The sharp Sobolev type inequalities in the Lorentz-Sobolev spaces in the hyperbolic spaces

摘要

Let W-1 L-p,L-q(H-n), 1 <= q, p < infinity denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces H-n. Our aim in this paper is three-fold. First of all, we establish a sharp Poincare inequality in W-1 L-p,L-q(H-n) with 1 <= q <= p which generalizes the result in [41] to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincare-Sobolev type inequalities in W-1 L-p,L-q(H-n) with 1 <= q <= p < n which generalize the results in [45] to the setting of Lorentz-Sobolev spaces. Finally, we provide the improved Moser-Trudinger type inequalities in W-1 L-n,L-q(H-n) in the critical case p = n with 1 <= q <= n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Polya-Szego type principle in W-1 L-p,L-q(H-n) with 1 <= q <= p which maybe is of independent interest. (C) 2020 Elsevier Inc. All rights reserved.