Weighted and Logarithmic Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and applications

The classical Caffarelli–Kohn–Nirenberg inequalities, originally established in Euclidean space in the 1980s, provide a unified framework for interpolation between Sobolev and Hardy inequalities. Their extension to stratified (or homogeneous Carnot) Lie groups began in the early 2000s, motivated by subelliptic analysis and geometric measure theory, revealing rich interactions between group structure, dilation symmetry, and functional inequalities. In this paper, we establish the weighted and logarithmic Caffarelli–Kohn–Nirenberg type inequalities on a stratified Lie group. As a consequence, we can apply them to prove the weighted ultracontractivity of positive strong solutions to the equation: dα · ∂u/∂t = ℒₚ((dα · u)m), where ℒₚ f = ∇H ( |∇H f|p−2 · ∇H f ) is a p-sub-Laplacian, d is a homogeneous norm associated with a fundamental solution of the sub-Laplacian, α ∈ ℝ, and 1 < p < Q.