Derek Gustafson
We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincaré inequality holds. We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.
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In these notes, we present versions of trace theorems for Sobolev spaces over an interval in the real line, and also a one-dimensional version of the well-known Poincare inequality.
Centro-affine Poincaré inequality: Unconditional convex bodies
研究关于\(C^{\infty}_+\)无条件凸体及特定光滑函数的中心仿射庞加莱不等式，证明其常数为\(n\)。
A sharp weighted anisotropic Poincaré inequality for convex domains
该研究属于数学分析领域，证明了凸域上一类加权各向异性Poincaré不等式最优常数的下界，具有理论意义。
In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain does not allow balls of arbitrarily large radius (finite ball condition). We prove that such a result can not be true in the `nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all ei…
From the Brunn-Minkowski inequality to a class of generalized Poincaré-type inequalities for torsional rigidity
本文利用Brunn-Minkowski不等式的凹性，在$C^{2}_{+}$类凸体边界上建立了一类关于扭转刚度的广义Poincaré型不等式。
In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the result of Payne-Weinberger. A lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Dirichlet boundary condition is also proved.
We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.
For a bounded three-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditi…
The classical Poincaré inequality establishes that for any bounded regular domain $Ω\subset \R^N$ there exists a constant $C=C(Ω)>0$ such that $$ \int_Ω |u|^2\, dx \leq C \int_Ω |\nabla u|^2\, dx \ \ \forall u \in H^1(Ω),\ \int_Ω u(x) \, dx=0.$$ In this note we show that $C$ can be taken independently of $Ω$ when $Ω$ is in a certain class of domains. Our result generalizes previous results in this direction.
Let $Ω$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $μ_1(Ω)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $Ω$; we prove that $μ_1(Ω) \ge 1$. The result is sharp since equality sign is achieved when $Ω$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(Ω,dγ_N)$, where $γ_N$ is the $N$-dimensional Gaussian measure.
Uniform Poincaré inequality in o-minimal structures
该研究在o-极小结构框架下定义了可定义区域的迹，证明了迹意义下边界零值的Sobolev函数满足Poincaré不等式，且可定义区域族的体积有界时不等式常数一致有界，拓展了Poincaré不等式的适用范围。
We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.
Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators
该研究提出基于轨迹的系统方法，证明含任意多个Hörmander交换子的次椭圆方程弱非负下解的庞加莱不等式，进而推导弱哈纳克不等式与赫尔德正则性。
The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights
本文证明完全单项式权重高斯测度满足ICD(2,∞)条件，从而得到最优常数2的尖锐Poincaré不等式及Brascamp--Lieb稳定性，并借助Laguerre谱分解导出精确亏损恒等式和梯度稳定性。
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up…
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^…
We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequali…
We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobole…
Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications
本文定义非局部半球梯度、散度、旋度算子，研究有界域上的函数空间，证明非局部Poincaré不等式，并应用于对流扩散、线弹性及Helmholtz分解。
Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to id…
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
Weighted graphs in the sense of John and a global Poincaré inequality
本文研究了加权图的条件，以保证全局波尔查诺不等式的成立，该条件是约翰1961年提出的条件的离散类比。
We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity.
We derive connections between optimal domain specific constants figuring in the Friedrichs-Velte inequality for conjugate harmonic functions, in the Babuška-Aziz inequality for the divergence and in the improved Poincaré inequality for the gradient. With the same method we obtain for spatial domains an improved Poincaré inequality for the rotation in connection with the corresponding Babuška-Aziz inequality.
The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, we have proved that if the $Q'$-curvature is nonnegative, and the integral of $Q'$-curvature is below the dimensional bound $c_1'$, then we have the isoperimetric inequality. In this paper, we manage to drop the condition on the nonnegativity of the $Q'$-curvature. We prove that the volume form $e^{4u}$ is a strong $A_\infty$ weight. As a corollary, we prove the Sobolev-Poincaré inequality on a class of CR-manifolds with integrable $Q'$-curvature.
Stability of the Poincaré-Korn inequality
本文在矩约束下解决了 Carrapatoso 等人关于 Poincaré-Korn 不等式锐常数高斯最优性的问题，并证明接近最优常数的测度在定量意义上接近标准高斯分布。
We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator $Δ_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in $\mathbb H^n$. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for $Δ_g$ in $\mathbb H^n$. As a consequence, we obtain the Poincaré--Sobolev inequality for the seco…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particular to CR Sasakian manifolds with Tanaka-Webster-Ricci curvature bounded from below and Carnot groups of step two.
We consider a frequency localized Bernstein inequality for the fractional Laplacian operator which has wide applications in fluid dynamics such as dissipative surface quasi-geostrophic equations. We use a heat flow reformulation and prove the inequality for the full range of parameters and in all dimensions. A crucial observation is that after frequency projection the zero frequency part of the Lévy semigroup does not participate in the inequality and therefore can be freely adjusted. Our proof is based on this idea and a careful perturbation of the Lévy semigroup near the zero frequency which…
A Poincaré formula for differential forms and applications
该研究针对带非空边界的紧黎曼流形上的微分形式，证明了新型广义庞加莱型不等式，推导了边界等距浸入欧氏空间时的曲率相关不等式及余维一极限情形，还在非零平行形式存在条件下得到新的Ros型不等式。
Weighted Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators
本文刻画加权 Poincaré 不等式的最佳常数与极大化函数，并据此改进含 \(L^p\) 余项的 Hardy 不等式，方法无需对称重排，适用于 Heisenberg 群、Carnot 群及多类退化椭圆算子。
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
As the classical $(p,q)$-Poincaré inequality is known to fail for $0 < p < 1$, we introduce the notion of weighted multilinear Poincaré inequality as a natural alternative when $m$-fold products and $1/m < p$ are considered. We prove such weighted multilinear Poincaré inequalities in the subelliptic context associated to vector fields of Hörmader type. We do so by establishing multilinear representation formulas and weighted estimates for multilinear potential operators in spaces of homogeneous type.
We prove a unique continuation property for the fractional Laplacian $(-Δ)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincaré-type inequalities for the operator $(-Δ)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equatio…
In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $Ω$, the first non-trivial Neumann eigenvalue $μ_1(Ω)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $Ω$, we show that $μ_1(Ω)=1$ if and only if $Ω$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
We improve the constant $\fracπ{2}$ in $L^1$-Poincaré inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\fracπ{2}}$. For Hamming cube the sharp constant is not known, and $\sqrt{\fracπ{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1\le \fracπ{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and…
We study the Korn-Poincaré inequality: \|u\|_{W^{1,2}(S^h)} < C_h \|D(u)\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \nabla u, and we assume the tangential boundary conditions: u\vec n^h = 0 on \partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <π/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition…
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
Logarithmic Sobolev, Hardy and Poincaré inequalities on the Heisenberg group
本文在Heisenberg群上系统证明含显式常数的Sobolev、Hardy、Nash、对数Gross及广义Poincaré等不等式，并实现无限维推广。
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