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Register a new userSharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four
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We establish sharp Hardy-Adams inequalities on hyperbolic space $mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $alpha>0$ there exists a constant $C_{alpha}>0$ such that [ int_{mathbb{B}^{4}}(e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2})dV=16int_{mathbb{B}^{4}}frac{e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2}}{(1-|x|^{2})^{4}}dxleq C_{alpha}. ] for any $uin C^{infty}_{0}(mathbb{B}^{4})$ with [ int_{mathbb{B}^{4}}left(-Delta_{mathbb{H}}-frac{9}{4}right)(-Delta_{mathbb{H}}+alpha)ucdot udVleq1. ] 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.
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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 Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincare-Sobolev and Hardy-Sobolev-Mazya 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 Gellers operators and ``Octonionic Gellers 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 Poincare-Sobolev and Hardy-Sobolev-Mazya 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 Poincare-Sobolev, Hardy-Sobolev-Mazya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.
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 Poincare-Sobolev, Hardy-Sobolev-Mazya, 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 Poincare-Sobolev, Hardy-Sobolev-Mazya 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.
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.
We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carath{e}odory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Mazya 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-Mazya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Mazya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.
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