TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 2, February 2010, Pages 673–685 S 0002-9947(09)04809-0 Article electronically published on September 15, 2009
A DEGENERATE SOBOLEV INEQUALITY FOR A LARGE OPEN SET IN A HOMOGENEOUS SPACE
SCOTT RODNEY
Abstract. In current literature, existence results for degenerate elliptic equa- tions with rough coefficients on a large open set Θ of a homogeneous space (Ω,d) have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on Θ of the form 1 1 (1) |w(x)|2σdx 2σ ≤ C Q(x, ∇w(x))dx 2 , Θ Θ holding for w ∈ Lip0(Θ) and some σ ∈ (1, 2]. However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1): (2) 1 1 1 1 1 |v(x)|2σ 2σ ≤ Cr Q(x, ∇v(x))dx + |v(x)|2dx 2 , | | | | | | Br Br Br Br Br Br holding for v ∈ Lip0(Br), with σ as above. The main result of this work shows that the global Sobolev inequality (1) can be obtained from the local Sobolev inequality (2) provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form Q(x, ·). This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on Θ and a maximum principle.
1. Introduction This work addresses difficulties in obtaining existence and global estimates of weak solutions to second order linear subelliptic Dirichlet problems in divergence form with rough coefficients. Specifically, a local to global result for degenerate Sobolev inequalities is presented. This is achieved via a new process involving weak solutions, with global estimates, to a 1-parameter family of degenerate Dirichlet problems and a maximum principle. Let Ω Rn be open with n ≥ 3. Denote by Q(x, ξ)=ξ Q(x)ξ a bounded, measurable, non-negative semi-definite quadratic form on Ω × Rn. We assume that the Carnot-Carath´eodory control distance d, defined by Q on Ω, defines a homoge- neous space structure in Ω in the sense of [15] and [12]. We consider equations of the form: (3) Xu = ∇ Q(x)∇u + HRu + S Gu + Fu = f + T g ∈ q ∈ q/2 where the coefficient functions H, G, g Lloc(Ω) and F, f Lloc (Ω) for q>2σ , 1 1 with + =1whereσ is as in (12). The vector fields R, S, T are assumed σ σ
Received by the editors October 1, 2007. 2000 Mathematics Subject Classification. Primary 35Hxx.
c 2009 American Mathematical Society Reverts to public domain 28 years from publication 673
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to be subunit with respect to Q(x, ·) in Ω. Recall that a vector field V (x)= n ∂ j=1 vj (x) , identified with the vector v(x)=(v1(x), ..., vn(x)), is said to be ∂xj subunit with respect to the quadratic form Q in Ω if 2 (4) v(x) · ξ ≤Q(x, ξ) for every x ∈ Ωandξ ∈ Rn. The operator X is assumed to satisfy the following negativity condition analogous to that of [4] in the elliptic case, that is, (5) (Fv− GSv) ≤ 0 Ω ∈ ∞ ≥ for all v C0 (Ω) with v 0inΩ. Remark 1.1. The negativity condition is sufficient for existence of weak solutions to Dirichlet problems involving the operator X; see (3). Indeed, if one sets H = G = g =0andF = c, c a constant, with Q the identity quadratic form on Rn, then one obtains the elliptic equation (6) ∆u + cu = f. In this case the negativity