We find a simple sufficient criterion on a pair of nonnegative weight functions $V(x)$ and $W(x) $ on a Carnot group $\mathbb{G},$ so that the general weighted $L^{p}$ Hardy type inequality
$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$
is valid for any $φ ∈ C_{0}^{∞ }(\mathbb{G})$ and $p>1.$ It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on $\mathbb{G}.$ We also present some new results on two-weight $L^{p}$ Hardy type inequalities with remainder terms on a bounded domain $Ω$ in $\mathbb{G}$ via a differential inequality.
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