Correction to: On the $$L^{r}$$ L r Hodge theory in complete non compact Riemannian manifolds
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Mathematische Zeitschrift
CORRECTION
Correction to: On the Lr Hodge theory in complete non compact Riemannian manifolds Eric Amar1 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Correction to: Math. Z. (2017) 287:751–795 https://doi.org/10.1007/s00209-017-1844-9 Our aim is to correct the proofs of Lemma 5.2 and Lemma 5.3 of original article. We use the notation of original article’s [Section 5]. For Lemma 5.2, the correction is straightforward: we replace the false (in general) estimate Au L r (Bx (R)) ≤ ∇g L ∞ (Bx (R)) R u L r (Bx (R)) + η()∇u L r (Bx (R)) , by the right one Au L r (Bx (R)) ≤ ∇g L ∞ (Bx (R)) uW 2,r (Bx (R)) , which is proved in original article. The remainder of the computation is exactly the same and concludes the proof of Lemma 5.2. In order to prove Lemma 5.3, we shall need the following tools. Lemma 0.1 Let B R := B(0, R) be the ball in Rn with center 0 and radius R ≤ 1 and B R = B(0, 3R/4). Suppose we have, with a constant C depending only on n, r , and the C 1 bound of the coefficients of ϕ , ∀v ∈ W 2,r (B1 ) vW 2,r (B ) ≤ C v L r (B1 ) + ϕ v L r (B ) . 1
1
Let u ∈ L r (B R ) be such that ϕ u ∈ L r (B R ). Then, u ∈ W 2,r (B R ) and uW 2,r (B ) ≤ c1 R −2 u L r (B R ) + c2 ϕ u L r (B ) , R
R
where the constants c1 , c2 depend only on n, r , and the
C 1 -bound
of the coefficients of ϕ .
The original article can be found online at https://doi.org/10.1007/s00209-017-1844-9.
B 1
Eric Amar [email protected] Université de Bordeaux, Talence, France
123
E. Amar
Proof We start with R = 1 and B := B(0, 1). We have by assumption vW 2,r (B ) ≤ C v L r (B1 ) + ϕ v L r (B ) , ∀v ∈ W 2,r (B1 ) 1
1
the constants C depending only on n, r and the C 1 bound of the coefficients of ϕ . It remains to make the simple change of variables y = Rx, dm(y) = R n dm(x), v(x) := u(Rx) and to notice that ∂ j v(x) = R∂ j (u)(Rx), ∂i2j v(x) = R 2 ∂i2j (u)(Rx) in the integrals defining the L r -norm, to get the result.
Lemma 0.2 Let ϕ be a second-order elliptic matrix operator with C ∞ coefficients operating on p-forms defined in U ⊂ Rn . Let B := B(0, R) be a ball in Rn , set B := B(0, 3R/4), and suppose that B U . Then we have an interior estimate: there are constants c1 , c2 depending only on n = dimR M, r and the C 1 norm of the coefficients of ϕ in B¯ such that (0.1) ∀v ∈ W p2,r (B) vW 2,r (B ) ≤ c1 R −2 v L r (B) + c2 ϕ v L r (B) . Proof For 0-forms, this lemma is exactly [1, Theorem 9.11] plus Lemma 0.1 to get the dependency in R. For p-forms, we cannot avoid the use of deep results on elliptic systems of equations. Let v be a p-form in B ⊂ Rn . We use the interior estimates in [2, Section 6.2, Theorem 6.2.6]. In our context of a second-order elliptic system, and with our notation, with r > 1, we get vW 2,r (B ) ≤ c1 R −2 v L r (B) + c2 ϕ v L r (B) , ∃C > 0 ∀v ∈ W p2,r (B) already including the dependency in R. The constants c1 , c2 depend only on r , n := dim M, and th
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