The $C^{\a}$ regularity of a class hypoelliptic ultraparabolic equations
aa r X i v : . [ m a t h . A P ] M a y The C α regularity of a class of hypoellipticultraparabolic equations WANG Wendong and ZHANG Liqun ∗ Institute of Mathematics, AMSS, Academia Sinica, Beijing
Abstract
We obtained the C α continuity for weak solutions of a class of ul-traparabolic equations with measurable coefficients of the form ∂ t u = P m i,j =1 X i ( a ij ( x, t ) X j u ) + X u . The result is proved by simplifyingand generalizing our earlier arguments for the C α regularity of homo-geneous ultraparabolic equations.keywords: Hypoelliptic, ultraparabolic equations, C α regularity We are concerned with the regularity of a class of ultraparabolic equa-tions. One of the typical example of the ultraparabolic equations is thefollowing equation(1 . ∂ u∂ t + y ∂ u∂ x − u ∂ u∂ y = 0 , which is of strong degenerated parabolic type equations, more precisely, anultraparabolic type equation. On the other hand, the equation (1.1), if we ∗ The research is partially supported by the Chinese NSF under grant 10325104. Email:[email protected] and [email protected] u , has the divergent form. The recent paperof Pascucci and Polidoro [18], has proved that the Moser iterative methodstill works for a class of ultraparabolic equations with measurable coefficientswhich are called homogeneous Kolmogorov-Fokker-Planck equations(or KFP-equations). By the same technique, Cinti, Pascucci, Polidoro [3] consider anonhomogeneous KFP-equations, and Cinti, Polidoro [4] deal with a moregeneral ultraparabolic equation which we will concentrate on in this paper.Their result shows that for a non-negative sub-solution u of the ultraparabolicequation they considered, the L ∞ norm of u is bounded by the L p norm( p ≥ C α class, then u issmooth. The second author [23] has proved C α property of weak solutionsby Kruzhkov’s approach for homogeneous KFP-equations, and the authorsdeal with nonhomogeneous KFP-equations in [20]. By simplifying the cut-offfunction and generalizing our earlier arguments, we are able to prove the C α regularity for weak solutions of more general ultraparabolic equations. Weprove a Poincar´e type inequality for non-negative weak sub-solutions of (1.2).Then we apply the inequality to obtain a local priori estimate which impliesthe H¨older estimates.Consider a class of ultraparabolic operator on R N +1 :(1 . Lu ≡ m X i,j =1 X i ( a ij ( x, t ) X j u ) + X u − ∂ t u = 0 , where ( x, t ) = z ∈ R N +1 , 1 ≤ m ≤ N , and X j ’s are smooth vector fields on R N , for j = 0 , , · · · , m .We follow the notations as in [4]. Let A = ( a ij ) m × m , and A be the2dentity matrix of m × m . Put Y = X − ∂ t , and denote(1 . L := m X k =1 X k + Y A curve γ : [0 , T ] → R N +1 is called L - admissible , if it is absolutely contin-uous and satisfies γ ′ ( s ) = m X k =1 λ k ( s ) X k ( γ ( s )) + Y ( γ ( s )) , a . e . in [0 , T] , for suitable bounded measurable functions λ ( s ) , . . . , λ m ( s ) . We make the following assumptions on the operator L :[ H ] : the coefficients a ij , 1 ≤ i, j ≤ m , are real valued, measurablefunctions of ( x, t ). Moreover, a ij = a ji ∈ L ∞ ( R N +1 ) and there exists a λ > λ m X i =1 ξ i ≤ m X i,j =1 a ij ( x, t ) ξ i ξ j ≤ λ m X i =1 ξ i for every ( x, t ) ∈ R N +1 , and ξ ∈ R m . X = P Ni =1 b j ( x ) ∂ x j with