condition becomes c ≤ 0, which is known to be sufficient for existence of weak solutions to equations of this type; see [4]. In a subsequent paper the existence theory for Dirichlet problems of the form (7) Xu = f + T g in Θ u = φ on ∂Θ where Θ Ω is open will be treated. To obtain existence and uniqueness of weak solutions, weak in the sense of [5], the Fredholm Alternative will be used. Thus, we are required to show that the solution of the zero problem (8) Xu = 0 in Θ u =0on∂Θ is unique and zero a.e. and so a maximum principle is needed. To obtain such a maximum principle it is sufficient to generalize the weak maximum principle of [4], the source of the idea for this work. The main ingredient for this generalization is a global Sobolev inequality on Θ of the form 1 1 2σ 2 (9) |w|2σ ≤ C Q(x, ∇w) , Θ Θ
holding for all w ∈ Lip0(Θ) for some σ ∈ (1, 2]. In the case where X is elliptic we n have σ = , the classical sharp Sobolev exponent. In the current literature one n − 2 is most often provided with a local version of the Sobolev inequality: 1 1 1 | |2σ 2σ ≤ 1 Q ∇ 2 (10) | | w Cr | | (y, w) , Br(x) Br (x) Br(x) Br (x)
holding for all w ∈ Lip0(Br(x)), 0
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inequality of the form (9). Rather, one obtains a weaker form of inequality (9). This inequality, referred to as a global weak Sobolev inequality, is of the form 1 1 1 2σ 2 2 (11) |w|2σ ≤ C Q(x, ∇w) + |w|2 , Θ Θ Θ
holding for all w ∈ Lip0(Θ). Inequality (11) is insufficient, using the techniques of [4], for the goal of establishing existence of weak solutions to Dirichlet problems of the form (7), due to a breakdown in a generalization of the proof of the weak maximum principle of [4]. In [12] it is shown that interior regularity of weak solu- tions to (7) is obtained via a weak local Sobolev inequality. It is assumed in [12], and shall be for the rest of this work, that ∃σ ∈ (1, 2] and C, δ > 0 such that if 0 (12) 1 1 2σ |w|2σ |B (x)| r Br (x) 1 1 ≤ 1 Q ∇ 2 1 | |2 2 C r | | (y, w) + | | w Br(x) Br (x) Br(x) Br (x) for all w ∈ Lip0(Br(x)) where Br(x) is the Carnot-Carath´eodory control ball of radius 0 The Poincar´e Inequality: ∃Cp > 0 such that for x ∈ Ωand0 holding for all w ∈ Lip(Br(x)). 1 Lastly, the setting in which the following takes place is the Hilbert space QH0 (Θ) 1,2 Q = WQ (Θ). Due to the degeneracy of the quadratic form , the gradient of an License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 676 SCOTT RODNEY ∈ 1 2 element u QH0 (Θ) may not be uniquely determined by its projection onto L . This strange behavior does not have strong implications for this work but does play a role in regularity theory; see [13]. 