Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations
Abstract
Lu, Guozhen
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Publicacions Matem`atiques, Vol 40 (1996), 301–329. EMBEDDING THEOREMS INTO LIPSCHITZ AND BMO SPACES AND APPLICATIONS TO QUASILINEAR SUBELLIPTIC DIFFERENTIAL EQUATIONS Guozhen Lu1 Abstract This paper proves Harnack’s inequality for solutions to a class of quasilinear subelliptic differential equations. The proof relies on various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields X1,... ,X msatisfying H¨ormander’s condition. The nonlinear subelliptic equations under study include the important p-sub-Laplacian equation, e.g., m j=1 X∗ j|Xu|p−2Xju=A|Xu|p+B|Xu|p−1+C|u|p−1+D, 1<p<∞ where |Xu|=m j=1 |Xju|21 2and Ais a constant; B,Cand Dcan be in appropriate function spaces. We note that Acan be nonzero. 1. Introduction One of the main purposes of this paper is to show various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields satisfying H¨ormander’s condition. The other, more importantly, is to apply some of our new theorems proved here to study the local regularity of certain classes of nonlinear subelliptic PDE formed by vector fields. These nonlinear subelliptic equations studied here include the important p-sub-Laplacian as a special case. 1Results of this paper were presented at the 890th AMS special session at Lexington, Kentucky in March, 1994. The author is supported in part by the National Science Foundation grant #DMS-9315963.
302 G. Lu Let Ω be a bounded, open and pathconnected domain in Rn, and let X1,... ,X mbe a collection of C∞real vector fields defined in a neighbourhood of the closure Ω of Ω. For a multi-index α=(i1,... ,i k), denote by Xαthe commutator [Xi1,[Xi2,... ,[Xik−1,X ik]],... ,] of length k=|α|. Throughout this paper we assume that the vector fields satisfy H¨ormander’s condition: there exists some positive integer ssuch that {Xα}|α|≤sspan the tangent space of Rdat each point of Ω. We can define a metric as follows: An admissible path γis a Lipschitz curve γ:[a, b]→Ω such that there exist functions ci(t), a≤t≤b, satisfying m i=1 ci(t)2≤1 and γ(t)=m i=1 ci(t)Xi(γ(t)) for almost every t∈[a, b]. Then a natural metric on Ω associated to X1,... ,X mis defined by (ξ,η) = min{b≥0:∃an admissible path γ:[0,b]→Ω such that γ(0) = ξ, and γ(b)=η}. The metric ball is defined by B(ξ,r)={η:(ξ,η)<r}. This metric is equivalent to the various other metrics defined in the work of NagelStein-Wainger [NSW]. Note that the Lebesgue measure is doubling with respect to the metric balls as shown in [NSW]. Thus (Ω,) is a homogeneous space. By the Rothschild-Stein lifting theorem (see [RoS]), the vector fields {Xi}m i=1 on Ω ⊂Rdcan be lifted to vector fields {˜ Xi}m i=1 in ˜ Ω=Ω× T⊂Rd×RN−d, where Tis the unit ball in RN−dby adding extra variables so that the resulting vector fields are free, i.e., the only linear relation between the commutators of order less than or equal to sat each point of ˜ Ω are the antisymmetric and Jacobi’s identity. Let G(m, s) be the free Lie algebra of steps with mgenerators, that is the quotient of the free Lie algebra with mgenerators by the ideal generated by the commutators of order at least s+ 1. Then {Xα}|α|≤sare free if and only if d= dim G(m, s). We also define Q=s j=1 jmjwhere mjis the number of linearly independent commutators of length j. This integer Qis called the homogeneous dimension associated with the vector fields. We now define the Sobolev space W1,p(Ω) to be the completion of C∞(Ω) under the norm ||f||W1,p(Ω) =Ω |f|p1/p +Ω |Xf|p1/p , where |Xf|expresses m i=1 |Xif|21 2. We also define W1,p 0(Ω) as the completion of C∞ 0(Ω) under the above norm ||·|| W1,p(Ω). Let us review briefly the known results on embedding theorems, especially Poincar´e type inequality for vector fields satisfying H¨ormander’s
