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Qiaohua Yang

Publications and source records attributed to Qiaohua Yang.

At least 19 recordsLinked to original sources
arXiv2026

Sharp trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere

We establish sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending the corresponding Euclidean results of Einav-Loss, Beckner, and Bez-Machihara-Sugimoto to these non-Euclidean settings. In the limiting case, sharp trace Beckner-Onofri inequalities are also established on the CR sphere. The proofs are based on a duality argument due to Bez-Machihara-Sugimoto, together with the Frank-Lieb sharp form of the Hardy-Littlewood-Sobolev inequalities on the Heisenberg group and the CR sphere. The same approach also yields trace Beckner-Onofri inequalities on the standard sphere.

math.AP↗
arXiv2026

Sharp quantitative integral inequalities for general conformally invariant extensions

In this paper, we develop a refined analysis of hypergeometric functions to establish sharp quantitative integral inequalities for a general family of conformally invariant extension operators and their adjoints. Our results extend the recent work of Frank, Peteranderl, and Read \cite{Frank&Peteranderl&Read} to the full admissible parameter range under the natural index constraints.

math.AP↗
arXiv2025

A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball

Frank et al. (J. Funct. Anal., 2022) stated that there is no relation between the reversed Hardy-Littlewood-Sobolev (HLS) inequalities and reverse Sobolev inequalities. However, we demonstrate that reverse Sobolev inequalities of order $γ\in(\frac{n}{2},\frac{n}{2}+1)$ on the $n$-sphere can be readily derived from the reversed HLS inequalities. For the case $γ\in(\frac{n}{2}+1,\frac{n}{2}+2)$, we present a simple proof of reverse Sobolev inequalities by using the center of mass condition introduced by Hang. In addition, applying this approach, we establish the quantitative stability of reverse Sobolev inequalities of order $γ\in(\frac{n}{2}+1,\frac{n}{2}+2)$ with explicit lower bounds. Finally, by using conformally covariant boundary operators and reverse Sobolev inequalities, we derive Sobolev trace inequalities on the unit ball.

math.AP↗
arXiv2024

Sharp fractional Sobolev and related inequalities on H-type groups

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers $\mathcal{L}_{s}(0<s<1)$ of the sublaplacian on H-type groups. From these inequalities we derive a sharp log-Sobolev inequality by considering a limiting case and a sharp Sobolev trace inequality. The later extends to this context the result of Frank, González, Monticelli and Tan (Adv. Math, 2015).

math.AP↗
arXiv2024

Conformally Covariant Boundary Operators and Sharp Higher Order CR Sobolev Trace Inequalities on the Siegel Domain and Complex Ball

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain ${\mathcal U}^{n+1}$ with the Heisenberg group $\mathbb{H}^{n}$ as its boundary and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$ with the complex sphere $\mathbb{S}^{2n+1}$ as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli-Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain ${\mathcal U}^{n+1}$ and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$.In particular, we generalize the Sobolev trace inequalityfor $γ\in (0, 1)$ in the CR setting by Frank-González-Monticelli-Tan to the case for all $γ\in (0, n+1)\backslash \mathbb{N}$. The family of higher order conformally covariant boundary operators we define are naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain ${\mathcal U}^{n+1}$ and complex ball $\mathbb{B}_{\mathbb{C}^{n+1}}$. Finally, we give an explicit solution to the scattering problem on the complex hyperbolic ball. More precisely, we obtain an integral representation and an expansion in terms of special functions for the solution to the scattering problem.

math.AP↗
arXiv2023

Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities

Using the scattering theory on the hyperbolic space $\mathbb{H}^n$, we give the explicit formulas of the fractional GJMS operators $P_γ$ for all $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ on $\mathbb{H}^n$.These $P_γ$ for $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ are neither conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ nor on $\mathbb{B}^n$ in $\mathbb{R}^{n}$ though $P_γ$ are conformal to $(-Δ)^γ$ via half space model and ball model of hyperbolic spaces when $γ\in\mathbb{N}$. To circumvent this, we introduce another family of fractional operators $\tilde{P}_γ$ on $\mathbb{H}^n$ which are conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ and $\mathbb{B}^n$. It is worthwhile to note that $\tilde{P}_γ\not =P_γ$ unless $γ$ is an integer. We establish the fractional Poincaré-Sobolev inequalities associated with both $P_γ$ and $\tilde{P}_γ$ on $\mathbb{H}^n$. In particular, when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$, we prove that the sharp constants in the $γ$-th order of Poincaré-Sobolev inequalities on the hyperbolic space associated with $P_γ$ and $\tilde{P}_γ$ coincide with the best $γ$-th order Sobolev constant in the $n$-dimensional Euclidean space $\mathbb{R}^n$. We also establish fractional Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_+$ and $\mathbb{B}^n$ and prove that the sharp constants in the $γ$-th order Hardy-Sobolev-Maz'ya inequalities on half space $\mathbb{R}^{n}_+$ and unit ball $\mathbb{B}^n$ are the same as the best $γ$-th order Sobolev constants in $\mathbb{R}^n$ when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourier analysis on hyperbolic spaces and delicate analysis of special functions.

