  In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and
dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy
inequality a Poincar\'e inequality. The proof of the weighted Hardy inequality
is based on the method of vector fields firstly introduced by Mitidieri
\cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$
that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an
application of our weighted Hardy inequality, we prove a non-existence result
for a p-Kolmogorov parabolic equation.