Harnack inequality for subelliptic equations 303 condition. We refer the interested reader to, e.g., [CDG1], [FGW] and [L1], for the embedding theorems of Sobolev type (i.e., the functions under consideration are assumed to be with compact support). For embedding theorems on groups, we refer the reader to [FS], [Kra], [Va] and [VS-CC]. For nonsmooth vector fields, extensive study has been given in [Fr], [FrL], [FrS] and [FGuW]. Theorem. Let E⊂⊂ Ω,1≤p<∞, then there exist some q= q(p)≥pand constants r0>0,C>0,c≥1, such that for any metric balls B=B(x,r)with cB =B(x,cr)⊂Ω,x∈E, and any f∈Lip1(B), the following inequality holds 1 |B|B |f−fB|q1/q ≤Cr1 |B|B m i=1 |Xif|p1/p provided 0<r<r 0, where C,c,r0depend only on E,Ω,fBmay be taken to be 1 |B|Bf. Such an inequality was first proved by D. Jerison [Jer] for all 1 ≤p< ∞and q=p. The same inequality in the setting of subelliptic operators was proved by Jerison and Sanchez-Calle in [JeS]. After the work of [Jer] and [JeS], the author of the present paper improved the result in [J] for p>1 and extend it to weighted case ([L1]-[L2]). Especially, when 1 <p<Q, it is shown in [L1] and [L2] that qcan be taken as 1≤q≤Qp Q−p. We remark here that by the Rellich-Kondrachov compact embedding theorem for vector fields satisfying H¨ormander’s condition (see, e.g., [L4]) and together with a well-known compactness argument (see, e.g., [L5]), one can recapture the proof of the Poincar´e inequality with 2Bon the right side for all 1 ≤q< pQ Q−pexcept the endpoint q=Qp Q−p. However, such a Poincar´e inequality usually involves a constant Cpossibly depending on the ball Bin general. When p=Q, the following inequality was shown in [L3] that for all balls Bwith cB ⊂Ω: 1 |B|B exp A|f−fB| || m i=1 |Xif|||Lp(B)Q Q−1dx ≤C where A>0, C>0 and c≥1 are absolute constants provided that f∈Lip1(B) is not constant. All the Poincar´e type inequalities proved so far are with the restriction 1 p−1 q≤1 Q. However, if we consider embedding theorems on the
304 G. Lu Campanato-Morrey spaces, we will get inequalities with larger differences 1 p−1 q. To state the theorems proved in [L3], we briefly define the Campanato-Morrey spaces as follows: Let now fB=1 |B|Bf(y)dy be the average over the ball Bof the function f. We define the following two types of Campanato-Morrey norms: Fix any R>0. Let Lp,λ(Ω) be the spaces of all functions f∈Lp loc(Ω) such that ||f||Lp,λ(Ω) = sup Bρ(B)λ|B|−1B |f−fB|p1 p <∞ where the sup is taken over all the balls B=B(x,r) with cB =B(x,cr)⊂ Ω with x∈E⊂⊂ Ω for some subset Kand ρ(B)=r(the radius of the ball B)≤R. It is easy to see that two elements of Lp,λ can be identified if they only differ by a constant. We also define the space Mp,λ of functions f∈Lp loc(Ω) such that ||f||Mp,λ(Ω) = sup Bρ(B)λ|B|−1B |f|p1 p <∞, where the sup is taken in the same sense as above. Then one of the main theorems proved in [L3] is the following: Theorem. Given any f∈W1,p loc (Ω) the following is true: ||f||Lp∗,λ(Ω) ≤C|| m i=1 |Xif|||Mp,λ(Ω) where 0<λ≤Q,1<p<λand p∗=λp λ−p, provided that the number R>0is small enough in the definition of the spaces Lp,λ(Ω) and Mp,λ(Ω). We note in the above that 1 p−1 p∗=1 λcan be taken much larger than the known gap in the Poincar´e inequality, which is known to be true so far for 1 Q. Recently, Franchi, Wheeden and the author showed in [FLW] that a Poincar´e inequality holds when p= 1 and q=Q Q−1(when p=q= 1, the result was due to Jerison [Jer]). We mention that this endpoint result for p= 1 contains certain important geometric information. Indeed, applying this Poincar´e inequality, we also derived a relative isoperimetric
