1 Correction to: Proc. Indian Acad. Sci. (Math. Sci.) 127 (2017) 833–855 https://doi.org/ 10.1007/s12044-017-0357-0

Abstract We correct the proof of the Sobolev-type inequality in [2] for \(1<p<2\) (called the Beckner inequality).

Keywords. Finsler geometry; Ricci curvature; Sobolev inequality.

Mathematics Subject Classification. Primary: 53C60, Secondary: 58C40, 58J35.

In [2, Theorem 5.6], we state the following Sobolev-type inequality on a Finsler manifold (M, F) equipped with a measure \(\mathfrak {m}\).

Theorem 1

Assume that \(\mathrm {Ric}_N \ge K>0\) for some \(N \in [n,\infty )\) and \(\mathfrak {m}(M)=1\). Then we have

$$\begin{aligned} \frac{\Vert f\Vert _{L^p}^2 -\Vert f\Vert _{L^2}^2}{p-2} \le \frac{N-1}{KN} \int _M F^2(\varvec{\nabla }f) \,\mathrm {d}\mathfrak {m}\end{aligned}$$
(1)

for all \(1 \le p \le 2(N+1)/N\) and \(f \in H^1(M)\).

The proof in [2] is, however, incorrect for \(1<p<2\) (precisely, the final approximation procedure requires \(p>2\)). Instead, we can apply the argument in [1] to show (1) for \(1<p<2\) (such an inequality is called the Beckner inequality). Furthermore, the argument in [1] gives the following generalization of Theorem 1.

Theorem 2

Assume that \((M,F,\mathfrak {m})\) is compact and satisfies \(\mathrm {Ric}_N \ge K>0\) for some \(N \in (-\infty ,-2)\) and \(\mathfrak {m}(M)=1\). Then we have

$$\begin{aligned} \frac{\Vert f\Vert _{L^p}^2 -\Vert f\Vert _{L^2}^2}{p-2} \le \frac{N-1}{KN} \int _M F^2(\varvec{\nabla }f) \,\mathrm {d}\mathfrak {m}\end{aligned}$$

for all \(1 \le p \le (2N^2+1)/(N-1)^2\) and \(f \in H^1(M)\).

We refer to a forthcoming book [3] for details and further discussions.