On general Caffarelli-Kohn-Nirenberg type inequalities involving non-doubling weights in the case of $p=1$

We study the Caffarelli-Kohn-Nirenberg type inequalities in the case of $p=1$ and generalize them adopting weight functions $w(\mathrm{\mid }x\mathrm{\mid })$ on $R^n$ with $w(t)$ in $W(R_{+})$. Here $W(R_{+})$ is a general class of weight functions on $R_{+}$ including non-doubling weights like $e^{1\mathrm{/}t}$ and $e^{-1\mathrm{/}t}$.