE SCHAUDER ESTIMATES 7 This document mostly keeps to the viewpoint of separately dealing with G and R to get results on the product space. Therefore we define a new dilation mapping δ′ : G R G R s × → × given by δ′(x,t) = (δ (x),s2t). s s With that being said, the parabolic distance is defined. Definition 8. Let (x,t),(y,s) G R. The parabolic distance is ∈ × d ((x,t),(y,s)) = (d(x,y)2 + t s )1/2. p | − | Because G is stratified, one can always find such a homogeneous norm d(x,y) and a dilation mapping δ (x) such that d(δ (x),δ (y)) = sd(x,y). Both the gauge distance and CC-distance s s s fulfill this requirement. This is the appropriate distance for the dilation chosen since d (δ′(x,t)) = (d(δ x)2 + s2t )1/2 = sd (x,t) p s s p as desired. For sake of simplicity, we will use (x(cid:12),t)((cid:12)y,s)−1 to denote the parabolic distance | (cid:12) (cid:12) | between the points (x,t) and (y,s), and (x,t) to be the parabolic distance between (x,t) and | | the origin. Using the quasi-triangle inequality for d(x,y) on the group (with constant A), the analogue can be shown for d with the same constant. p Proposition 9. There exists a constant, A depending on the group, G, such that for all points (x,t),(y,s),(z,τ) G R the inequality holds. ∈ × (3.1) (x,t)(y,s)−1 A (x,t)(z,τ)−1 + (z,τ)(y,s)−1 . | | ≤ | | | | Proof. If we can show that (cid:0) (cid:1) d(x,y)2+ t s C d(x,z)2 + t τ +d(z,y)2+ τ s , | − | ≤ | − | | − | then we are done since (cid:0) (cid:1) d(x,y)2+ t s C d(x,z)2 + t τ +d(z,y)2 + τ s | − | ≤ | − | | − | C((cid:0)d(x,z)2 + t τ +2 d(x,z)2 + t (cid:1)τ d(z,y)2 + τ s ≤ | − | | − | | − | +d(z,y)2 + τ s ) p p | − | 2 = C d(x,z)2 + t τ + d(z,y)2 + τ s . | − | | − | (cid:16)p p (cid:17) Taking the square root of both sides, would give the desired inequality d(x,y)2+ t s √C d(x,z)2 + t τ + d(z,y)2+ τ s . | − | ≤ | − | | − | If A2 > 1pin the quasi-triangle inequ(cid:16)aplity for d(x,y), we getp (cid:17) d(x,y)2+ t s A2d(x,z)2 + t τ +A2d(z,y)2 + τ s | − | ≤ | − | | − | = A2 d(x,z)2 + t τ +d(z,y)2+ τ s | − | | − | +(1 A2) t τ +(1 A2) τ s (cid:0) (cid:1) − | − | − | − | A2 d(x,z)2 + t τ +d(z,y)2+ τ s . ≤ | − | | − | If A2 < 1, we actually have a true tr(cid:0)iangle inequality by adding positive(cid:1)additional terms (1 A2)d(x,z)2 and (1 A2)d(z,y) to the right hand side. (cid:3) − − 8 PRICE We also use the following form of reverse triangle inequality, (3.2) (x,t) A (y,s) A (x,t)(y,s)−1 . | |− | | ≤ | | Definition 10. Choosing d(x,z) to be the gauge distance defined earlier, the set (x,t) = (z,τ) : t τ < r2 and d(x,z) < r r Q | − | is known as the parabolic cylinder.(cid:8) (cid:9) Let B (x) denote the CC-ball on G, then it is easy to see that (x,t) = rQ+2 B (0) r r 1 |Q | | | where the measures indicated are Lebesgue measures. A simple computation gives B (x) = r B (0) rQ, and consequently (x,t) = rQ+2 B (0) . The Jacobian determinant of δ|: G | G 1 r 1 r | | |Q | | | → is simply rQ, so the calculation follows by a change of variable. See [2] for this calculation and other properties of the CC-balls. Properties of the