Harnack inequality for subelliptic equations 305 inequality ([FLW]). A new representation formula was derived in [FLW] which improves the one obtained in [L1]. Results in [FLW] also sharpen the exponents given in those inequalities in [L1]-[L2]. One of the main goals of this paper is to show some new embedding theorems for H¨ormander’s vector fields which will complement the theorems mentioned above. The current theorems shown here together with the previously known ones will give a fairly complete picture of embedding theorems for vector fields of H¨ormander’s type. More importantly, we will employ these new theorems to prove a Harnack inequality for a certain class of quasilinear subelliptic differential equations formed by vector fields satisfying H¨ormander’s condition. We first state the embedding theorems. From now on, we use frequently |Xf|to express m i=1 |Xif|21 2. Theorem 1.1. Suppose p>Q. Then there exists some constant c≥1 such that for any f∈W1,p(Ω), for any ball BRwith cBR⊂Ωwe have sup x,y∈BR |f(x)−f(y)|≤C|BR|1 Q−1 p||Xf||Lp(cBR). Furthermore f∈C0,γ loc (Ω) (the local nonisotropic Lipschitz space), where γ=1−Q p, in the sense that for any compact subset K⊂Ω sup x,y∈K,x=y |f(x)−f(y)| ρ(x,y)γ≤C||Xf||Lp(Ω), provided that one of the metric balls B(x,(x,y)) and B(y,(y, x)) is contained in Ω. The embedding W1,p(Ω) →C0,β loc (Ω) is compact provided β<γ. Remark. If we assume f∈W1,p 0(Ω), p>Q, then we can show f∈C0,γ(Ω), i.e., sup x,y∈Ω,x=y, |f(x)−f(y)| ρ(x,y)γ≤C||Xf||Lp(Ω). Theorem 1.2. Given any 1≤p<∞and c≥1. Suppose Kand 0<α≤1are two positive constants. Let f∈W1,p(Ω) satisfy BR |Xf|p(x)dx ≤Kp|BR|R(−1+α)p,
306 G. Lu for all balls BR⊂Ω, then f∈C0,α loc (Ω) and for any ball BRwith cBR⊂Ω we have sup x,y∈BR |f(x)−f(y)|≤CKRα where C=C(Q, α). Moreover, for any compact subset K⊂Ωthere exists r0>0, we have sup x,y∈K,x=y,(x,y)≤r0 |f(x)−f(y)| ρ(x,y)α≤CK. Theorem 1.3. Given any 1≤p<∞and c≥1. Suppose f∈ W1,p(Ω) and also that there exists a positive constant Ksuch that BR |Xf|p(x)dx ≤Kp|BR|R−p, for all balls BR⊂Ω. Then there exist positive constants σand Csuch that for all balls BRwith cBR⊂Ω BR exp σ K|f−fB|(x)dx ≤C|BR|. We remark here that Theorems (1.2) and (1.3) do not involve the homogeneous dimension Q, both the theorems and proofs work in more general settings, say, for Grushin or nonsmooth vector fields (see [FGuW]). By employing the above theorems when 1 <p<∞, we shall establish certain Harnack inequalities for weak solutions, subsolutions, and supersolutions of quasilinear second order subelliptic partial differential equations of the form (1.4) m j=1 X∗ jAj(x,u,X 1u, X2u, . . . , Xmu)+B(x,u,X 1u, X2u,... ,X mu)=0 where X∗ jis the adjoint of Xj, which is not necessarily a vector field in general; u(x) is assumed to be in W1,p loc (Ω). As a special case of our theorems, we will be able to obtain the local regularity for the well-known sub-Laplacian. The Harnack inequality will be established under certain structural assumptions on the equation (1.4) (see Theorems (3.9), (3.13), (3.15) and Corollary (3.11) in Section 3). We now let x=(x1,... ,x n), η=(η1,... ,η m) denote vectors in Rnand Rmrespectively and Xu =(X1u,... ,X mu). Let A(x,u,η)=
Harnack inequality for subelliptic equations 307 (A1(x,u,η),... ,A m(x,u,η)) and B(x,u,η) be, respectively, vector and scalar measurable functions defined on Ω×R×Rm, where Ω is a domain in Rn. The structure of the equation (1.4) throughout this paper will be assumed to satisfy the following: (1.5) |A(x,u,η)|≤a0|η|p−1+(a1(x)|u|)p−1+(a3(x))p−1, η·A(x,u,η)≥|η|p−(a2(x)|u|)p−(a4(x))p, |B(x,u,η)|≤b0|η|p+b1(x)|η|p−1+(b2(x))p|u|p−1+(b3(x))p where 1 <p<∞,a0,b0are constants, ai(x), bi(x) are nonnegative measurable functions satisfying certain integrability properties which will be described in Section 3. Such type of equations when Xi=∂ ∂xi(i=1,2,... ,n)inRnhave been studied in the literature (see [Ser], [GiT], [Tru], [Zie]). We point out here that the equation (1.4) has been studied in [CDG1] when pis restricted to 1 <p≤Qunder the assumption of b0= 0. Our theorems proved in this paper include all 1 <p<∞and also b0= 0. Moreover, the results in [CDG1] require higher integrability conditions on the coefficients ai(x)(i=1,2,3,4), bj(x)(j=1,2,3) than the ones given here (see Section 3 for details). For example, by our theorems