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

  • Eric AmarEmail author
Correction

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].

For Lemma 5.2, the correction is straightforward: we replace the false (in general) estimate
$$\begin{aligned} {\left\| {Au}\right\| }_{L^{r}(B_{x}(R))}\le {\left\| {\nabla g}\right\| }_{L^{\infty }(B_{x}(R))} {\left\| {\Delta _{{\mathbb {R}}}u}\right\| }_{L^{r}(B_{x}(R))} +\eta (\epsilon ){\left\| {\nabla u}\right\| }_{L^{r}(B_{x}(R))}, \end{aligned}$$
by the right one
$$\begin{aligned} {\left\| {Au}\right\| }_{L^{r}(B_{x}(R))}\le {\left\| {\nabla g}\right\| }_{L^{\infty }(B_{x}(R))} {\left\| {u}\right\| }_{W^{2,r}(B_{x}(R))}, \end{aligned}$$
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 \({\mathbb {R}}^{n}\) with center 0 and radius \(R\le 1\) and \(B'_{R}=B(0,3R/4)\). Suppose we have, with a constant C depending only on n, r, and the \({\mathcal {C}}^{1}\) bound of the coefficients of \(\Delta _{\varphi }\),
$$\begin{aligned} \forall v\in W^{2,r}(B_{1})\qquad {\left\| {v}\right\| }_{W^{2,r}(B_{1}')}\le C\left( {\left\| {v}\right\| }_{L^{r}(B_{1})} +{\left\| {\Delta _{\varphi }v}\right\| }_{L^{r}(B_{1})}\right) . \end{aligned}$$
Let \(u\in L^{r}(B_{R})\) be such that \(\Delta _{\varphi }u\in L^{r}(B_{R})\). Then, \(u\in W^{2,r}(B_{R}')\) and
$$\begin{aligned} {\left\| {u}\right\| }_{W^{2,r}(B_{R}')}\le c_{1}R^{-2}{\left\| {u}\right\| }_{L^{r}(B_{R})} +c_{2}{\left\| {\Delta _{\varphi }u}\right\| }_{L^{r}(B_{R})}, \end{aligned}$$
where the constants \(c_{1},c_{2}\) depend only on n, r, and the \({\mathcal {C}}^{1}\)-bound of the coefficients of \(\Delta _{\varphi }.\)

Proof

We start with \(R=1\) and \( B:=B(0,1).\) We have by assumption
$$\begin{aligned} \forall v\in W^{2,r}(B_{1})\qquad {\left\| {v}\right\| }_{W^{2,r}(B_{1}')} \le C\left( {\left\| {v}\right\| }_{L^{r}(B_{1})} +{\left\| {\Delta _{\varphi }v}\right\| }_{L^{r}(B_{1})}\right) , \end{aligned}$$
the constants \(\displaystyle C\) depending only on nr 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

Let \(\Delta _{\varphi }\) be a second-order elliptic matrix operator with \({\mathcal {C}}^{\infty }\) coefficients operating on 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
$$\begin{aligned} \forall v\in W_{p}^{2,r}(B)\qquad {\left\| {v}\right\| }_{W^{2,r}(B')} \le c_{1}R^{-2}{\left\| {v}\right\| }_{L^{r}(B)} +c_{2}{\left\| {\Delta _{\varphi }v}\right\| }_{L^{r}(B)}. \end{aligned}$$
(0.1)

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 \(\displaystyle v\) be a 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
$$\begin{aligned} \exists C>0\ \ \forall v\in W_{p}^{2,r}(B)\qquad {\left\| {v}\right\| }_{W^{2,r}(B')} \le c_{1}R^{-2}{\left\| {v}\right\| }_{L^{r}(B)} +c_{2}{\left\| {\Delta _{\varphi }v}\right\| }_{L^{r}(B)}, \end{aligned}$$
already including the dependency in 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

Let (Mg) 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
$$\begin{aligned} \forall u\in W^{2,r}(B_{x}(R))\qquad {\left\| {u}\right\| }_{W^{2,r}(B_{x}(R/2))} \le c_{1}R^{-2}{\left\| {u}\right\| }_{L^{r}(B_{x}(R))} +c_{2}{\left\| {\Delta u}\right\| }_{L^{r}(B_{x}(R))}. \end{aligned}$$