smoothfunctions b j ( x );[ H ] : there exists a homogeneous Lie group G ≡ ( R N +1 , ◦ , δ µ ) such that(i) X , . . . , X m , Y are left translation invariant on G ,(ii) X , . . . , X m are δ µ -homogeneous of degree one and Y is δ µ -homogeneousof degree two;[ H ] : for every ( x, t ), ( ξ, τ ) ∈ R N +1 with t > τ , there exists an L - admissible path γ : [0 , t − τ ] → R N +1 such that γ (0) = ( x, t ), γ ( t − τ ) = ( ξ, τ ).The requirements of [ H ] and [ H ] ensure that the operator L satisfiesthe well-known H¨ormander’s hypoellipticity condition by Kogoj and Lan-conelli [8]. We refer to [4] and [8] for more details on the hypoelliptic typeoperator on R N +1 . 3he Schauder type estimate of (1.2) has been obtained, for example,Lunardi [12] and Manfredini [14]. Besides, the regularity of weak solutionshave been studied by Bramanti, Cerutti and Manfredini [2], Polidoro andRagusa [19], Manfredini and Polidoro [13] assuming a weak continuity on thecoefficient a ij . It is quite interesting whether the weak solution has H¨olderregularity under the assumption [ H ]. One of the approaches to the H¨olderestimates is to obtain the Harnack type inequality. In the case of ellipticequations with measurable coefficients, the Harnack inequality is obtainedby J. Moser [15] via an estimate of BMO functions due to F. John andL. Nirenberg together with the Moser iteration method. J. Moser [16] alsoobtained the Harnack inequality for parabolic equations with measurablecoefficients by generalizing the John-Nirenberg estimates to the paraboliccase. Also De Giorgi developed an approach to obtain the H¨older regularityfor elliptic equations. Another approach to the H¨older estimates is given byS. N. Kruzhkov [9], [10] based on the Moser iteration to obtain a local prioriestimate, which provides a short proof for the parabolic equations. Nash [17]introduced another technique relying on the Poincar´e inequality and obtainedthe H¨older regularity.Let X be the gradient with respect to the variables X , X , · · · , X m , and Xu = ( X u, X u, · · · , X m u ) T . We say that u is a weak solution if it satisfies(1.2) in the distribution sense, that is for any φ ∈ C ∞ (Ω), Ω is a open subsetof R N +1 , then(1 . Z Ω φY u − ( Xu ) T AXφ = 0 , where u , Xu , Y u ∈ L (Ω) . Our main result is the following theorem.
Theorem 1.1
Under the assumptions [ H ] ∼ [ H ] , the weak solution of(1.2) is H¨older continuous. Some Preliminary and Known Results
We follow the earlier notations to give some basic properties. For the moredetails of the subject, we refer to Cinti and Polidoro [4], Kogoj and Lanconelli[8], or Bonfiglioli, Lanconelli and Uguzzoni[1].We say a Lie group G = ( R N +1 , ◦ ) is homogeneous if a family of dilations( δ µ ) µ> exists on G and is an automorphism of the group: δ µ ( z ◦ ζ ) = δ µ ( z ) ◦ δ µ ( ζ ), for all z, ζ ∈ R N +1 and µ >
0, where δ µ = diag ( µ α , µ α , . . . , µ α N , µ ) , for 1 ≤ i ≤ N , α i is a positive integer, and α ≤ α ≤ · · · ≤ α N . Moreover, the dilation δ µ induces a direct sum decomposition on R N , and R N = V ⊕ , · · · , ⊕ V k . If we denote x = x (1) + x (2) + · · · + x ( k ) with x ( j ) ∈ V j ,then δ µ ( x, t ) = ( D µ x, µ t ) , where D µ ( x (1) + x (2) + · · · + x ( k ) ) = ( µx (1) + µ x (2) + · · · + µ k x ( k ) ) . Let Q = dimV + 2 dimV + · · · + kdimV k , then the number Q + 2 is usually called the homogeneous dimension of( R N +1 , ◦ ) with respect to the dilation δ µ .A real function f ( x ) defined on R N is called δ µ - homogeneous of degree m ∈ R , if f ( x ) does not vanish identically and, for every x ∈ R N and µ > f ( δ µ ( x, λ m f ( x ) . X is called δ µ - homogeneous of degree m ∈ R , if for every φ ∈ C ∞ ( R N ), x ∈ R N , and µ >