in Section 3 the solutions of, e.g., the following very simple equation for all 1 <p<∞ satisfies a uniform Harnack inequality: m j=1 X∗ j|Xu|p−2Xju=A|Xu|p+B|Xu|p−1+C|u|p−1+D, where again |Xu|=m j=1 |Xju|21 2,Ais a constant; B,Cand Dare in appropriate function spaces which will be specified below. We should also mention that when 1 <p≤Qwe shall assume the solutions are a priori bounded (when b0= 0 such an assumption can be dropped, see Section 3) while when p>Qthe local boundedness and H¨older continuity of the solutions follows by the embedding Theorem (1.1) proved in this paper without obtaining Harnack inequality first. However, one still needs to prove the Harnack inequality for p>Qbecause H¨older continuity of the solutions does not lead to this. We also remark that the proofs of the Harnack inequalities for the solutions of the equation (1.4) rely on Sobolev embedding theorems (see for example [L1], [FGW]) and embedding theorems into Lipschitz and BMO spaces proved here. We will also need to adapt the well-known Moser’s iteration argument [Mos] to our nonlinear subelliptic case. For
308 G. Lu elliptic Euclidean case, we refer the interested reader to [LaU], [Mos], [Nas], [Ser], [GiT], [Tru], [Zie] and references therein. Our Harnack inequalities extend to the subelliptic context results due to J. Serrin, N. Trudinger, Ladyzhenskaya and Ural’tseva (see [Ser], [Tru], [LaU]). We also mention that subelliptic variational problems have been studied by Xu in [X1]. The organization of the paper is as follows: Section 2 contains the proofs of the embedding theorems which will be needed in proving the Harnack inequality. Section 3 devotes the proof of the Harnack inequality, H¨older continuity, and estimates of the solutions at the boundary. We will use the letters C,c, etc., to denote the absolute constants and may differ from line to line. 2. Proofs of Theorems (1.1), (1.2) and (1.3) We recall again that by the Rothschild-Stein lifting theorem (see [RoS]) the vector fields {Xi}m i=1 on Ω ⊂Rdcan be lifted to vector fields {˜ Xi}m i=1 in ˜ Ω=Ω×T⊂Rd×RN−d, where Tis the unit ball in RN−d. There is also a metric ˜:˜ Ωט Ω→R+associated with the lifted vector fields ˜ X1,... , ˜ Xm. We note that the Lebesgue measure of the ball |˜ B(ξ,r)|≈rQ, where Qis the homogeneous dimension of G, and ˜ B(ξ,r) is the metric ball in (˜ Ω,˜). Thus (˜ Ω,˜) is a homogeneous space in the sense of Coifman and Weiss. We should mention the proofs given in this section are not the simpliest ones. The following lemma is necessary in order to show Theorem (1.1). Lemma 2.1. Given any metric ball ˜ B⊂˜ Ωand any Lipschitz continuous function ˜ f∈Lip1(˜ Ω). Then there exist constants c≥1and C≥1 such that for any ξ∈˜ Band any constant C0the following is true: |˜ f(ξ)−˜ f˜ B|≤Cc˜ B Mm i=1 |˜ Xi˜ f|+|˜ f−C0|χc˜ B(η) ˜(ξ,η)Q−1dη where ˜ f˜ B=1 |˜ B|˜ B˜ f(η)dη, and ˜(ξ,η)is the metric distance associated to the lifted vector fields {˜ Xi}m i=1;M(g)is the Hardy-Littlewood maximal function for g. This lemma was essentially proved in [L1] (Lemma (3.2) in [L1]). In [L1] it was shown that there is a constant C˜ Bsuch that |˜ f(ξ)−C˜ B|≤c˜ B Mm i=1 |˜ Xi˜ f|+|˜ f|χc˜ B(η) ˜(ξ,η)Q−1dη.
Harnack inequality for subelliptic equations 309 But we can show such a constant C˜ Bcan be replaced by ˜ f˜ B. Moreover, if we replace the function ˜ fby ˜ f−C0, we will get Lemma (2.1). Actually in the proof below we will take C0=˜ fc˜ B. Remark. In the above representation formula, it contains the HardyLittlewood maximal function and also the zero order term |˜ f−C0|. Such a formula is good enough for most Lpestimates for p>1 as demonstrated in [L1]-[L3]. The proof of Theorem (1.1) given below by using this formula is interesting itself when we get rid of the Maximal function by using the boundedness of the maximal function in Lpnorm and control the terms containing the zero-order term |˜ f−C0|by using the known Poincar´e inequality