Stein–Weiss inequalities for the fractional integral operators in Carnot groups and applications

Abstract

In this article we consider the fractional integral operator I α on any Carnot group 𝔾 (i.e. nilpotent stratified Lie group) in the weighted Lebesgue spaces L p,ρ(x)β (𝔾). We establish Stein–Weiss inequalities for I α, and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator I α from the spaces L p,ρ(x)β (𝔾) to L q,ρ(x)−γ (𝔾), and from the spaces L 1,ρ(x)β (𝔾) to the weak spaces WL q,ρ(x)−γ (𝔾) by using the Stein–Weiss inequalities. In the limiting case , we prove that the modified fractional integral operator is bounded from the space L p,ρ(x)β (𝔾) to the weighted bounded mean oscillation (BMO) space BMOρ(x)−γ (𝔾), where Q is the homogeneous dimension of 𝔾. As applications of the properties of the fundamental solution of sub-Laplacian ℒ on 𝔾, we prove two Sobolev–Stein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As another application, we prove the boundedness of I α from the weighted Besov spaces to .