Jonas Azzam
We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.
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We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
The main result of this paper supports a conjecture by C. Pérez and E. Rela about a very recent result of theirs on self-improving theory. Also, we extend the conclusions of their theorem to the range $p<1$. As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We improve some already known results. Finally, we also explore analog inequalities in the context of metric spaces …
We prove fractional Sobolev-Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.
We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, $s$-John domains and Hölder-$α$ domains, and discuss their optimality.
We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $μ$ under some mild regularity conditions on the measure $μ$. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.
Sharp constants for an inequality of Poincaré type is studied. The problem is solved by using optimal control theory.
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness…
We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.
One-sided Rellich inequalities, Regularity problem and uniform rectifiability
本文证明在边界一致 n-可整流的有界 corkscrew 域上，单侧 Rellich 不等式成立，并证明该不等式在平面情形下刻画了边界的一致可整流性；同时证明此类域上正则性问题在弱 L^1 中可解，并由此说明一般椭圆算子的 Dirichlet 问题可解性不蕴含正则性问题可解性。
The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.
In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincaré inequalities for all values of $p \ge 1$.
We show some non-standard Poincaré type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions and fractional integrals. We also provide a sharper result by using extrapolation techniques.
Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.
We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-Δ_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-Δ_{\mathbb H^N}$. We find the optimal constant in the resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at …
We prove Poincaré and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincaré inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse Hölder inequalities for weak solutions of the degenerate systems. As a means to prove the Poincaré inequalities we prove that the Riesz potential and frac…
We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.
We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincaré inequalities. The key point is the study of an optimal weighted Wirtinger inequality.
This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure $(p,p)$-Poincaré inequality for $1<p<\infty$ improves to a $(p,p-\varepsilon)$-Poincaré inequality for some $\varepsilon>0$ under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalit…
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 study generalized Poincaré inequalities. We prove that if a function satisfies a suitable inequality of Poincaré type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we get a unified approach to proving that the maximal operator is bounded on Sobolev, Lipschitz and BMO spaces.
Poincaré inequalities on hyperbolic-type spaces
该研究针对一类非双倍双曲型度量测度空间，通过离散化得到spiderweb图，结合转移原理建立了全局$L^p$-庞加莱不等式，为相关空间的分析提供了关键工具。
We study the validity of (q,p)-Poincaré inequalities, q<p, on domains in R^n which satisfy a quasihyperbolic boundary condition, i.e. domains whose quasihyperbolic metric satisfies a logarithmic growth condition. In the present paper, we show that the quasihyperbolic boundary condition domains support a (q,p)-Poincaré inequality whenever p>p_0, where p_0 is an explicit constant depending on q, on the logarithmic growth condition, and on the boundary of the domain.
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.
We study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the identity, semigroups or mean value operators) that have off-diagonal decay in some range. Our results provide a unified theory that is applicable to the classical Poincaré inequalities and furthermore it includes oscillations defined in terms of semigroups associated with second order elliptic operators as those in the Kato conjecture. In this latter situation…
We prove here some dimension free Poincaré-type inequalities on Hamming cube for functions with different spectral properties and for fractional Laplacians. In this note the main attention is paid to estimates in $L^1$ norm on Hamming cube. We build the examples showing that our assumptions on spectral properties of functions cannot be dropped in general.
Weighted fractional Poincaré inequalities via isoperimetric inequalities
本文利用分数阶等周不等式证明加权分数阶 Poincaré-Sobolev 不等式，改进 Bourgain-Brezis-Mironescu 估计与 Meyers-Ziemer 定理，并推广含 $A_p$ 权的结果。
In this paper we prove that if $Ω\in\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \inf_{a\in \mathbb{R}}\| (f(x)-a) w_1(x) \|_{L^q(Ω)} \le C \|\nabla f(x) d(x)^αw_2(x) \|_{L^p(Ω)} $$ where $f$ is a locally Lipschitz function on $Ω$, $d(x)$ denotes the distance of $x$ to the boundary of $Ω$, the weights $w_1, w_2$ satisfy certain cube conditions, and $α\in [0,1]$ depends on $p,q$ and $n$. This result generalizes previously known weighted inequalities, which can also be obtained with our approach.
We prove a local $L^p$-Poincaré inequality, $1\leq p < \infty$, on noncompact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal $L^2$-Poincaré inequality with respect to suitable finite measures on the group.
The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. F…
An interpolation result for $A_1$ weights with applications to fractional Poincaré inequalities
本文刻画了任意区域上加权 $L^1$ 与 $W^{1,1}$ 空间的实插值空间，并以此推导出对分数阶参数 $s$ 具有尖锐依赖性的加权分数阶 Poincaré 不等式，证明其等价于梯度的加权 Poincaré 不等式。
From generalized Poincaré to Poincaré-Sobolev inequalities via self-improving methods
本文通过自改进方法，改进了广义Poincaré不等式的主要结果，并推导出更优的Poincaré-Sobolev不等式，特别适用于轴平行立方体。
Degenerate Poincaré-Sobolev inequalities via fractional integration
本文通过分数积分方法改进了波尔雅-索波列夫不等式，解决了关于$A_1$常数的猜想，并扩展到高阶导数和分数不等式。
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