A NOTE ON A POINCARE TYPE INEQUALITY FOR SOLUTIONS TO SUBELLIPTIC EQUATIONS

We prove Poincare type inequalities for solutions to certain classes o f quasilinear subelliptic equations, including the well-known p-Sublap lacian. A notable feature in these inequalities is to replace the usua l f(B), the average of f over a metric ball B, by f(x(0)) for x(0) is an element of B. Result of this kind was considered earlier by Ziemer [18] in the classic case. We mention that our endpoint result, even in the classic case, is not obtainable through the compactness argument.