from Lpto Lp. (See the similar argument in [L3].) Of course, the proof can be much simplified by using the new representation formula obtained in [FLW]. We thought the proof of Theorem (1.1) given below may have its own interest. Before we start to prove the Theorems (1.1), (1.2) and (1.3), we briefly explain how the proofs go. We will first prove the theorems for free vector fields {˜ Xi}and the functions ˜ fdefined on ˜ Ω. Secondly, for any function fdefined on Ω which satisfies the assumptions in the theorems associated with the vector fields {Xi}, we define the new function ˜ f(ξ)= ˜ f(x,t)= f(x) for x∈Ω and ξ∈˜ Ω and we prove for so defined ˜ fit satisfies the conditions associated with the lifted vector fields {˜ Xi}. Thirdly, we then show the conclusions of the theorems for so defined ˜ fwill lead to the conclusions for the original function f. We also mention that on the nilpotent Lie group some similar results to our Theorem (1.1) were derived in [Fol], [SC], [Cou] and [Kra]. Proof of Theorem (1.1): We first show that the theorem holds for ˜ f∈Lip1(˜ Ω). The general case follows by an argument of approximation. Given any ball ˜ B⊂˜ Ω, and any ˜ f∈W1,p(˜ Ω) Lip1(˜ Ω), p>Q. Let ˜ f˜ B= 1 |˜ B|˜ B˜ f(η)dη. Taking C0=˜ fc˜ Bin Lemma (2.1), then by Lemma (2.1), for any ξ∈˜ B, |˜ f(ξ)−˜ f˜ B|≤Cc˜ B Mm i=1 |˜ Xi˜ f|+|˜ f−˜ fc˜ B|χc˜ B(η) ˜(ξ,η)Q−1dη ≤Cc˜ BM m i=1 |˜ Xi˜ f|+|˜ f−˜ fc˜ B|χc˜ B(η) p dη1/p ·c˜ B 1 ˜(ξ,η)(Q−1)pdη1/p
316 G. Lu Lemma 3.7. Suppose that u(x)∈W1,p 0(Ω),f(x)∈LQ,ρα(Ω) if p< Q;f(x)∈Lt loc(Ω) if p=Q;f(x)∈Lp(Ω) if p>Q. Then for any 4>0 (3.8) ||fu||p,Ω≤4||Xu||p,Ω+C(p, Q, α, Ω,||f||)4−β||u||p,Ω, where β=β(p, Q)>0if p>Qand β=β(p, Q, α)>0if p≤Q. Remark. When p<Q, if we only assume f∈LQ loc(Ω) but assume the LQnorm is small then this lemma still holds as one can see from the proof given below. Proof: We first assume p<Q. Given each fixed small enough r>0. Then we can find a partition of unity of the domain Ω. More precisely, there exists a finite sequence of metric balls Bi=B(xi,r), i= 1,2,... ,M and functions ηi(x), i=1,2,... ,M such that supp{ηi}⊂ Bi,|Xηi|≤Cr−1,Ω⊂M i=1 Biand M i=1 ηp i(x) = 1 for all x∈Ω. Thus if we set ui(x)=u(x)ηi(x) for i=1,... ,M Ω f(x)pu(x)pdx= M i=1 Bi fp(x)up(x)ηp i(x)dx = M i=1 Bi fp(x)up i(x)dx ≤ M i=1 Bi fQ(x)dxp/Q ·Bi u pQ Q−p i(x)dxQ−p Q ≤ M i=1 Bi fQ(x)dxp/Q ·Bi |Xui|p(x)dx ≤Crαp M i=1Bi ηp i(x)|Xu|p(x)dx+Bi up(x)|Xηi(x)|pdx ≤Crαp Ω |Xu|p+Crαp−pΩ |u|p. We note that we have used the following Sobolev inequality since ui(x) has support in Bi(see, for example, Theorem C in [L1]) Bi ui(x)pQ Q−pdxQ−p Qp ≤CBi |Xui|p1 p . If we replace the constant Crαp by 4>0 we will get our proof. We note the precise constant C(r, p, Q, α) can be calculated.
Harnack inequality for subelliptic equations 317 Now let p>Q, we use the same partition of unity as above. Then Ω f(x)pu(x)pdx = M i=1 Bi fp(x)up(x)ηp i(x)dx = M i=1 Bi fp(x)up i(x)dx. We note again supp{ui}⊂Biand p>Q, then we have by Theorem (1.1) in Section 1, ui(x)∈L∞(Bi) and its norm is bounded by Cr1−Q/p||Xui||p,Bi≤Cr1−Q/p||Xui||p,Ω. Therefore, Ω f(x)pu(x)pdx ≤ M i=1 Bi fp(x)·rp−Q||Xui||p p,Ω ≤Crp−Q·||f||p p,Ω·||Xu||p p,Ω+C|f||p,Ωr−Q||u||p,Ω. Then by setting 4=Crp−Q·||f||p p,Ωwe will get the proof. When p=Q,uiis exponentially integrable as shown in [L3] and especially in Lt loc for all t>Q. We now assume f∈Lt loc(Ω) for some t>Q, then arguing as above Ω f(x)Qu(x)Qdx = M i=1 Bi fQ(x)uQ(x)ηp i(x)dx= M i=1 Bi fQ(x)uQ i(x)dx ≤ M i=1 Bi ft(x)dxQ/t ·Bi u tQ t−Q i(x)dxt−Q t ≤ M i=1 Bi ft(x)dxQ/t ·rQ(t−Q)/tBi |Xui|Q(x)dx ≤CrQ(t−Q)/t||f||Q t,Ω M i=1 Bi ηQ i(x)|Xu|Q(x)dx +Bi uQ(x)|Xηi(x)|Qdx ≤CrQ(t−Q)/t||f||Q t,ΩΩ |Xu|Q+C(r, p, Q, t)Ω |u|Q. Taking CrQ(t−Q) t||f||Q t,Ω=4, we will get the desired result. All the results proved in this paper will be of local nature. We will simply denote a ball of radius ρas Bρand drop the center in the notation because the centers are not important here.