1 To define the degenerate Sobolev space QH0 (Θ) for an open bounded subset Θ of the homogeneous space (Ω,d), we first define the related space L2(Θ, Q). The following development of the space L2(Θ, Q) is due to E. Sawyer and R.L. Wheeden in [13]. Define the form weighted vector space L2(Θ, Q) to consist of all measurable vector valued functions h(x)=(h (x), ..., h (x)) defined a.e. on Θ for which 1 n 1 2 (14) ||h||L2(Θ,Q) = Q(x, h(x))dx < ∞. Θ 2 If one identifies two functions h, g ∈L(Θ, Q)if||h − g||L2(Θ,Q) =0,then(14) defines a norm on the vector space L2(Θ, Q). As is shown in [13], with the norm (14) L2(Θ, Q) is a Hilbert space with respect to the inner product (15) (16) (w, v)Q = w(x)v(x)dx+ < ∇w, ∇v>L2(Θ,Q) . Θ This inner product generates a norm on Lip0(Θ) given by 1 1 2 2 (17) ||w|| 1 =(w, w) = ||w|| + ||∇w||L2 Q . QH0 (Θ) Q 2 (Θ, ) 1 1,2 We define the space QH0 (Θ) = WQ (Θ)asthecompletionofLip0(Θ) under the 1 norm (17). QH0 (Θ) is then a Hilbert space with respect to the inner product (16); see [7] for more details. 1 The space QH0 (Θ) consists of equivalence classes of Cauchy sequences of Lip- schitz functions. The equivalence class represented by a Cauchy sequence {uj }⊂ Lip0(Θ) will be denoted by [{uj }]. Due to the nature of the norm (17), if {wj }⊂ 2 Lip0(Θ) is Cauchy with respect to (17), then {wj } is also Cauchy in L (Θ) and 2 {∇wj } is Cauchy in L (Θ, Q). Thus for the equivalence class [{wj }]thereare 2 2 unique functions wL2 ∈ L (Θ),w∇ ∈L(Θ, Q) associated to it. As discussed in J 1 [13] this reasoning gives rise to a Hilbert space isomorphism between QH0 (Θ) 2 ×L2 Q W1,2 and a closed subspace of L (Θ) (Θ, ) denoted by Q (Θ). This Hilbert space isomorphism is defined by (18) J ([{uj }]) = (uL2 ,u∇). ∈ 1 Remark 1.2. The gradient of a function u QH0 (Θ)isamemberofthespace L2 Q J 1 W1,2 (Θ, ). Due to the isomorphism of QH0 (Θ) into Q (Θ) one identifies u = (uL2 ,u∇). One may be tempted to believe that u∇ is uniquely determined by 2 the “L -part” of u, uL2 . As can be seen in an example due to Fabes, Kenig and Serapioni in [1], this is in fact not the case. This does not affect the work to be 1 → 2 presented here, but we shall need to consider the natural map I : QH0 (Θ) L (Θ) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A DEGENERATE SOBOLEV INEQUALITY IN A HOMOGENEOUS SPACE 677 ◦J W1,2 defined as I = P ,whereP is projection of Q (Θ) onto the first component, such that (19) I([{uj }]) = uL2 . Due to the example of Fabes, Kenig and Serapioni, the natural map may not be one to one; however, as will be seen below, I is a compact mapping, which is the essential property to be exploited here. Some care has been taken to make a distinction between an equivalence class { } 1 ∈W1,2 [ uj ]ofQH0 (Θ) and a pair (uL2 ,u∇) Q (Θ). However, what follows will contain some abuses of notation to simplify equations and inequalities. Indeed, u 1 will be used to denote an element [{uj }] ∈ QH (Θ), and this same u will also be 2 used to denote its L -part uL2 . The difference will be taken as understood in the context presented. Further, by the definitions given above the Sobolev inequality 1 Q (12) extends by density to QH0 (Br) functions. The quadratic form also gives rise to a pointwise inner product and norm which are useful for simplifying notation. For 2 v, w ∈L (Θ, Q)wewrite holds for all w ∈ Lip0(Θ) provided the following conditions hold: 1. the weak local Sobolev inequality (12); 2.thePoincar´einequality(13). The