Regularity near the characteristic boundary for sub-laplacian operators

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372 DIMITER VASSILEV the supremum is achieved, that is, for every n there exist a h n ∈ G such that ( ) ∫ 1 (3-12) ˆQ n = |u r n (g)| d H(g). n B 1/rn (h n ) The concentration functions Q n of the dilated and translated sequence def (3-13) v n = rn −Q/p∗ δ r −1τ gn u n satisfy ∫ (3-14) Q n (1) = dν n B 1 (e) and Q n (1) = 1 2 , where we have set g n = δ dn h −1 n . The homogeneity properties of the metric are essential for proving (3-14). From the definition of v n and (2-8), ∫ ∫ (3-15) |v n (h)| p∗ d H(h) = B r (g) ∫ = ∫ = ∫ = ∫ = {d(g,h)
EXISTENCE AND REGULARITY FOR SUB-LAPLACIAN EQUATIONS 373 Since I λ = λ p/p∗ I , we see that I λ is strictly subadditive, that is, (3-19) I 1 for every α ∈ (0, 1). Assume first that the compactness case in Lemma 3.3 holds when applied to the sequence dν n . Let g n be points as in the lemma and, for ε > 0, choose R = R(ε), such that ∫ (3-20) dν n ≥ 1 − ε for every n. B R (g n ) If ε < 1 / 2 , then ∫ B R (g n ) dν n > 1 / 2 . Since by construction ∫ B 1 (e) dν n = 1 / 2 while ∫ G dν n = 1, we see that B 1 (e) and B R (g n ) have a nonempty intersection, and thus there exist a ball B 2R+1 (e) ⊃ B R (g n ). This implies that ∫ ∫ (3-21) dν n ≥ B 2R+1 (e) dν n ≥ 1 − ε for every n, B R (g n ) and therefore the condition (1) of Lemma 3.3 holds with g n ≡ e for every n. By taking ε → 0, we proved that ∫ (3-22) dν = 1. G If we look at the sequence (dµ n ) now, we have dµ n ⇁ dµ and ∫ G dµ n → I , and thus ∫ G dµ ≤ I . On the other hand, Lemma 3.5 gives dν n ⇁ dν =|v| p∗ + ∑ j d j δ gj , dµ n ⇁ dµ ≥|Xv| p d H + ∑ j e j δ gj , for certain d j , e j ≥ 0 satisfying (3-23) I d j p/p ∗ ≤ e j . We shall prove that all d j ’s are zero, and thus ∫ G |v|p∗ d H = 1. Let ∫ α def = |v| p∗ d H < 1. Since ∫ G dν = 1, we have ∑ d j = 1 − α. From ∫ G dµ ≤ I we have ∫ |Xv| p d H ≤ I − ∑ e j . Now (3-23) gives ∫ I = I 1 ≥ G G G |Xv| p d H + ∑ e j ≥ I α + ∑ I d j p/p ∗ ≥ I α + ∑ I dj .
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Regularity near the characteristic boundary for sub-laplacian operators

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