parabolic balls and parabolic cylinders are nicely explained in [7]. Sometimes it is convenient to use the notation (x,t) = B (x) Λ (t) where Λ (t) = r r r r τ R : t τ < r2 . Q × { ∈ | − | } We can also define a parabolic ball to be the set B ((x ,t ),r) := (y,s) : (x ,t )(y,s)−1 < r . p o o o o | | Its size is comparable to the parabolic cy(cid:8)linder, but they are not the(cid:9)same set. 3.2. Group Polynomials. In this section, definitions and terminology regarding group poly- nomials are given followed by several results regarding Taylor polynomials. For more details see [21]. Let l N and consider a multi-index, I = [(i ,k ),(i ,k ),...,(i ,k )], where 1 k r and 1 1 2 2 l l j 1 i ∈k . Derivatives of a smooth function, f, defined in G will be denoted as≤ ≤ ≤ j ≤ mj XIf = X X ...X f i1,k1 i2,k2 il,kl wheretheorderofthederivative is I = l k .Throughoutthispaper, weareonlyconcerned | | j=1 j with derivatives appearing in the first layer of the stratification meaning the order will simply P be the number of vector fields applied to the function. Definition 11. A group polynomial on G R is a function that can be expressed in exponential × coordinates as P(x,t) = a xItβ I,β I,β X where I = (i )k=1...r and β and a are real numbers, and j,k j=1...mk I,β xI = xij,k, j,k j=1...mYk;k=1...r or equivalently, P(x,t) = a (x,t)d I,β d X where (x,t)d = xItβ and I +2β = d. | | The homogeneous degree of the monomial xI is given by r mk I = ki , j,k | | k=1 j=1 XX POINTWISE SCHAUDER ESTIMATES 9 and the homogeneous degree of (x,t)d is (as expected) d = I + 2β. We will notate the set | | of polynomials of homogeneous degree not exceeding d as . To avoid tedious language, the d P homogeneous degree will simply be called the degree of the polynomial. Definition 12. If h C∞(G R) and k is a positive integer, then the kth order Taylor ∈ 0 × polynomial P of h at the origin is the unique polynomial of homogeneous order at most k such k that XIDlP (0,0) = XIDlh(0,0) t k t for all I +2l k. | | ≤ We will need several results regarding Taylor polynomials and their remainders. The upcom- ing three of which appear in [21, pp.33-35]. Theorem 13. (Stratified Mean Value Theorem) Let G be stratified. There exists constants c > 0 and b > 0 such that for every f C1 and for all x,y G, ∈ ∈ f(xy) f(x) c y sup X f(xz) j | − | ≤ | | | | |z|≤b|y| 1≤j≤m1 where X is in the first layer of the stratification, and is a homogeneous norm on G. j |·| Theorem 14. (Stratified Taylor Inequality) Let G be stratified. For each positive integer k, there exists a constant c > 0 such that for every f Ck and for all x,y G, k ∈ ∈ f(xy) P (y) c y k sup XIf(xz) XIf(x) x k | − | ≤ | | | − | |z|≤bk|y| |I|=k where P (y) is the left Taylor Polynomial of f at x of homogeneous degree k and b is as in the x Stratified Mean Value Theorem. Corollary 15. If k 1, then there exists positive constants C and b (independent of k) such k that for every f C≥k+1(G) and all x,y G there holds ∈ ∈ f(xy) P (y) C y k+1 sup XIf(xz) x k | − | ≤ | | | | |z|≤bk|y| |I|=k+1 where P (y) is the kth order left