Sobolev inequality on Riemannian manifolds (Q816631)

Let \(M\) be an \(n\)-dimensional complete Riemannian manifold, \(\rho\) be the geodesic distance on \(M\), and \(d\mu\) the Riemannian measure. Denote by \(B(x,r)\) the geodesic ball of center \(x\in M\) and radius \(r>0\), and by \(V(x,r)\) its Riemannian volume. One says that \(M\) satisfies the doubling volume property if there exists a constant \(D_0\) such that \(V(x,2r)\leq D_0V(x,r)\) for all \(x\in M\), \(r>0\). Denote \(\nu = \log_2D_0\). In the present paper, the author proves a necessary and sufficient condition for the Sobolev inequality \(\|f\|_q\leq C_{n,\nu,p,q} (\|\nabla f\|_p+\| f\|_p)\) \((2\leq p<q<\infty)\) on a Riemannian manifold \(M\) satisfying the doubling volume property and an on-diagonal heat kernel estimate.