0, it holds X ( φ ( δ µ ( x, µ m ( Xφ )( δ µ ( x, . The norm in R N +1 , related to the group of translations and dilation tothe equation is defined by || ( x, t ) || = r, if r is the unique positive solution to the equation x r α + x r α + · · · + x N r α N + t r = 1 , where ( x, t ) ∈ R N +1 \ { } and by [ H ] and H¨ormander’s hypoellipticity con-dition, we attain α = · · · = α m = 1 , < α m +1 ≤ · · · ≤ α N . And || (0 , || = 0. Obviously k δ µ ( x, t ) k = µ k ( x, t ) k , for all ( x, t ) ∈ R N +1 . The quasi-distance in G is d ( z, ζ ) := k ζ − ◦ z k , ∀ z, ζ ∈ R N +1 , where(2 . ζ − ◦ z = ( S ( x, t, ξ, τ ) , t − τ )and S ∈ R N is smooth (see [8]). Moreover, for every compact domain K ∈ R N +1 , there exists a positive constant C K such that(2 . C − K | z − ζ | ≤ d ( z − ζ ) ≤ C K | z − ζ | k , ∀ z, ζ ∈ K | · | denotes the usual Euclidean modulus (see for instance, Prop 11.2in [7]).The ball at a point ( x , t ) is defined by B r ( x , t ) = { ( x, t ) | || ( x , t ) − ◦ ( x, t ) || ≤ r } , and B − r ( x , t ) = B r ( x , t ) ∩ { t < t } . For convenience, we sometimes use the cube instead of the balls. The cubeat point (0 ,
0) is given by C r (0 ,
0) = { ( x, t ) | | t | ≤ r , | x | ≤ r α , · · · , | x N | ≤ r α N } . It is easy to see that there exists a constant Λ such that C r Λ1 (0 , ⊂ B r (0 , ⊂ C Λ r (0 , , where Λ only depends on Q and N .We recall L = P m k =1 X k + Y, whose fundamental solution Γ ( · , ζ ) withpole in ζ ∈ R N +1 is smooth out of the diagonal of R N +1 × R N +1 , has thefollowing properties:(2 . i ) Γ ( z, ζ ) = Γ ( ζ − ◦ z,
0) = Γ ( ζ − ◦ z ) , ∀ z, ζ ∈ R N +1 , z = ζ ;( ii ) Γ ( z, ζ ) ≥ , and Γ ( x, t, ξ, τ ) > t > τ ;( iii ) R R N Γ ( x, t, ξ, τ ) dx = R R N Γ ( x, t, ξ, τ ) dξ = 1 , if t > τ ;( iv ) Γ ( δ µ ◦ z ) = µ − Q Γ ( z ) , ∀ z = 0 , µ > .
4) Γ ( z, ζ ) ≤ C || ζ − ◦ z || − Q , z, ζ ∈ R N +1 (see [4] or [8]).A weak sub - solution of (1.2) in a domain Ω is a function u such that u , Xu , Y u ∈ L loc (Ω) and for any φ ∈ C ∞ (Ω), φ ≥ . Z Ω φY u − ( Xu ) T AXφ ≥ . A result of Cinti and Polidoro obtained by using the Moser’s iterativemethod (see Prop 4.4 in [4]) states as follows.
Lemma 2.1
Let u be a non-negative weak sub-solution of (1.2) in Ω . Let ( x , t ) ∈ Ω and B − r ( x , t ) ⊂ Ω and p ≥ . Then there exists a positiveconstant C which depends only on the operator L such that, for < r ≤ .
6) sup B − r ( x ,t ) u p ≤ Cr Q +2 Z B − r ( x ,t ) u p , provided that the last integral converges. We make use of a classical potential estimates (see (1.11) in [5]) here toprove the Poincar´e type inequality.
Lemma 2.2
