Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

Abstract

In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1 and the Stein-Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein-Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein-Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein-Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).

Stein–Weiss inequalities with the fractional Poisson kernel

“…For more results about proving precise estimates for Stein-Weiss functionals in conjunction with the study of Housdorff-Young and Pitt's type inequalities and their multilinear versions, we refer the reader to the works of Beckner [1,2,5,6,7,8]. We note that the existence of extremal functions for the Stein-Weiss inequalities in the case p < q under the assumption α + β ≥ 0 has been established by Chen, Lu and Tao [19], which extends Lieb's result under the stronger assumption that α ≥ 0 and β ≥ 0, using the concentration-compactness of Lions ( [34,35]).…”

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

“…In [16], Lieb used the method of rearrangement technique and symmetrization to prove the existence of extremals of (1.4) in some cases. Later in [6], the authors extended Lieb's result on the Heisenberg group under certain assumptions. We refer the readers to the articles [4,5,11,12] to understand more about Stein-Weiss inequalities and its extremal functions.…”

Reversed Hardy-Littlewood-Sobolev inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$

“…For the case p = q, Han [23] used the concentration-compactness principles to study the existence of extremal functions of (1.5). Recently, Han, Lu, Zhu [24] and Chen, Lu, Tao [6] established two classes of weighted HLS inequalities on Heisenberg group and proved the existence of extremal functions by the concentrationcompactness principles.…”