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

TL;DR

This paper extends Caffarelli-Kohn-Nirenberg inequalities to the case of p=1 with general weight functions, including non-doubling weights, broadening their applicability in analysis.

Contribution

It generalizes existing inequalities by incorporating a wide class of non-doubling weights for the case p=1.

Findings

Established new inequalities with non-doubling weights

Included weights like e^{1/t} and e^{-1/t}

Extended the scope of Caffarelli-Kohn-Nirenberg inequalities

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 $e^{-1/t}$.