Let ( R N +1 , ◦ ) is a homogeneous Lie group of homogeneous di-mension Q + 2 , α ∈ (0 , Q + 2) and G ∈ C ( R N +1 \ { } ) be a δ µ -homogeneousfunction of degree α − Q − . If f ∈ L p ( R N +1 ) for some p ∈ (1 , ∞ ) , then G f ( z ) ≡ Z R N +1 G ( ζ − ◦ z ) f ( ζ ) dζ , is defined almost everywhere and there exists a constant C = C ( Q, p ) suchthat (2 . || G f || L q ( R N +1 ) ≤ C max || z || =1 | G ( z ) | || f || L p ( R N +1 ) , where q is defined by q = 1 p − αQ + 2 . orollary 2.1 Let f ∈ L ( R N +1 ) , and recall the definitions in [3] Γ ( f )( z ) = Z R N +1 Γ ( z, ζ ) f ( ζ ) dζ , ∀ z ∈ R N +1 , and Γ ( X j f )( z ) = − Z R N +1 X ( ζ ) j Γ ( z, ζ ) f ( ζ ) dζ , ∀ z ∈ R N +1 , where j = 1 , · · · , m , then exists a positive constant C = C ( Q ) such that (2 . k Γ ( f ) k L k ( R N +1 ) ≤ C k f k L ( R N +1 ) , and (2 . k Γ ( X j f ) k L k ( R N +1 ) ≤ C k f k L ( R N +1 ) , where ˜ k = 1 + Q − , k = 1 + Q , and j = 1 , · · · , m . We may consider the local estimate at a ball centered at (0 , a ij is constant.The key point in our argument is to obtain a Poincar´e type inequality. Thenby using the Poincar´e type inequality, we prove the following Lemma 3.5which is essential in the oscillation estimates in Kruzhkov’s approaches inparabolic case. Then the C α regularity result follows easily by the standardarguments. We follow the same route as [23] and [20], but the idea is moresimple and technical. We give them together for completeness.For convenience, we let x ′ = ( x , · · · , x m ) and x = ( x ′ , x ). We considerthe estimates in the following cube, instead of B − r , C − r = { ( x, t ) | − r ≤ t < , | x ′ | ≤ r, | x m +1 | ≤ ( λ ′ r ) α m , · · · , | x N | ≤ ( λ ′ r ) α N } , λ ′ > K r = { x ′ | | x ′ | ≤ r } ,S r = { x | | x m +1 | ≤ ( λ ′ r ) α m , · · · , | x N | ≤ ( λ ′ r ) α N } . Let 0 < α, β < t and h , we denote N t,h = { ( x ′ , x ) | ( x ′ , x ) ∈ K βr × S βr , u ( · , t ) ≥ h } . By the homogeneousness of X j , j = 1 , · · · , m , we can deduce X j = m X i =1 C ( j ) i ∂ x i + X i>m C ( j ) i ( x ) ∂ x i , where C ( j ) i is a constant for i ≤ m and C ( j ) i ( x ) is a polynomial of homoge-neous degree α i − i > m . Similarly X = X i>m b i ( x ) ∂ x i , where b i ( x ) is a polynomial of homogeneous degree α i −
2. In the followingdiscussions, we always assume r ≪
1, and that the constants C ( j ) i ( i ≤ m )and the coefficients of these polynomial functions b j ( x ) and C ( j ) i ( x )( i > m )are bounded by λ , since we can choose λ as a large constant. Moreover, allconstants dependant on m , k or Q will be denoted by dependence on B . Lemma 3.1
Suppose that u ( x, t ) ≥ be a solution of equation (1.2) in C − r centered at (0 , and mes { ( x, t ) ∈ C − r , u ≥ } ≥ mes ( C − r ) , then there exist constants α , β and h , < α, β, h < which only depend on B , λ and N such that for almost all t ∈ ( − αr , , mes {N t,h } ≥ mes { K βr × S βr } . roof: Let v = ln + ( 1 u + h ) , where h is a constant, 0 < h <
1, to be determined later. Then v at pointswhere v is positive, satisfies(3 . m X i,j =1 X i ( a ij ( x, t ) X j v ) − ( Xv ) T AXv + X v − ∂ t v = 0 . Let η ( s ) be a smooth cut-off function so that η ( s ) = 1 , for s < βr,η ( s ) = 0 , for s ≥ r. Moreover, 0 ≤ η ≤ | η ′ | ≤ − β ) r .Now we let η = η ( | x ′ | ) and η = Π j>m η j , where η j = η ( λ ′ | x j | αj ) for j > m .Multiplying η η to (3.1) and integrating by parts on K r × S r × ( τ, t )(3 . R K βr R S βr v ( t, x ′ , x ) dxdx ′ + λ R tτ R K r R S r η η | Xv | dxdx ′ dt ≤ C ( B,λ,N ) β Q (1 − β ) (1 + λ ′− + λ ′− ) | S βr | | K βr | + R tτ R K r R S r η η X vdxdx ′ dt + R K r R S r v ( τ, x ′ , x ) dxdx ′ ≤ C ( B,λ,N ) β Q (1 − β ) | S βr | | K βr | + R tτ R K r R S r η η X vdxdx ′ dt + R K r R S r v ( τ, x ′ , x ) dxdx ′ , a.e. τ, t ∈ ( − r , . Let I B ≡ Z K r Z S r η η X j>m b j ( x ) ∂ x j vdxdx ′ , . | I B | = | R K r R S r η P j>m ( b j ( x ) ∂ x j η ) vdxdx ′ |≤ C ( λ, N ) ln( h − ) R K r R S r P j>m | η ′ ( | x j | αj λ ′ ) | λ ′ | x j | αj − ( λ ′ r ) α j − ≤ C ( λ,N )(1 − β ) r λ ′ β − Q | S βr || K βr | ln( h − ) . Integrating by t to I B , we have(3 . R tτ R K r R S r η η X vdxdx ′ dt ≤ C ( λ,N )(1 − β ) λ ′ β − Q | S βr || K βr | ln( h − ) . We shall estimate the measure of the set N t,h . Let µ ( t ) = mes { ( x ′ , x ) | x ′ ∈ K r , x ∈ S r , u ( · , t ) ≥ } . By our assumption, for 0 < α < r mes ( S r ) mes ( K r ) ≤ Z − r µ ( t ) dt = Z − αr − r µ ( t ) dt + Z − αr µ ( t ) dt, that is Z − αr − r µ ( t ) dt ≥ ( 12 − α ) r mes ( S r ) mes ( K r ) , then there exists a τ ∈ ( − r , − αr ), such that(3 . µ ( τ ) ≥ ( 12 − α )(1 − α ) − mes ( S r ) mes ( K r ) . By noticing v = 0 when u ≥ , we have(3 . Z K r Z S r v ( τ, x ′ , x ) dxdx ′ ≤
12 (1 − α ) − mes ( S r ) mes ( K r ) ln( h − ) . Now we choose α (near zero), β (near one), and λ ′ large enough such that(3 . C ( λ, N )(1 − β ) λ ′ β − Q + 12 β Q (1 − α ) ≤ , . R K βr R S βr v ( t, x ′ , x ) dxdx ′ ≤ [ C ( B,λ,N ) β Q (1 − β ) + ln( h − )] mes ( K βr × S βr ) . When ( x ′ , ¯ x ) / ∈ N t,h , , we haveln( 12 h ) ≤ ln + ( 1 h + h ) ≤ v, then ln( 12 h ) mes ( K βr × S βr \ N t,h ) ≤ Z K βr Z S βr v ( t, x ′ , x ) dxdx ′ . Since C + ln( h − )ln( h − ) −→ , as h → , then there exists constant h such that for 0 < h < h and t ∈ ( − αr , mes ( K βr × S βr \ N t,h ) ≤ mes ( K βr × S βr ) . Then we proved our lemma.Let χ ( s ) be a smooth function given by χ ( s ) = 1 if s ≤ θ Q r,χ ( s ) = 0 if s > r, where θ Q < is a constant. Moreover, we assume that0 ≤ − χ ′ ( s ) ≤ − θ Q ) r , | χ ′′ ( s ) | ≤ Cr , and for any β , β , with θ Q < β < β < , we have | χ ′ ( s ) | ≥ C ( β , β ) r − > , β r ≤ s ≤ β r. For x ∈ R N , t ≤
0, we set Q = { ( x ′ , ¯ x, t ) | − r ≤ t ≤ , x ′ ∈ K rθ , | x j | ≤ ( rθ ) α j , j = m + 1 , · · · , N } . We define the cut off functions by φ ( x, t ) = χ ([ N X j = m +1 θ α j x j r α j − Q − C tr Q − ] Q ) ,φ ( x, t ) = χ ( θ | x ′ | ) , (3 . φ ( t, x ) = φ ( t, x ) φ ( x, t ) , where C > C r Q − ≥ | P j>m θ α j b j ( x ) x j r Q − α j | , for all z ∈ Q . Remark 3.1
By the definition of φ and the above arguments, it is easy tocheck that, for θ , r small enough and t ≤ (1) φ ( z ) ≡ , in B − θr ,(2) supp φ T { (x , t) | x ∈ R N , t ≤ } ⊂ Q ,(3) there exists α > θ, which depends on C , such that { ( x, t ) | − α r ≤ t < , x ′ ∈ K βr , ¯ x ∈ S βr } ⊆ supp φ, (4) < φ ( z ) < , for z ∈ { ( x, t ) | − α r ≤ t ≤ − θr , x ′ ∈ K βr , ¯ x ∈ S βr } . Lemma 3.2
Under the above notations, we have
Y φ ( z ) ≤ , for z ∈ Q . roof: Let [ P Nj = m +1 θ αj x j r αj − Q − C tr Q − ] be denoted by [ · · · ]. Then Y φ = χ ′ ([ · · · ] Q ) Q [ · · · ] Q − [ C r Q − + P j>m (2 θ α j b j ( x ) x j r Q − α j )]For the term b j ( x ) x j r Q − α j , since | b j ( x ) | ≤ C ( λ, N )( rθ ) α j − , we obtain | X j>m θ α j b j ( x ) x j r Q − α j | ≤ C ( λ, N ) θ r Q − . We can choose a positive constant C > , such that C ( λ, N ) θ < C , then Y φ ( z ) ≤ z ∈ Q ) holds.We sometimes abuse the notations of B − r and C − r , since there are equiva-lent. Now we have the following Poincar´e’s type inequality. Lemma 3.3