318 G. Lu Theorem 3.9. Suppose that u(x)is a nonnegative weak solution of (1.4) in a metric ball B3ρ⊂Ωwith 0≤u<M in B3ρ. Then (3.10) max Bρ u(x)≤Cmin Bρ u(x)+m(ρ), where C=C(p, Q, a0,b 0M,λρ). For the standard Harnack inequality stated below to hold, we need to assume that a3(x), a4(x), b3(x)=0. Corollary 3.11. Suppose that u(x)is a nonnegative weak solution of (1.4) in a metric ball B3ρ⊂Ωwith 0≤u<M in B3ρ. Assume that a3(x),a4(x),b3(x)=0. Then (3.12) max Bρ u(x)≤Cmin Bρ u(x) where C=C(p, Q, a0,b 0M,λρ). The special case of our theorem, i.e., b0= 0 has been found in [CDG1] when 1 <p≤Q, but with stronger assumptions on the coefficients ai(x) and bj(x). In this case b0= 0, we do not need to assume the boundedness of u(x) provided that the functions in the structure conditions (3.1) do not depend on M(since b0M= 0). We treat all the cases 1 <p<∞ here in a unified way. One of the main features is the availability of the new embedding theorem proved in this paper. For the weak supsolutions of (1.4) we have the following weak Harnack inequality. Theorem 3.13. Suppose that u(x)is a weak supsolution of (1.4) in a metric ball B3ρ⊂Ωwith 0≤u<M in B3ρ. Then (3.14) ρ −Q γ||u(x)||γ,B2ρ≤Cmin Bρ u(x)+m(ρ) for any γ<Q(p−1) Q−pif p≤Q,γ≤∞if p>Qand where C= C(p, Q, a0,b 0M,λρ). For the weak subsolutions of (1.4) we have the following estimate:
Harnack inequality for subelliptic equations 319 Theorem 3.15. Suppose that u(x)is a weak subsolution of (1.4) in a metric ball B3ρ⊂Ωwith 0≤u<M in B3ρ. Then (3.16) max Bρ u(x)≤Cρ −Q γ||u(x)||γ,B2ρ+m(ρ) for any γ>p−1, where C=C(p, Q, a0,b 0M,λρ). We remark here that Theorem (3.15) also holds for p= 1 as one can see from the proof below. It is clear that Theorem (3.9) is a consequence of Theorems (3.13) and (3.15). The proofs of the above theorems adapt the well-known iteration argument of Moser [Mos]. More closely related arguments can be found in [Ser], [GiT], [Tru] and citeZie. We now define the functional (3.17) φ(s, h)=1 |Bh|Bh |u|sdx1 s ,s=0,h>0. Thus (3.18) φ(∞,ρ) = max Bρ u(x), φ(−∞,ρ) = min Bρ u(x). Consequently, the inequalities (3.10), (3.14) and (3.16) may be written as (3.19) φ(∞,ρ)≤C(φ(−∞,ρ)+m(ρ)) , φ(γ,2ρ)≤C(φ(−∞,ρ)+m(ρ)) , φ(∞,ρ)≤C(φ(γ,2ρ)+m(ρ)) . Before we prove all the Harnack inequalities we first make the following reductions. We define a2(x)=a2(x)+m(ρ)−1a4(x) b2(x)=b2(x)+m(ρ)1 p−1b3(x) a1(x)=a1(x)+m(ρ)−1a3(x) u(x)=u(x)+m(ρ). Thus u(x) will satisfy an equation of the form (1.4) m j=1 X∗ jAj(x, u, X1u, X2u, . . . , Xmu)+B(x, u, X1u, X2u,... ,X mu)=0
320 G. Lu where A(x,u,η) and B(x,u,η) satisfies the following conditions: (3.20) |A(x,u,η)|≤a0|η|p−1+(a1(x)|u|)p−1, η·A(x,u,η)≥|η|p−(a2(x)|u|)p, |B(x,u,η)|≤b0|η|p+b1(x)|η|p−1+b2(x)p|u|p−1. Therefore this reduces the structure conditions to the cases a3(x)= b3(x)=a4(x) = 0, i.e., m(ρ) = 0. For simplicity we will also drop the “bar” from A,B,u(x), a1(x), a2(x), b2(x) and simply write A,B,u(x), a1(x), a2(x), b2(x). Proof of Theorem (3.9): We assume with no loss of generality that u(x)≥4>0. We select a test function in (3.2) (3.21) φ(x)=ξp(x)uq(x)e(sgn q)b0u(x), where q= 0 and ξ(x)≥0, ξ(x)∈C∞ 0(B3ρ) will be specified later. By (3.21), we have (3.22) Xφ(x)=(sgn q)ξpe(sgn q)b0u(b0uq+|q|uq−1)Xu+pξp−1uqe(sgn q)b0uXξ, where Xf =(X1f,... ,Xmf) is the subelliptic gradient vector for the given function f. Substituting (3.21) and (3.22) into (3.2) we get (3.23) (sgn q)B ξpe(sgn q)b0u(b0uq+|q|uq−1)Xu ·A(x,u,Xu) +pB ξp−1uqe(sgn q)b0uXξ·A(x,u,Xu)−B ξqe(sgn q)b0uuqB(x,u,Xu) ≤0ifuis a subsolution, (≥0ifuis a supersolution.) The above ·stands for the inner product. In the following calculations, it will be understood that q>0 when u satisfies the hypothesis of Theorem (3.15) and that q<0 when usatisfies the hypothesis of Theorem (3.13). By employing the structure condition (3.20) and together with (3.23), we get (3.24) B3ρ e(sgn q)b0uξp(b0uq+|q|uq−1)|Xu|p ≤B3ρ e(sgn q)b0uξp(b0u+|q|)ap 2up+q−1 +pB3ρ e(sgn q)b0uξp−1|Xξ|(a0|Xu|p−1+ap−1 1up−1)uq +B3ρ e(sgn q)b0uξp(b0|Xu|p+b1|Xu|p−1+bp 2up−1)uq.
