New weighted Hardy's inequalities with application to non-existence of global solutions

In this article, we prove a weighted Hardy inequality for $1<p<+\mathrm{\infty }$ and dimension $d\ge 1$. 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 $1<p<+\mathrm{\infty }$ and $d\ge 1$ 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.