Let w be a non-negative weak sub-solution of (1.2) in B − . Thenthere exists a constant C , only depends on B, λ and N , such that for r <θ < . Z B − θr ( w ( z ) − I ) ≤ Cθ r Z B − rθ | Xw | , where I is given by (3 . I = sup B − θr [ I ( z ) + C ( z )] , and (3 . I ( z ) = Z B − rθ [ − Γ ( z, · ) wY φ ]( ζ ) dζ ,C ( z ) = Z B − rθ [ m X j =1 | X j φ | Γ ( z, · ) w ]( ζ ) dζ , where Γ is the fundamental solution, and φ is given by (3.10). roof: We represent w in terms of the fundamental solution of Γ , i.e. ϕ ( z ) = − Z R N +1 Γ ( z, ζ ) L ϕ ( ζ ) dζ , ∀ ϕ ∈ C ∞ ( R N +1 ) . By an approximation and the support of φ and Γ , for z ∈ B − θr , we have(3 . w ( z ) = R B − rθ [ h A X ( wφ ) , X Γ ( z, · ) i − Γ ( z, · ) Y ( wφ )]( ζ ) dζ = I ( z ) + I ( z ) + I ( z ) + C ( z ) , where I ( z ) are given by (3.13) and I ( z ) = Z B − rθ [ h ( A − A ) Xw, X Γ ( z, · ) i φ − Γ ( z, · ) h ( A + A ) Xw, Xφ i ]( ζ ) dζ ,I ( z ) = Z B − rθ [ h AXw, X (Γ ( z, · ) φ ) i − Γ ( z, · ) φY w ]( ζ ) dζ .C ( z ) = Z B − rθ [ h A Xφ, X Γ ( z, · ) i w + Γ ( z, · ) h A Xw, Xφ i ]( ζ ) dζ Note that supp φ T { τ ≤ } ⊂ Q ⊂ B − rθ , z ∈ B − θr and h A Xφ, X Γ ( z, · ) i vanishes in a small neighborhood of z . Integrating by parts we have C ( z ) = R B − rθ [ P m j =1 | X j φ | Γ ( z, · ) w ]( ζ ) dζ . From our assumption, w is a weak sub-solution of (1.2), and φ is a testfunction of this semi-cylinder. In fact, we let˜ χ ( τ ) = τ ≤ , − nτ ≤ τ ≤ /n, τ ≥ /n. Then ˜ χ ( τ ) φ Γ ( z, · ) can be a test function (see [4]). As n → ∞ , we obtain φ Γ ( z, · ) as a legitimate test function, and I ( z ) ≤
0. Then in B − θr ,0 ≤ ( w ( z ) − I ) + ≤ I ( z ) = I + I .
16y Corollary 2.1 we have(3 . || I || L ( B − θr ) ≤ C ( λ, N ) θr || I || L
2+ 4 Q ( B − θr ) ≤ C ( B, λ, N ) θr || Xw || L ( B − rθ ) . Similarly for I , || I || L ( B − θr ) ≤ |B − θr | − Q − Q +4 || I || L k ( B − θr ) ≤ C ( B, λ, N ) θ r || | Xw | | Xφ | || L ( B − rθ ) , where | Xφ | = | χ ′ ( θ | ξ ′ | ) θX ( | ξ ′ | ) | ≤ C ( B, λ, N ) θr , and | Xφ | ≤ | χ ′ | Q [ · · · ] Q − X ≤ i ≤ m ,j>m | C ( i ) j ( x ) θ α j x j r α j − Q | ≤ C ( B, λ, N ) θ Q r − , thus || I || L ( B − θr ) ≤ C ( B, λ, N ) θ r || Xw || L ( B − rθ ) . Then we proved our lemma.Now we apply Lemma 3.3 to the function w = ln + hu + h . If u is a weaksolution of (1.2), obviously w is a weak sub-solution. We estimate the valueof I given by Lemma 3.3. Lemma 3.4
Under the assumptions of Lemma 3.3, there exist constants λ , r and r < θ . λ only depends on constants α , β , λ , B , N , and φ , < λ < , such that for r < r (3 . | I | ≤ λ ln( h − ) . Proof:
We first come to estimate C ( z ) and as before, denote x = ( x ′ , ¯ x, t )and ζ = ( ξ ′ , ¯ ξ, τ ). Note z ∈ B − θr , we have | C ( z ) |≤ R B − rθ [ P m j =1 | X j φ | Γ ( z, · ) w ]( ζ ) dζ ≤ r sup ξ ∈ supp( Xφ ) P m j =1 | X j φ | ln( h − ) . ( By (iii) in (2 . | X j φ | , | X j φ | ≤ | X j φ | + 2 | X j φ X j φ | + | X j φ | , where | X j φ | = | θχ ′ ( θ | ξ ′ | ) ∂ ξ j | ξ ′ || ≤ θr − , | X j φ | ≤ Cθ − Q r − and | X j φ | = | χ ′ Q [ · · · ] Q − ( X i>m C ( j ) i ( ξ )2 ξ i θ α i r α i − Q ) | ≤ C ( B, λ, N ) θ Q r − , moreover, | X j φ | ≤ C ( B, λ, N ) θ Q r − . Hence(3 . | C ( z ) | ≤ C ( B, λ, N ) θ Q ln( h − ) = C ( B, λ, N ) θ α ln( h − )where α = Q > X = P j>m b j ( x ) ∂ x j , we know Y φ = φ Y φ . Now we let w ≡ z ∈ B − θr (3.14) gives,(3 .