Harnack inequality for subelliptic equations 321 We note the term b0B3ρ e(sgn q)b0uξpuq|Xu|p can be dropped from both sides of (3.24). After calculation and H¨older’s inequality and together with the estimate 0 ≤b0u<b 0M, we can bootstrap the terms involving |Xu|and we will get (3.25) B3ρ ξpuq−1|Xu|p ≤C(1 + |q|−1)pB3ρ(4a1+a2+b1+b2)pξp+4p 1−p|Xξ|pup+q−1 for any given 0 <4≤1. Set f(x)=(4a1+a2+b1+b2), then by Lemma (3.7) (ξplays the role of uthere) and the assumptions on ai(x) and bj(x)(i,j=1,2) we get (3.26) B3ρ ξpuq−1|Xu|p≤C(1 + |q|−1)pB3ρ (ξp+|Xξ|p)up+q−1 where Cdepends on λρ (see definition of λat the beginning of this section) and etc. We now let (3.27) v(x)=ut(x) where pt =p+q−1 for q=1−p; log u(x) for q=1−p. Thus (3.26) can be written as (3.28) ||ξXv||p,B3ρ≤C|t|(1 + |q|−1)||(ξ+|Xξ|)v||p,B3ρfor q=1−p, 0; C||ξ+|Xξ|||p,B3ρfor q=1−p. We consider the case q=1−pin (3.28). By Sobolev embedding lemma (Theorem C in [L1]), and the exponential integrability when p=Qand Theorem (1.1) when p>Qin Section 1, we get (3.29) ||ξv||χp,B3ρ≤C|t|(1 + |q|−1)ρ|B3ρ|1 χp −1 p||(ξ+|Xξ|)v||p,B3ρ where χ=Q Q−pif p<Q, and χcan be arbitarily large if p=Q, and χ=∞if p>Q. Let now r1,r2satisfy ρ≤r1,r2≤2ρand select ξ(x)
322 G. Lu as a cut-off function such that ξ(x)=1onBr1and ξ(x) = 0 outside Br2 and |ξ(x)|≤C(r2−r1)−1. The existence of such a cut-off function was proved in [L1]. (3.30) ||v||χp,Br1≤C|t|(1 + |q|−1)(r2−r1)−1ρ|B3ρ|1 χp −1 p||v||p,Br2. We note here that r1,r2,ρare comparable, and also note that the Lebesgue measure is doubling with respect to the metric balls by the work of [NSW]. Thus by taking t-th root of both sides of (3.30) and setting s=pt =p+q−1, we will get the following for positive s (3.31) φ(χs, r1)≤C|t|(1 + |q|−1)(r2−r1)−1p/s φ(s, r2), while for negative swe get (3.32) φ(χs, r1)≥C|t|(1 + |q|−1)(r2−r1)−1p/s φ(s, r2). We now fix some s0>0 and define s=sj=χjs0,r j=(1+2 −j)ρ, j =0,1,2,... We assume s0is so selected that no sjwill coincide with p−1 for otherwise s=sj=p−1 and q= 0. Therefore 1 + |q|−1<Cfor all j. By (3.31) we obtain (3.33) φ(sj+1,r j+1)≤[C(2χ)j]pχ−j s0φ(sj,r j) ≤Cχ−j[C(2χ)p/s0]jχ−jφ(s0,2ρ)≤Cφ(s0,2ρ). We have used the fact that χ>1 and then the corresponding series in the above converges. If we let j→∞we will get (3.34) φ(∞,ρ)≤Cφ(s0,2ρ). It is clear then for any s0=γ>p−1, (3.34) holds and then we have shown Theorem (3.15). Actually, Theorem (3.15) also holds when p=1 because in the above proof s0is allowed to be any positive number. Suppose now that u(x) is a supersolution, (3.33) holds for any s0>0 and sj<p−1 and thus (3.35) φ(γ,2ρ)≤Cφs0,5ρ 2
Harnack inequality for subelliptic equations 323 for any s0>0, γ<Q(p−1) Q−pif p≤Qand γ≤∞if p>Q. We note that the iteration of (3.32) will lead to (3.36) φ−s0,5ρ 2≤Cφ(−∞,ρ) for any s0>0. Therefore, if we can show that there exists some s0>0 such that φs0,5ρ 2≤Cφ−s0,5ρ 2 then we will have proved Theorem (3.13). We now let Brbe any ball contained in Bρ0and choose ξ(x) such that ξ(x)=1onBrand 0 outside B2rand |ξ(x)|≤Cr−1. Then we get ||Xv||p,Br≤CrQ−p p, where vis as in (3.28) when q=p−1. Thus Theorem (3.9) and (3.13) will follow from Theorem (1.3) in Section 1. One application of the above theorem is the H¨older continuity of the weak solutions of (1.4). Theorem 3.37. Suppose that u(x)is a weak