main obstacle to proving theorem (1.1) is a weak maximum principle requir- 1 → ing the following compactness result concerning the natural map I : QH0 (Θ) L2(Θ), defined by I(u)=u. This result is similar to theorem 3.4 in [3] but does not require the context of Lipschitz vector fields. Theorem 1.2. Let Θ Ω be open with (Ω,d) a homogeneous space. Let the qua- Q · 1 → dratic form (x, ) be as in Theorem ( 1.1). Then the natural map I : QH0 (Θ) L2(Θ) is compact provided the local Poincar´einequality(13) holds in Ω. Theorem (1.2) evolved substantially through discussion of [3] with S. K. Choi of the Singapore National University and R. Wheeden of Rutgers University and the author thanks them for their comments. Theorem (1.2) can be shown to hold in a much broader context and is the subject of current work joint with the authors just mentioned. This work may be found in the near future under the title ”A Compactness Theorem for Generalized Sobolev Spaces”. The last step in obtaining the strong Sobolev inequality (9) is to exploit the close connection between the existence of weak solutions to a family of Dirichlet problems related to the principal part of the operator X and Sobolev inequalities. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 678 SCOTT RODNEY Corollary 1.1. Let 0 <τ<1 be given and let q>2σ where σ is as in (12).If q the local weak Sobolev inequality (12) holds in Ω and f ∈ L 2 (Θ), then there exists a weak solution of the homogeneous Dirichlet problem ∇ Q(x)∇u − τu = f in Θ (21) u =0on ∂Θ. Further, if ||f||q/2 =1, ∃C = C(σ, q, Θ,d) > 0 so that this weak solution satisfies the global estimate (22) sup |u|≤C Θ where C is independent of τ. Theorem 1.1 will be achieved via Corollary 1.1, using the fact that the constant in (22) is independent of τ>0 in a slight modification of the proof of the local Sobolev inequality appearing in [12], which was achieved via Lq-subellipticity for the homogeneous Dirichlet problem. 2. Proofs { } 1 Proof of Theorem 1.2: Let fn be a sequence in the unit ball of QH0 (Θ). Then 1 { } from the definition of the space QH0 (Θ)wehavethat fn is a bounded sequence in L2(Θ) and so contains a weakly convergent subsequence in L2(Θ) which we 2 relabel as {fn}. Our goal is to show that {fn} is Cauchy in L (Θ) giving that {fn} converges in norm. Let >0 be given and let 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A DEGENERATE SOBOLEV INEQUALITY IN A HOMOGENEOUS SPACE 679 Thus, we may fix r so that 0 2 n Since χ j ∈ L (Ω) as Ω R we may choose, using the weak convergence of {f }, Br n m, n sufficiently large so that (25) II ≤ . { } 2 1 Thus fn is Cauchy in L (Θ). Therefore, any bounded sequence in QH0 (Θ) contains a subsequence convergent in L2(Θ) giving that I is indeed a compact mapping. Proof of Corollary 1.1 To begin, we obtain a weak Sobolev inequality on the set Θ via a standard partition of unity argument. The proof of the following lemma is omittedhereasitiselementary. Lemma 2.1. Let Θ be an open set with Θ Ω and assume that the local weak Sobolev inequality (12) holds in