18) 1 = R B − rθ [ − φ Γ ( z, · ) Y φ ]( ζ ) dζ + C ( z ) | w =1 . By Lemma 3.2,(3 . − φ Γ ( z, · ) Y φ ≥ , we only need to prove − φ Γ ( z, · ) Y φ has a positive lower bound in a domainwhich w vanishes, and this bound independent of r and small θ . So we canfind a λ , < λ <
1, such that this lemma holds and λ is independent of r and small θ. We observe that the support of χ ′ ( s ) is in the region θ Q r < s < r ,thus for some β ′ <
1, the set B − β ′ r \B −√ θr with θr /C ≤ | t | ≤ α r is containedin the support of φ φ ′ . Then we can prove that the integral of (3.19) on asubset of the domain B − β ′ r \ B −√ θr is lower bounded by a positive constant.18or z ∈ B − θr , 0 < α ≤ α and set(3 . ζ ∈ Z = { ( ξ, τ ) | − α r ≤ τ ≤ − α r , ξ ′ ∈ K βr , ¯ ξ ∈ S βr , w ( ξ, τ ) = 0 } , then | Z | = C ( α , β, B, λ, N ) r Q +2 by Lemma 3.1. We note that when ζ =( ξ, τ ) ∈ Z and θ is small, w ( ζ ) = 0 , φ ( ζ ) = 1, | χ ′ ([ · · · ] Q ) | ≥ C ( α , B, λ, N ) r − > . Consequently R Z [ − φ Γ ( z, · ) Y φ ]( ζ ) dζ = − R Z φ Γ ( z, · ) χ ′ ([ · · · ] Q ) Q [ · · · ] Q − [ C r Q − + P j>m (2 θ α j b j ( ξ ) ξ j r Q − α j )] dζ ≥ C ( B, λ, α , N ) R Z r Q − [ r Q ] Q − r − Γ ( ζ − ◦ z, dζ ≥ C ( B, λ, α , N ) R Z r − Γ ( ζ − ◦ z, dζ = C ( B, λ, α, β, N ) = C > , where we have used Γ ( z, ζ ) ≥ Cr − Q , as τ ≤ − α r and z ∈ B − θr . In fact, by(iv) in (2.3) one getΓ ( z, ζ ) = r − Q Γ ( S ( x, t, ξ, τ ) , t − τr ) , where α ≤ t − τr ≤ , we have Γ ( z, ζ ) ≥ C ( α ) r − Q . Then we can choose a small θ which is fixed from now on and r < θ , such that(3 . | I | ≤ (1 − C + C θ α ) ln( h − ) + C θ α ln( h − ) ≤ λ ln( h − )where 0 < r < r , 0 < λ <
1, depends on α , β , B , λ , N , and φ . Lemma 3.5
Suppose that u ( x, t ) ≥ is a solution of equation (1.2) in B − r centered at (0 , and mes { ( x, t ) ∈ B − r , u ≥ } ≥ mes ( B − r ) . Then there xist constant θ and h , < θ, h < which only depend on B , λ , λ and N such that u ( x, t ) ≥ h in B − θr . Proof:
We consider w = ln + ( hu + h ) , for 0 < h <
1, to be decided. By applying Lemma 3.3 to w and scaling, wehave − Z B − θr ( w − I ) ≤ C ( B, λ, N ) θβr |B − θr | Z B − βr | Xw | . Let ˜ u = uh , then ˜ u satisfies the conditions of Lemma 3.1. We can get similarestimates as (3.2), (3.5), (3.7) and (3.8), hence we have(3 . C ( B, λ, N ) θr |B − θr | R B − βr | Xw | ≤ C ( B, λ, N ) θr |B − θr | [ C ( B,λ,N ) β Q (1 − β ) + ln( h − )] mes ( K βr × S βr ) ≤ C ( θ, B, N, λ ) ln( h − ) , where θ has been chosen. By Lemma 2.1, there exists a constant, still denotedby θ , such that for z ∈ B − θr ,(3 . w − I ≤ C ( B, λ, N )(ln( h − )) . Therefore we may choose h small enough, so that C (ln( 1 h )) ≤ ln( 12 h ) − λ ln( 1 h ) , then (3.16) and (3.23) gives sup B − θr h u + h ≤ h , B − θr u ≥ h , then we have finished our proof. Proof of Theorem 1.1.
