Joaquim Martin, Mario Milman
Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the Pó% lya-Szegö and Faber-Krahn principles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give complete details on an alternative formulation of the Polya-Szego principle that was mentioned in Remark 1 of our paper "Isoperimetry and Symmetrization for Logarithmic Sobolev inequalities". We also provide an alternative proof to a result in the same paper.
Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.
We characterize Poincaré inequalities in metric spaces using rearrangement inequalities
We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincar'e inequality for the Gaussian measure.
We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probablity measures between Gaussian and exponential. We show a kind of reverse Polya-Szego principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimat…
We develop a new method to obtain symmetrization inequalities of Sobolev type. Our approach leads to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces.
Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities
本文通过对数索博列夫不等式，建立了满足CD(0,∞)的加权黎曼流形乘积的渐近等周剖面在小体积下的尖锐极限恒等式，并推广到非光滑子空间，同时得到大偏差与中偏差渐近。
We obtain symmetrization inequalities in the context of Fractional Hajlasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations
In a previous paper we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in $W_{0}^{1,1}(Ω)$. In this paper we extend our method to Sobolev functions that do not vanish at the boundary.
We find sufficient conditions for a probability measure $μ$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d μ} \Bigr) d μ\le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d μ+ B \int_{\R^d} f^2 d μ, $$ where $F$ is concave and $c$ (a cost function) is convex. We show that under broad assumptions on $c$ and $F$ the above inequality holds if for some $δ>0$ and $ε>0$ one has $$ \int_{0}^ε Φ\Bigl(δc\Bigl[\frac{t F(\frac{1}{t})}{{\mathcal I}_μ(t)} \Bigr] \Bigr) dt < \infty, $$ where ${\mathcal I}_μ$ is the isoperimetric function of $μ$ and $Φ…
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S.
Large Deviations Principle for Isoperimetry and Its Equivalence to Nonlinear Log-Sobolev Inequalities
本文通过建立等周问题与非线性对数Sobolev不等式的等价性，揭示了等周极小解在不同几何结构空间中的行为差异。
Anisotropic symmetrization, convex bodies, and isoperimetric inequalities
本文利用各向异性等周不等式、Sobolev 函数精细性质及 Brunn-Minkowski 理论，给出各向异性 Pólya-Szegö 型不等式无需逼近论证的新证明，并实现对极值函数的刻画。
Elementary proofs of sharp isoperimetric inequalities on a normed space $(\mathbb{R}^n,||\cdot||)$ equipped with a measure $μ= w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding equality cases. When $p \in (0,\infty]$ and in addition $w^p$ is assumed concave, the result is an immediate corollary of the Borell-Brascamp-Lieb extension of the classical Brunn-Minkowski inequality, providing an elementary proof of a recent result of Cabré-Ros Oton-Serra. When $p \in (-1/n,0)$, the relevant property turns out to be a novel "complemented Brunn-Minko…
We introduce a notion of "gradient at a given scale" of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that generalize the quasi-isometries). We prove that for a Riemmanian manifold satisfying a local Poincare inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry…
Oscillation inequalities, lower ball growth and Sobolev embeddings on metric measure spaces
研究度量测度空间上球函数下界、重排振荡与Sobolev型嵌入的关系，证明下界估计等价于点态对称化不等式，并导出赋范振荡不等式和Sobolev嵌入。
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
本文证明了具有显式常数且大维数极限行为正确的Sobolev不等式尖锐定量稳定性，并据此推导出显式与维数无关常数的Gaussian log-Sobolev不等式最优定量稳定性估计。
Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals…
In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…
Sharp anisotropic singular Trudinger-Moser inequalities in the entire space
本文在全空间Sobolev型空间中，利用凸对称化与Schwarz对称化的联系，建立了涉及各向异性Dirichlet范数的精确奇异Trudinger-Moser不等式，并讨论了其在多种约束条件下的精确性及精确增长性。
Sharp Quantitative Forms of the Hardy Inequality on Cartan-Hadamard Manifolds via Sobolev-Lorentz Embeddings
本文在满足中心等周不等式的Riemann流形上，利用Sobolev-Lorentz嵌入技术，推导出Hardy不等式的精确形式，并建立了流形与欧几里得空间之间Hardy亏量的对应关系。
The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams inequality of order two with the exact growth condition in $\mathbb H^n$. Then we use it to derive a sharp Adams-type inequality and an Adachi-Tanaka-type inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of $\mathbb H^n$, which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。