Ω.Then∃C = C(σ, Θ) > 0 such that 1 1 1 1 | |2σ 2σ ≤ 1 ∇ ∇ 2 1 | |2 2 (26) | | w C | | wQ(x) w + | | w Θ Θ Θ Θ Θ Θ for any w ∈ Lip0(Θ). Consider the operator associated to L = ∇ Q(x)∇ given by Lτ u = ∇ Q(x)∇u − τu and the associated Dirichlet problem Lτ u = f in Θ (27) u =0on∂Θ. By assumption, f ∈ Lq/2(Θ) for some q>2σ , σ being the dual of the exponent σ appearing in (12). The associated bi-linear form for this Dirichlet problem acting 1 × 1 on QH0 (Θ) QH0 (Θ) is given by: (28) Lτ (u, v)= < ∇u, ∇v>Q +τ uv. Θ Θ Thus, a weak solution of (27) must then satisfy: (29) Lτ (u, v)=− fv Θ ∈ 1 1 for any v QH0 (Θ). The linear functional acting on QH0 (Θ) given by Φ(v)= − fv is clearly continuous since ||f||2 ≤ C||f||q/2 and ||v||2 ≤||v||QH1(Θ).Fur- Θ L 1 ther, the bi-linear form τ is both bounded and coercive on QH0 (Θ). Indeed, the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 680 SCOTT RODNEY pointwise inequality < ∇u, ∇v>Q≤ ||∇u||Q||∇v||Q, holding almost everywhere in Θ, together with H¨older’s inequality gives Lτ (u, v) ≤ ||u||Q||v||Q + ||u||2||v||2 Θ 1/2 1/2 ≤ ||∇ ||2 ||∇ ||2 || || || || u Q v Q + u 2 v 2 Θ Θ (30) ≤ 2||u||QH1(Θ)||v||QH1(Θ) and hence boundedness of Lτ . The coercivity of Lτ is immediate since 0 <τ<1. Indeed, || ||2 ||∇ ||2 || ||2 τ u QH1(Θ) = τ u Q + τ u 2 Θ ≤ ||∇ ||2 || ||2 u Q + τ u 2 Θ (31) = Lτ (u, u). Therefore, by the Lax-Milgram theorem there exists a unique weak solution to (27). For the second part of the theorem we follow the method of proof outlined in Theorem 8.15 of [4]; this argument also appears in [5]. If u is a weak solution to ∈ 1 + ∈ 1 (27), then by definition u QH0 (B). It is clear that we then have u QH0 (Θ), where u+ is defined as in [7], so that u+ is a valid test function for u, i.e. (32) (∇u+) Q∇u = − (fu+). Θ Θ This equality is key and should be kept in mind for what will follow. Set w = u+ +1. For β ≥ 1 we define, for N>1, HN :[1, ∞) → R+ by ⎧ ⎨ 0ifs<1 β − ≤ ≤ HN (s)=⎩ s 1if1s N βNβ−1(s − N)+N β − 1ifs>N. 1 ∞ ∈ ∞ This function is clearly C ([1, )) with HN L and HN (1) = 0. The lemma on ∈ 1 admissible compositions in [12] and [7] then gives HN (w) QH0 (Θ). Next, define the test function GN :[1, ∞) → R+ by x 2 GN (x)= (HN (s)) ds, 1 which satisfies the key inequality ≤ (33) GN (x) xGN (x). ∈ 1 We require that GN QH0 (Θ) in order to be able to test the weak solution against it. However, since GN (1) = 0 we have by the lemma on admissible compo- ∈ 1 sitions in [12] and [7] that indeed G(w) QH0 (Θ). The weak Sobolev inequality License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A DEGENERATE SOBOLEV INEQUALITY IN A HOMOGENEOUS SPACE 681 (26) then gives: 1 2 1 σ C |H (w)|2σ ≤ ||∇H (w)||2 + ||H (w)||2 by (26) 1 N | | N Q N 2 |Θ| σ Θ Θ Θ C2 = (H (w))2(∇w) Q∇w + ||H (w)||2 |Θ| N N 2 Θ 2 C ∇ ∇ || ||2 = | | ( GN (w)) Q u + HN (w) 2 . Θ Θ The last equality is due to the fact that ∇w = ∇u+ = ∇u where ∇u+ =0andif + u =0wehavethatw = 1, giving GN (w) = 0. Thus, using the fact that u is a weak solution to (27), 1 2 1 σ C |H (w)|2σ ≤ |f|G (w)+(1− τ)||H (w)||2 . 