We may assume that M = sup B − r (+ u ) =sup B − r ( − u ), otherwise we replace u by u − c , since u is bounded locally.Then either 1 + uM or 1 − uM satisfies the assumption of Lemma 3.5, and wesuppose 1 + uM does it, thus Lemma 3.5 implies existing h > B − θr (1 + uM ) ≥ h , i.e. u ≥ M ( h − Osc B − θr u ≤ M − M ( h − ≤ (1 − h Osc B − r u, which implies the C α regularity of u near point (0 ,
0) by the standard iterationarguments. By the left invariant translation group action, we know that u is C α in the interior. References [1] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,
Stratified Lie Groupsand Potential Theory for their Sub-Laplacians , Springer-VerlagBerlin Heidelberg, 2007[2] M. Bramanti, M. C. Cerutti and M. Manfredini. L p estimates forsome ultraparabolic equations , J. Math. Anal. Appl., 200 (2) 332-354 (1996).[3] C. Cinti, A. Pascucci and S. Polidoro, Pointwise estimates for so-lutions to a class of non-homogenous Kolmogorov equations , Math-ematische Annalen, Volume 340, n.2, pp.237-264, 2008[4] C. Cinti and S. Polidoro,
Pointwise local estimates and Gaussianupper bounds for a class of uniformly subelliptic ultraparabolic op-erators , J. Math. Anal. Appl. 338 (2008) 946-969215] G. B. Folland,
Subellitic estimates and function space on nilpotentLie groups , Ark. Math., 13 (2): 161-207, (1975).[6] M. Di Francesco and S. Polidoro,
Harnack inequality for a class ofdegenerate parabolic equations of Kolmogorov type . Adv. Diff. Equ.11, 1261C1320 (2006)[7] P. Hajlasz and P. Koskela,
Sobolev met Poincar ´ e , Mem. Amer.Math. Soc. 145 (2000) x+101.[8] A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality fora class of hypoelliptic ultraparabolic equations , Mediterr. J. Math.1 (2004) 51C80.[9] S. N. Kruzhkov,
A priori bounds and some properties of solutions ofelliptic and parabolic equations , Math. Sb. (N.S.) 65 (109) 522-570,(1964).[10] S. N. Kruzhkov,
A priori bounds for generalized solutions of second-order elliptic and parabolic equations , (Russian) Dokl. Akad. NaukSSSR 150 748–751, (1963).[11] E. Lanconelli and S. Polidoro,
On a class of hypoelliptic evolutionoperaters , Rend. Sem. Mat. Univ. Politec. Torino, 52,1 (1994), 29-63, 1994[12] A. Lunardi,
Schauder estimates for a class of degenerate elliptic andparabolic operators with unbounded coefficients in R N . Ann. ScuolaNorm. Sup. Pisa Cl. Sci. 24(4), 133-164 (1997) 23.[13] M. Manfredini and S. Polidoro, Interior regularity for weak solu-tions of ultraparabolic equations in the divergence form with dis-continuous coefficients , Boll Unione Mat. Ital. Sez. B Artic. Ric.Mat. (8), 1 (3) 651-675, (1998).2214] M. Manfredini,
The Dirichlet problem for a class of ultraparabolicequations . Adv. Diff. Equ. 2, 831-866 (1997) 24.[15] J. Moser,
On Harnack’s theorem for elliptic differential equations ,Comm. Pure Appl. Math. 14 577–591 (1961).[16] J. Moser,
A Harnack inequality for parabolic differential equations ,Comm. Pure Appl. Math. 17 101–134 (1964).[17] J. Nash,
Continuity of solutions of parabolic and elliptic equations ,Amer. J. Math., 80, 931-954, (1958).[18] A. Pascucci and S. Polidoro,
The moser’s iterative method for aclass of ultraparabolic equations , Commun. Contemp. Math. Vol. 6,No. 3 (2004) 395-417.[19] S. Polidoro and M. A. Ragusa,
H¨older regularity for solutions ofultraparabolic equations in divergence form , Potential Anal. 14 no.4, 341–350 (2001).[19] W. Wang and L. Zhang,
The C α regularity of a class of non-homogeneous ultraparabolic equations , arXiv:math.AP/0711.3411.[21] Z. P. Xin and L. Zhang On the global existence of solutions to thePrandtl’s system , Adv. in Math. 181 88-133 (2004).[22] Z. P. Xin, L. Zhang and J. N Zhao,
Global well-posedness for thetwo dimensional Prandtl’s boundary layer equations , preprint.[23] L. Zhang,
The C α reglarity of a class of ultraparabolic equationsreglarity of a class of ultraparabolic equations