Let $(M,g)$ be a complete $n$-dimensional Riemannian manifold with $\operatorname{Ric}_g\ge -(n-1)K g$ for $K\ge 0$. For every $R_0>0$ there exists $C=C(n,K,R_0)$ such that every geodesic ball $B_g(x,r)$ with $0<r\le R_0$ satisfies
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\begin{align*}
\int_{B_g(x,r)} |f-f_{B_g(x,r)}|^2\,d\operatorname{vol}_g
\le C r^2\int_{B_g(x,r)} |\nabla f|_g^2\,d\operatorname{vol}_g
\end{align*}
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for every real-valued map $f: \overline{B_g(x,r)} \to \mathbb{R}$ with $f\in C^1(\overline{B_g(x,r)};\mathbb{R})$.