# Correction to: On the \(L^{r}\) Hodge theory in complete non compact Riemannian manifolds

## 1 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].

*false (in general) estimate*

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

*C*depending only on

*n*,

*r*, and the \({\mathcal {C}}^{1}\) bound of the coefficients of \(\Delta _{\varphi }\),

*n*,

*r*, and the \({\mathcal {C}}^{1}\)-bound of the coefficients of \(\Delta _{\varphi }.\)

### Proof

*n*,

*r*and the \({\mathcal {C}}^{1}\) bound of the coefficients of \(\Delta _{\varphi }.\) 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 \(\partial _{j}v(x)=R\partial _{j}(u)(Rx),\ \partial ^{2}_{ij}v(x)=R^{2}\partial ^{2}_{ij}(u)(Rx)\) in the integrals defining the \(L^{r}\)-norm, to get the result. \(\square \)

### Lemma 0.2

*p*-forms defined in \(U\subset {\mathbb {R}}^{n}.\) Let \(B:=B(0,R)\) be a ball in \({\mathbb {R}}^{n}\), set \(B':=B(0,3R/4)\), and suppose that \(B\Subset U.\) Then we have an interior estimate: there are constants \(c_{1},c_{2}\) depending only on \(n=\dim _{{\mathbb {R}}}M,\ r\) and the \({\mathcal {C}}^{1}\)norm of the coefficients of \(\Delta _{\varphi }\) in \(\bar{B}\) such that

### 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.

*p*-form in \(B\subset {\mathbb {R}}^{n}.\) 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

*R*.

The constants \(c_{1},c_{2}\) depend only on *r*, \(n:=\dim M\), and the bounds and moduli of continuity of the coefficients of the matrix \(\Delta _{\varphi }.\)

In particular, if the coefficients of \(\Delta _{\varphi }\) are close to those of \(\Delta _{{\mathbb {R}}}\) in the \({\mathcal {C}}^{1}\) norm, the constants \(c_{1},c_{2}\) are close to the ones obtained for \(\Delta _{{\mathbb {R}}}.\)\(\square \)

We are now in position to prove Lemma 5.3, which is the following.

### Lemma 5.3

*M*,

*g*) be a riemannian manifold. For \(x\in M,\ \epsilon >0,\) we take a \(\epsilon \)-admissible ball \(B_{x}(R).\) We have a local Calderon Zygmund inequality on the manifold

*M*: there are constants \( c_{1},c_{2}\) depending only on \(n=\dim _{{\mathbb {R}}}M,\ r\), and \(\epsilon \) such that

### Proof

*p*-form

*u*read in the chart \((V,\varphi ),\)

*n*,

*r*, and \(\epsilon \).

*U*and the canonical measure \(dv_{g}\) on \(B_{x}(R)\) are equivalent; more precisely, because of condition 1), we have\((1-\epsilon )^{n}\le \left| {\det g}\right| \le (1+\epsilon )^{n},\) and the measure \(dv_{g}\) read in the chart \(\varphi \) is \(dv_{g}={\sqrt{\left| {\det g_{ij}}\right| }}\,d\xi ,\) where \(d\xi \) is the Lebesgue measure in \({\mathbb {R}}^{n}.\) So the Lebesgue estimates and the Sobolev estimates up to order 2 on

*U*are valid in \(B_{x}(R),\) up to a constant depending only on

*n*,

*r*and \(\displaystyle \epsilon \) by Lemma 4.2 in original article. In particular,

*R*for 3

*R*/ 2, this proves Lemma 5.3. \(\square \)

## Notes

## References

- 1.Gilbarg, D., Trudinger, N.: Elliptic Partial Differential equations, volume 224 of Grundlheren der mathematischen Wissenschaften. Springer, (1998)Google Scholar
- 2.Morrey, C.B.: Multiple Integrals in the calculus of variations, volume 130 of Die Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, New York (1966)Google Scholar