1 N | | N N 2 |Θ| σ Θ Θ Θ Since w ≥ 1, taking limits in all of the above we have by (30) and (33) that 1 2 | |2σ σ ≤ C | | 2 − || ||2 HN (w) σ−1 f w GN (w)+(1 τ) HN (w) 2 Θ |Θ| σ Θ 2 C | | 2 || ||2 = σ−1 f (wHN (w)) + HN (w) 2 . |Θ| σ Θ Now HN and HN are increasing in N and (34) holds for every N. The monotone convergence theorem applied to each side of the equation then yields: 2 || β − ||2 ≤ C | | · β−1 2 || β − ||2 w 1 2σ σ−1 f (w βw ) + w 1 2 | | σ Θ Θ 2 2β 2 2β (34) ≤ C(Θ) β ||w || q + β ||w || q , q−2 q−2 q since w ≥ 1and1< due to the choice of q>2σ . Taking square roots and q − 2 using the homogeneity of the Lp norm, we obtain the reverse H¨older estimate: β β (35) ||w ||2σ ≤ Mβ||w || 2q q−2 σ(q − 2) where M = M(Θ). Setting χ = > 1, since q>2σ , we rewrite the above q as 1 β (36) ||w||βχq∗ ≤ Mβ ||w||βq∗ 2q ∗ ∗ where q∗ = . Hence, (36) says that w ∈ Lβq implies w ∈ Lβχq which is a q − 2 stronger inclusion since χ>1. Following the argument found in [5] Theorem 3.1 we obtain: + + (37) sup u ≤ sup w ≤ C(σ, q, Θ) ||u ||2 +1 . y∈Θ y∈Θ Replacing u+ with u− in all of the above yields the inequality − − (38) sup u ≤ C(σ, q, Θ) ||u ||2 +1 , y∈Θ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 682 SCOTT RODNEY giving (39) ||u||L∞(Θ) ≤ C(σ, q, Θ, )(||u||2 +1). To finish the proof we bound ||u||2 independently of τ. Thus, we seek to show that if uτ is a weak solution to the problem (27), then ||uτ ||2 ≤ C where C is independent of τ. Suppose the inequality is not true. Then ∃ a sequence {τk}k and a sequence { } of Q-weak solutions vτk of the Dirichlet problems Lτk u = f in Θ (40) u =0on∂Θ || || || || →∞ { } with f q/2 =1(q>2σ ) such that vτk 2 . This forces the sequence τk to have limit zero. Indeed, if we assume τk → τ>0 we obtain from (31) that for each k ∈ N, τ ||v ||2 ≤L(v ,v ) k τk 2 τ τk τk ≤| | vτk f Θ ≤|| || (41) vτk 2. || || ≤ 1 ∞ || || →∞ → This yields vτk 2 < , contradicting vτk 2 .Thusτk 0. Set τk 1 β = (so β → 0ask →∞), w = β v and f = β f.Thenw is a k || || k k k τk k k k vτk 2 Q-weak solution of the Dirichlet problem Lτk u = fk in Θ (42) u =0on∂Θ. Now, we have that ||wk||2 =1foreachk and so by the compactness of the natural 1 → 2 map I : QH0 (Θ) L (Θ), from Theorem 1.2, we have that there is a subsequence { }∞ → 2 wk k=0 (relabeled so that we keep the same index) such that wk w in L and || || → 1 ∈ 1 w 2 = 1. Further, we also have wk w in QH0 (Θ), giving w QH0 (Θ) so that we may test w against itself. This is due to the inequality ||∇ ||2 ≤L wk Q τk (wk,wk) Θ (43) ≤||wk||2||fk||q/2 = βk. Thus we obtain w as a Q-weak solution of the Dirichlet problem Lv = ∇ Q(x)∇v = 0 in Θ (44) v =0on∂Θ. By the Poincar´e inequality (13) we then obtain that on any ball Br with r>0 sufficiently small: | − |2 (45) w wBr =0, Br giving that w is constant on any ball Br with r>0 sufficiently small and hence that w is constant in Θ. Without loss of generality we assume w = 1 in Θ. To obtain a contradiction we now show that (45) implies w =0inL2(Θ). Let Γ Ωbean open set so that Θ¯ ⊂ Γ ⊂ Γ¯ Ω. Let {wj }j ⊂ Lip0(Θ) be a sequence representing 1 { } ⊂ \ w in QH0 (Θ). Define