ON THE HARNACK INEQUALITY FOR HARMONIC FUNCTIONS ON COMPLETE RIEMANNIAN MAINFOLDS

First it is shown that on the complete Riemannian manifold with nonnegative Ricci curvature $\overline M$ the Sobolev type inequality $\||\nabla u|{|_2} \geqslant {C_{n,\alpha }}||u|{|_{2\alpha }}(\alpha \geqslant 1)\]$, for all $u \in H^2_1(\overline M)$ holds if and only if $V_x(r)=Vol(B_x(r))\geq C_nr^n$ and $\alpha=\frac{n}{n-2}$. Let M be a complete Riemannian manifolds which is uniformly equivalent to $\overline M$, and assume that $V_x(r)\geq C_nr^n$ on $\overline M$. Then it is prioved that the John-Nirenberg inequality, holds on M. Finally, based on the Sobolev inequality and John-Nirenberg inequality, the Harnack inequality for harmonic functions on M is obtained by the method of Moser, arid consequently some Liouville theorems for harmonic functions and harmonic maps on M are proved.