A Sawyer-type sufficient condition for the weighted Poincaré inequality

Abstract

In this paper we prove the Poincaré-type weighted inequality

$$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} \nabla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$

for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain $\Omega$; here the weights v, $\omega$ are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincaré inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability $v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}$ with $r>1$ for $q=p$ or the reverse doubling condition on the function v for $q>p$ . In other words, a Sawyer type sufficient condition on weight functions is established.