Proof

We transcribe the problem in \({\mathbb {R}}^{n}\) by use of a coordinates chart \((V,\varphi ).\) The Hodge laplacian is the second-order elliptic matrix operator \(\Delta _{\varphi }\) with \({\mathcal {C}}^{\infty }\) coefficients operating in \(\varphi (V)\subset {\mathbb {R}}^{n}.\) By the choice of a \(\epsilon \)-admissible ball \(B_{x}(R),\) and with \(R':=3R/4,\) we have
$$\begin{aligned} U':=\varphi (B_{x}(R'))\subset B_{e}(0,(1+\epsilon )R'),\ U:=\varphi (B_{x}(R))\subset B_{e}(0,(1+\epsilon )R)\subset \varphi (V). \end{aligned}$$
We apply Lemma 0.2 to the euclidean balls \(B':=B_{e}(0,(1+\epsilon )R')\) and \( B:=B_{e}(0,(1+\epsilon )R)\), and we get, with \(u_{\varphi }\) the p-form u read in the chart \((V,\varphi ),\)
$$\begin{aligned} {\left\| {u_{\varphi }}\right\| }_{W^{2,r}(B')} \le c_{1}R^{-2}{\left\| {u_{\varphi }}\right\| }_{L^{r}(B)} +c_{2}{\left\| {\Delta _{\varphi }u_{\varphi }}\right\| }_{L^{r}(B)}. \end{aligned}$$
The fact that the coefficients of \(\Delta _{\varphi }\) are \(\epsilon \)-close to those of \(\Delta _{{\mathbb {R}}}\) in the \({\mathcal {C}}^{1}\) norm, by condition 2) in the definition of the \(\epsilon \)-admissible ball, implies that the constants \(c_{1},c_{2}\) depend only on n, r, and \(\epsilon \).
The Lebesgue measure on 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 nr and \(\displaystyle \epsilon \) by Lemma 4.2 in original article. In particular,
$$\begin{aligned} \forall x\in M\qquad \hbox {Vol}(B_{x}(R))\le (1+\epsilon )^{n/2}\nu _{n}R^{n}, \end{aligned}$$
(0.2)
where \(\nu _{n}\) is the euclidean volume of the unit ball in \({\mathbb {R}}^{n}.\)
So we get, with \(A:=B_{x}((1+2\epsilon )R)\supset \varphi ^{-1}(B)\) and \(A':=\varphi ^{-1}(B')\),
$$\begin{aligned} {\left\| {u}\right\| }_{W^{2,r}(A')}\le c_{1}R^{-2}{\left\| {u}\right\| }_{L^{r}(A)}+c_{2}{\left\| {\Delta u}\right\| }_{L^{r}(A)}). \end{aligned}$$
Now we notice that \(A'\supset B_{x}(R'),\) so a fortiori,
$$\begin{aligned} {\left\| {u}\right\| }_{W^{2,r}(B_{x}(R'))} \le c_{1}R^{-2}{\left\| {u}\right\| }_{L^{r}(B_{x}((1+2\epsilon )R)} +c_{2}{\left\| {\Delta u}\right\| }_{L^{r}(B_{x}((1+2\epsilon )R)}. \end{aligned}$$
Finally, choosing \(\epsilon \le 1/4,\) we get
$$\begin{aligned} {\left\| {u}\right\| }_{W^{2,r}(B_{x}(R'))} \le c_{1}R^{-2}{\left\| {u}\right\| }_{L^{r}(B_{x}(3R/2)} +c_{2}{\left\| {\Delta u}\right\| }_{L^{r}(B_{x}(3R/2)}, \end{aligned}$$
which is the local Calderon Zygmund inequality on \(B_{x}(3R/4)\subset B_{x}(3R/2)\subset M.\) So changing R for 3R / 2,  this proves Lemma 5.3. \(\square \)

Notes

References

  1. 1.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential equations, volume 224 of Grundlheren der mathematischen Wissenschaften. Springer, (1998)Google Scholar
  2. 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

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université de BordeauxTalenceFrance

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