Taylor polynomial of f at the point x. x Now results analogous to those above will be provided, incorporating the time derivatives and the parabolic distance. To this end, we will first define some special classes of functions. We will say that f C0(Ω) if f is continuous on the open set Ω G R with respect to the ∈ ∗ ∈ × parabolic distance, and we define f(x,t) f(x,τ) C1(Ω)= f C0(Ω) : X f C0(Ω) for i = 1,...,m and | − | < , t = τ . ∗ ∈ ∗ i ∈ ∗ 1 t τ 1/2 ∞ 6 (cid:26) | − | (cid:27) For any positive integer k 2, we define ≥ Ck(Ω) = f Ck−1(Ω) : X f Ck−1(Ω) for i = 1,...,m and D f Ck−2(Ω) . ∗ ∈ ∗ i ∈ ∗ 1 t ∈ ∗ Roughly speak(cid:8)ing, a function in the set Ck(Ω) will have continuous derivatives up to(cid:9)order k ∗ (with respect to the parabolic distance) as well as “half” derivatives in the t variable up to order k 1 in keeping with the idea that one time derivative is equivalent to two spatial derivatives. − 10 PRICE Lemma 16. Suppose g C1(G R), then for all (y,s) G R/(0,0) there exists a positive ∈ ∗ × ∈ × constant C such that g(y,s) g(0,0) C (y,s) sup X g(z,τ) ). i | − | ≤ | | | | |(z,τ)|≤b|(y,s)| i,=1,...,m1 Proof. Notice first that g(y,s) g(0,0) g(y,s) g(y,0) + g(y,0) g(0,0) . | − | ≤ | − | | − | The first term can be estimated using the “half” derivative while the second term requires the Stratified Mean Value Theorem. Together we get the following: g(y,s) g(0,0) C s 1/2 +C y sup X g(z,0) 1 i | − | ≤ | | | | | | |z|≤b|y| i,=1,...,m1 where 0 < τ < s. Here s isreferringto theEuclidean distanceand y isanyhomogeneous normonG. Clearly, | | | | s 1/2 (y,s) and y (y,s) giving | | ≤ | | | | ≤ | | g(y,s) g(0,0) C (y,s) sup X g(z,0) i | − | ≤ | | | | |z|≤b|y| i,=1,...,m1 Taking the appropriate supremum gives the conclusion. (cid:3) Now we give a parabolic version of the Stratified Taylor Inequality. Theorem 17. Suppose G is stratified. For every positive integer k, there exists positive con- stants C and b depending on k such that for all f Ck(G R) and all (x,t),(y,s) G R, ∈ ∗ × ∈ × f(y,s) P (y,s) C (y,s) k sup XIDlf(z,τ) XIDlf(0,0) | − k | ≤ k| | | t − t | |(z,τ)|≤bk|(y,s)| |I|+2l=k where P (y,s) is the kth order Taylor polynomial about the origin. k Proof. The method of proof is similar to the proof of the Stratified Taylor Inequality in [21]. Fix a (y,s) G R and let g(x,t) = f(x,t) P (y,s).By definition of the Taylor Polynomial k ∈ × − for all I +2l k, | | ≤ XIDlg(0,0) = XIDlf(0,0) XIDlP (y,s) = 0. t t − t k By induction on n, we will show for all 0 < n k ≤ (3.3) XJDpg(y,s) C (y,s) n sup XIDlf(z,τ) XIDlf(0,0) | t | ≤ n| | | t − t | |(z,τ)|≤bn|(y,s)| |I|+2l=k where J +2p = k n. The case n = k will give the conclusion. | | − To begin, we will see the case n = 0 is trivial since J +2p = k and | | XJDpg(y,s) = XJDpf(y,s) XJDpP (y,s) | t | | t − t k | = XJDpf(y,s) XJDpf(0,0) | t − t | sup XIDlf(z,τ) XIDlf(0,0) . ≤ | t − t | |(z,τ)|≤|(y,s)| |I|+2l=k Suppose (3.3) is true for n = k 1, and consider the case n = k. Using Lemma 16, we have − g(y,s) g(0,0) = g(y,s) | − | | | C (y,s) sup X g(z,τ) . i ≤ | | | | |z|≤b|y| i,=1,...,m1