Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case
Markos Fisseha Yimer, Lars Erik Persson, Michael Ruzhansky, Natasha Samko, Tsegaye Gedif Ayele
TL;DR
The paper extends Hardy–Hilbert type inequalities to homogeneous groups by developing a unified kernel framework and establishing sharp constants for kernels of prescribed homogeneity. It proves a base Hardy–Hilbert inequality on a homogeneous group with constant (Qπ)/sin(π/p) using spherical averages, and then develops the classical kernel theory with C_p^* characterizations, including equivalent Hardy-type and Hardy–Hilbert–type forms. The results are further generalized to general kernels on homogeneous groups of order −Q, yielding sharp constants C_p^* and Euclidean analogues, along with equivalence theorems connecting different formulations. Overall, the work broadens Hardy–Hilbert-type analysis to non-Euclidean, noncommutative settings and provides a comprehensive kernel-based framework for these inequalities.
Abstract
There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of several of the Hardy-Hilbert type inequalities in the classical case to a general kernel case. Finally, we generalize these results to the homogeneous group case.