math.AP↗
arXiv2023

Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincaré-Einstein Manifolds

In this paper we introduce conformally covariant boundary operators for Poincaré-Einstein manifolds satisfying a mild spectral assumption. Using these boundary operators we set up higher order Dirichlet problems whose solutions are such that, when applied to by our boundary operators, they recover the fractional order GJMS operators on the conformal infinity of the manifold. We moreover obtain all related higher order trace Sobolev inequalities on these manifolds. In conjunction with Beckner's fractional Sobolev inequalities on the sphere, we obtain as an application the sharp higher order Sobolev trace inequalities on the ball.

math.DG↗
arXiv2023

Trudinger-Moser and Hardy-Trudinger-Moser inequalities for the Aharonov-Bohm Magnetic field

The main results of this paper concern sharp constant of the Trudinger-Moser inequality in $\mathbb{R}^{2}$ for Aharonov-Bohm magnetic fields. This is a borderline case of the Hardy type inequalities for Aharonov-Bohm magnetic fields in $\mathbb{R}^2$ studied by A. Laptev and T. Weidl. As an application, we obtain the exact asymptotic estimates on best constants of magnetic Hardy-Sobolev inequalities. In order to achieve our goal, we introduce a new operator $T_{a}$ on the unit circle $\mathbb{S}^{1}$ and give the asymptotic estimates of the heat kernel $e^{tT_{a}}$ via the Poisson summation formula. Finally, we show that such Trudinger-Moser inequalities in the unit ball $\mathbb{B}^{2}$ can be improved via subtraction of an additional Hardy term to derive a Hardy-Trudinger-Moser inequality.

math.AP↗
arXiv2021

Sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the upper half spaces

In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities.

math.AP↗
arXiv2021

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $\mathbb{H}^n$ and as well as on the entire hyperbolic space $\mathbb{H}^n$. Among other techniques, one of our main novelties is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Maz'ya inequalities and careful study of delicate properties of Green's functions of $P_k-λ$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Green's functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.

math.AP↗
arXiv2021

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane

Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on these spaces which are of their independent interests. To this end, we need to identify and introduce the ``Quaternionic Geller's operators" and ``Octonionic Geller's operators" which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernel estimates and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient for us to establish, as a byproduct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works, completes our study of the factorization theorems, higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.

math.CA↗
arXiv2021

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain $\mathcal{U}^{n}$ and the unit ball $\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein.

math.CA↗
arXiv2019

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Green's functions of the operator $ -Δ_{\mathbb{H}}-\frac{(n-1)^{2}}{4}$ on the hyperbolic space $\mathbb{B}^n$ and operators of the product form are given, where $\frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Δ_{\mathbb{H}}$ on $\mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).

math.CA↗
arXiv2019

Sharp Sobolev trace inequalities for higher order derivatives

Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.

math.AP↗
arXiv2017

Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

We establish sharp Hardy-Adams inequalities on hyperbolic space $\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $α>0$ there exists a constant $C_α>0$ such that \[ \int_{\mathbb{B}^{4}}(e^{32π^{2} u^{2}}-1-32π^{2} u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32π^{2} u^{2}}-1-32π^{2} u^{2}}{(1-|x|^{2})^{4}}dx\leq C_α. \] for any $u\in C^{\infty}_{0}(\mathbb{B}^{4})$ with \[ \int_{\mathbb{B}^{4}}\left(-Δ_{\mathbb{H}}-\frac{9}{4}\right)(-Δ_{\mathbb{H}}+α)u\cdot udV\leq1. \] As applications, we obtain a sharpened Adams inequality on hyperbolic space $\mathbb{B}^{4}$ and an inequality which improves the classical Adams' inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.

math.AP↗
arXiv2017

Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces

By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.

math.AP↗