solution of (1.4) in Ω which is also locally bounded. Then u(x)is H¨older continuous in Ωand if Bρ0⊂Ωthen (3.38) oscBρu(x)≤Cρ ρ0αsup Bρ0 |u(x)|+m(ρ0), for all Bρ⊂Bρ0and some α>0, and C=C(p, Q, a0,b 0M). The proof of the above theorem is fairly standard and we omit the details. We now consider the estimates of the solutions at the boundary of the certain domains. Let Sbe a subset of ∂Ω and u(x)∈W1,p loc (Ω). Then we say that u≤D on Sif for every 4>0 there is a neighborhood of S, called MS, such that u≤D+4a.e. in Ω MS. With such a definition we may easily define the notions supSu, infSuand oscSu= supSu−infSu.
324 G. Lu We consider the equation (3.39) m j=1 X∗ jAj(x,u,X 1u, X2u, . . . , Xmu)+B(x,u,X 1u, X2u,... ,X mu)=0 under the following structure condition (for simplicity): (3.40) |A(x,y,η)|≤a0|η|p−1+a3 η·A(x,u,η)≥|η|p−a4, |B(x,u,η)|≤b0|η|p−bp 3. Then Theorem 3.41. Let u(x)be a weak solution of (3.39) in Ω.Let B=B3ρ(x0)and L= sup B∂Ω u(x), M= sup BΩ u(x). Then the function v(x)given by v(x)=M−sup(u, L)for x∈ΩB, M−Lfor x∈B\Ω will satisfy (3.42) ρ−Q p−1||v||p−1;B2ρ≤Cmin Bρ v+m(ρ), where C=C(p, Q, a0,b 0,M). We now introduce the notion of a “regular point” on the boundary ∂Ω. A point x0∈∂Ω is called “regular” if there exists a positive constant ρ0=ρ(x0) such that for all ρ≤r0 (3.43) |Bρ(x0)\Ω|≥θ0|Bρ(x0)|. If every point of ∂Ω is regular we say ∂Ω is regular, and it is called “uniformly regular” if ρ0and θ0can be selected independent of x0.
Harnack inequality for subelliptic equations 325 Corollary 3.44. Let u(x)satisfy the hypotheses of Theorem (3.41) and suppose that x0∈∂Ωis regular. Then for all ρ≤ρ0, (3.45) sup Bρ u(x)−L≤40(M−L)+Cm(ρ), where 40<1and Cis a positive constant depending on p,Q,a0,b0,M and θ0. We also state a theorem which is an extension of Theorem (3.37) to the boundary. Theorem 3.46. Suppose that u(x)is a weak solution of (3.39) in Ωwhich is also locally bounded. Let x0∈Ωbe regular. Then for any ρ≤ρ0,γ<1, (3.47) oscBρΩu(x)≤Cρ ρ0α(1−γ) +4(ρ), where Cand α>0depend on p,Q,a0,b0,supBρ0u,θ0and ρ∗and 4(ρ)=osc∂ΩBργρ1−γ 0 u. All the proofs of the above Theorems (3.41)-(3.46) follow by modifying the proofs of corresponding Theorems (3.9)-(3.15) and (3.37) and we omit the details. One needs to assume that all the vector fields are well defined and satisfying the H¨ormander’s condition in a larger domain Ω1 containing Ω so that all the embedding theorems hold for those balls considered in the theorems. After the paper was written and first circulated (with a slightly longer title) in February 1994, we learnt that some related work on Poincare estimates has also been obtained in [BM], [MS], [Cou2], [HK]. A Poincar´e type inequality with |f(x)−fB|replaced by |f(x)−f(x0)| for solutions to subelliptic quasilinear equations studied in the current paper has been given in [L5] for p≥1 and in [BKL] for p<1, among other things. We also became aware of the work [HH] for Harnack estimates on Carnot groups in conjunction with the quasiregular mappings, and the interior regularity for subelliptic systems [XZ], and isoperimetric inequality independently derived in [CDG2] similar to that in [FGW]. Acknowledgement. The author wishes to thank the referee for his many useful comments and remarks, which help and improve the exposition of the paper.