Hardy inequalities on metric measure spaces, III: the case q ≤ p ≤ 0 and applications

Abstract

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy–Littlewood–Sobolev and the Stein–Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q ≤ p < 0 ) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 ); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477 , 20210136 ( doi:10.1098/rspa.2021.0136 )), which treated the cases 1 < p ≤ q < ∞ and p > q , respectively.