Faber-Krahn inequalities and Sobolev-Orlicz inclusions (Q1366417)

Let \((M,g)\) be a complete Riemannian manifold and \(\phi:\mathbb{R}_+\to \mathbb{R}_+\) a convex function satisfying some integrability conditions at \(0\) and \(\infty\) for the function \(v(y)=\sup_{x\geq 0}\frac{\phi^{-1}[\phi(x)y]}{x}\). Then the following statements are equivalent: i) \(H_0^1(M)\to L^{\overline\phi}\), where \(L^{\overline\phi}\) is the Orlicz space with \(\overline\phi=\phi(x^2)\); ii) \((M,g)\) satisfies the isoperimetric Faber-Krahn inequality \(\lambda_1^D(\Omega)\geq\Lambda (\text{vol }\Omega)\), where \(\Lambda(v)=C^{te}\varphi^{-1}\left(\frac{C^{te}}{v}\right)\) and \(\varphi\) is a convex function conjugate to \(\phi\); iii) the minimal heat kernel of \((M,g)\) satisfies the inequality \(P(t,x, x)\leq C^{te}\varphi (\frac{C^{te}}{v})\). The proof uses some results and ideas from the papers by \textit{A. A. Grigorian} [Rev. Mat. Iberoam. 10, 395-452 (1994; Zbl 0810.58040)] and \textit{N. Varopoulos} [J. Funct. Anal. 63, 240-260 (1985; Zbl 0608.47047)]. Some equivalence between estimates for decaying of the heat kernel and inequalities on capacities and Green functions are also given.