Sharp Adams type inequalities for the fractional Laplace-Beltrami operator on noncompact symmetric spaces

TL;DR

This paper proves optimal Adams type inequalities for fractional Sobolev spaces on noncompact symmetric spaces, extending known results to a broader class of spaces using Fourier analysis.

Contribution

It establishes sharp Adams and Hardy-Adams inequalities for fractional Sobolev spaces on noncompact symmetric spaces, generalizing recent hyperbolic space results.

Findings

Sharp Adams inequalities for fractional Sobolev spaces on symmetric spaces.

Sharp Hardy-Adams inequalities on $W^{n/2, 2}(X)$ spaces.

Extension of hyperbolic space results to general noncompact symmetric spaces.

Abstract

We establish sharp Adams type inequalities on Sobolev spaces $W^{\alpha, n/\alpha}(X)$ of any fractional order $\alpha< n$ on Riemannian symmetric space $X$ of noncompact type with dimension $n$ and of arbitrary rank. We also establish sharp Hardy-Adams inequalities on the Sobolev spaces $W^{n/2, 2}(X)$. For the real hyperbolic spaces, such results were recently obtained by J. Li et al. (Trans. AMS, 2020). We use Fourier analysis on the symmetric spaces to obtain these results.