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 $e^{-1/t}$.