arXiv subjects

Chunxia Tao

Publications and source records attributed to Chunxia Tao.

4 recordsLinked to original sources

arXiv2026

Critical GJMS Equations on $\mathbb{H}^n \times \mathbb{S}^m$

Let $M=\mathbb{H}^n\times\mathbb{S}^m$, where $n\geq 2$, $m\geq 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\leq k 0$. We study$$P_kU-\lambda U=|U|^{q-2}U,\qquad q=\frac{2N}{N-2k},\qquad 0<\lambda\leq\Lambda_0,$$and attainment of the associated critical quotient $S_{\lambda,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<\lambda<\Lambda_0$, the inequality $S_{\lambda,k}(M) \Lambda_{\mathrm{loc}}$, where $\Lambda_{\mathrm{loc}}$ is explicit. If $N\geq2k+2$ and $S_{\Lambda_0,k}(M)<S_{N,k}$, attainment also holds at $\lambda=\Lambda_0$ in the threshold form completion. At the threshold, $L^2$-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving $\mathcal{A}=\mathbb{R}^{n-1}\times{0}$. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping $\mathcal{A}$ from normalized minimizing sequences. If $r_j^{(J)}$ denotes the remainder after the first $J$ extracted profiles, then$$\lim_{J\to\infty}\limsup_{j\to\infty}|r_j^{(J)}|_{L^q(\mathbb{R}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.

math.AP↗

arXiv2026

Existence and symmetry of extremals for the high order Hardy-Sobolev-Maz'ya inequalities

In this article, we establish the existence of an extremal function for the k-th order critical Hardy-Sobolev-Maz'ya (HSM) inequalities on the upper half space $\mathbb{R}^{n+1}_{+}$ when $k\ge 2$ and $n\geq 2k+2$: $$\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,\frac{2n}{n-2k}} \left(\int_{\mathbb{R}^{n}_{+}}|u|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}. $$ The analysis of this extremal problem is challenging due to the presence of the higher order derivatives, the lack of translation invariance, the inapplicability of rearrangement techniques on the upper half-space, and the presence of a Hardy singularity along the boundary. To overcome these difficulties, instead of directly considering the HSM inequality on the upper half space, we establish the existence of an extremal for its equivalent version: Poincar\'e-Sobolev inequality on the hyperbolic space. We develop a novel duality theory of the minimizing sequences, the concentration-compactness principle for radial functions in the hyperbolic setting, which combines with the Helgason-Fourier analysis and the Riesz rearrangement inequality on the hyperbolic space, to resolve the lack of compactness issue. As an application, we also obtain the existence of positive symmetric solutions for the high order Brezis-Nirenberg equation on the entire hyperbolic space associated with the GJMS operators $P_k$ (i.e., when $k\ge 2$): $$ P_{k}\left(f\right)-\alpha f=|f|^{p-2}f $$ at the critical situation $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ when either $2k+2\leq n$ and $p=\frac{2n}{n-2k}$ or $2k<n$ and $2<p<\frac{2n}{n-2k}$.

math.AP↗

arXiv2019

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

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).

math.CA↗

arXiv2019

Stein-Weiss inequalities with the fractional Poisson kernel

In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel (see Theorem 1.1): \begin{equation}\label{int1} \int_{\mathbb{R}^n_{+}}\int_{\partial\mathbb{R}^n_{+}}|ξ|^{-α}f(ξ)P(x,ξ,γ)g(x)|x|^{-β}dξdx\leq C_{n,α,β,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n_{+})}\|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{equation} where $P(x,ξ,γ)=\frac{x_n}{(|x'-ξ|^2+x_n^2)^{\frac{n+2-γ}{2}}}$, $2\le γ<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$ and $p,\ q'\in (1,\infty)$ and satisfy $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{α+β+2-γ}{n}=1$. Then we prove that there exist extremals for the Stein-Weiss inequality (0.1) and the extremals must be radially decreasing about the origin (see Theorem 1.5). We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler-Lagrange equations of the extremals to the Stein-Weiss inequality (0.1) with the fractional Poisson kernel (see Theorems 1.7 and 1.8). Our result is inspired by the work of Hang, Wang and Yan [29] where the Hardy-Littlewood-Sobolev type inequality was first establishedmwhen $γ=2$ and $α=β=0$ (see (1.5)). The proof of the Stein-Weiss inequality (0.1) with the fractional Poisson kernel in this paper uses our recent work on the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel [18] and the present paper is a further study in this direction.

math.AP↗