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

Abstract: We study the Caffarelli-Kohn-Nirenberg type inequalities in the case of $p=1$ and generalize them adopting weight functions $w(|x|)$ 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 $e^{1/t}$ and $e^{-1/t}$.