Existence of an extremal of Dunkl-type Sobolev inequality and of Stein-Weiss inequality for D-Riesz potential

Abstract

In this paper, we prove the existence of an extremal for the Dunkltype Sobolev inequality in case of p = 2. Also we prove the existence of an extremal of the Stein-Weiss inequality for the D-Riesz potential in case of r = 2.

Relative Nash-type and $L^2$-Sobolev inequalities for Dunkl operators and applications

“…One possible approach is to use pseudo-Poincaré inequalities. Such Inequalities were established by S. Adhikari, V. P. Anoop and S. Parui in [1] for p = 2 by Velicu in [14] for 1 ≤ p ≤ 2. The proofs developed in [1] and [14] are different in nature.…”

“…Such Inequalities were established by S. Adhikari, V. P. Anoop and S. Parui in [1] for p = 2 by Velicu in [14] for 1 ≤ p ≤ 2. The proofs developed in [1] and [14] are different in nature. The L 2 nature of the inequality established in [1] allows the Dunkl transform to be used and the Velicu result is based on the use of the carré-du-champ operator and semi-group techniques.…”

“…The proofs developed in [1] and [14] are different in nature. The L 2 nature of the inequality established in [1] allows the Dunkl transform to be used and the Velicu result is based on the use of the carré-du-champ operator and semi-group techniques.…”