w¯j j Lip0(Γ) wherew ¯j = wj in Θ andw ¯j = 0 in Γ Θ. 2 Setw ¯ = w in Θ and w = 0 in Γ \ Θ.Clearlywehavethat¯¯ wj → w¯ in L (Γ) and we License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A DEGENERATE SOBOLEV INEQUALITY IN A HOMOGENEOUS SPACE 683 also obtain Q(x, ∇w¯j) → 0asj →∞since (∇wj) Q(x)∇wj → (∇w) Q(x)∇w Γ in L1(Θ). Thus we obtain via the Poincar´e inequality thatw ¯ is constant in Γ. Indeed, from the Poincar´e inequality we have that for any ball B ⊂ Γ with radius sufficiently small, | − |2 ≤ ∇ ∇ w¯j w¯jB Cpr ( w¯j ) Q(x) w¯j B B (46) ≤ (∇w¯j ) Q(x)∇w¯j → 0asj →∞. Γ Sincew ¯ is zero in any ball of radius sufficiently small in Γ \ Θwethenhavethat¯ w¯ = 0 in Γ and hence w = 0 in Θ, which is a contradiction since ||w||L2(Θ) =1. 2 Proof of Theorem 1.1 The proof of this result mimics the proof of the local weak Sobolev inequality (12) in [12], with modifications since we use the global σ +1 q weak Sobolev inequality (11). Set η = ;thus1<η<σ.Letf ∈ L 2 (Θ); be 2 such that ||f|| q =1withq =2η > 2σ .Letu be the unique weak solution to the 2 Dirichlet problem (27) with τ>0 to be chosen in a moment. Thus, for v ∈ Lip0(Θ) we have: 2 2 2 v f = − < ∇v , ∇u>Q −τ uv Θ Θ Θ 2 = −2 v<∇v, ∇u>Q −τ uv Θ Θ 1/2 1/2 ≤ 2||∇ ||2 || ||2 2 2 v u Q v Q + τ uv . Θ B Θ Using the fact that u is a weak solution, the square of the first factor satisfies: 2||∇ ||2 ∇ 2 ∇ − ∇ ∇ v u Q = < (uv ), u>Q 2 Θ Θ 2 2 2 = −2 Absorbing the first term on the right into the left hand side and multiplying by 2 yields: 2 2||∇ ||2 ≤ | | ||∇ ||2 | | 2 2 2 v u Q 4 sup u v Q +2 sup u v f +2τ u v Θ Θ Θ Θ Θ Θ (48) 2 2 ≤ | | ||∇ ||2 | | 2 | | 2 4max sup u v Q , sup u v f, sup u τ v . Θ Θ Θ Θ Θ Θ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 684 SCOTT RODNEY No matter which term on the right is the maximum, (48) with (29) gives: 2 ≤ | | ||∇ ||2 | | 2 v f C sup u v Q + τ sup u v B Θ Θ Θ Θ 2 2 2 1 q q 2 1 q q 2 ≤ C|Θ| q |f| 2 ||∇v|| + τ |f| 2 v |Θ| Q |Θ| Θ Θ Θ Θ 1 1 1 1 η 1 η | | η | |η ||∇ ||2 | |η 2 (49) = C Θ | | f v Q + τ | | f v . Θ Θ Θ Θ Θ Θ Thus, we have: 1 1 η 1 |v|2η =sup v2f | | | | Θ Θ 1 | |η Θ Θ |Θ| Θ f =1 1 1 ≤ | | η ||∇ ||2 2 (50) C Θ | | v Q + τ v . Θ Θ Θ Taking square roots of both sides gives: 1 1 1 1 2η 2 1 2 | |2η ≤ ||∇ ||2 2 2 (51) | | v C v Q + τ v . Θ Θ Θ Θ At this point we are free to choose τ as close to zero as we like. We choose τ>0 sufficiently small so that we may absorb the L2 norm on the right hand side into the L2η normontheleft.Weareabletodothissince2≤ 2η. This yields the Sobolev inequality 1 1 1 | |2η 2η ≤ 1 ||∇ ||2 2 (52) | | v C | | v Q , Θ Θ Θ Θ holding for all v. Combining this result with the weak Sobolev inequality on Θ, (26), we obtain: 1 1 1 2σ 2 2η (53) |w|2σ ≤ C Q(x, ∇w) + C˜ |w|2η Θ Θ Θ 1 2 ≤ C Q(x, ∇w) , Θ for w ∈ Lip0(Θ). References 1. Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7, no. 1 (1982), 77–116. MR643158 (84i:35070) 2. C. Fefferman and D.H. Phong, The local regularity of solutions to degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77–116. 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