The Median Principle for Inequalities and Applications
Sever Silvestru Dragomir
本文应用中位数原理推导Gruss和Ostrowski型积分不等式，提供统一证明方法。
中位数原理被应用于Gruss型和Ostrowski型的不同积分不等式。
The median principle is applied for different integral inequalities of Gruss and Ostrowski type.
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
In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined, and in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point …
The Polya-Szego principle in the fractional setting: a glimpse on nonlocal functional inequalities
本文综述分数阶波利亚-塞格原理及其在非局部泛函不等式中的应用，展示对称化方法在分数阶情形的有效性及等周型不等式等精确结果，并讨论稳定性问题与各向异性、高斯情形下的推广。
We consider degenerated nonlinear PDE of elliptic type: $$ - \mathrm{div}(a(|x|)|\nabla w(x)|^{p-2} \nabla w(x)) + h(|x|,w(x),\langle\nabla w(x),\frac{x}{|x|}\rangle)=ϕ(w(x)), $$ where $x$ belongs to the ball in $\bf{R}^n$. Using the argument based on Opial-type inequalities, we investigate qualitative properties of their radial solutions, like e.g. maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.
Unique continuation inequalities for the Dunkl-Schrödinger equation via uncertainty principles
本文基于Dunkl变换的定量不确定性原理，建立了Dunkl-薛定谔方程在两个时间点的唯一持续不等式，证明了(\varepsilon,k)-薄集对Dunkl变换形成强 annihilating 对，并由此获得Dunkl-薛定谔方程解的定量唯一持续性质。
We prove a sharp $L^p$ weighted Hardy inequality involving boundary distance $δ$ for any domain $Ω\subsetneq \mathbb R^n$. The inequality may be improved substantially under the additional assumption that $-\log δ$ is subharmonic. Applications of these inequalities to the principle frequency of the $p-$Laplacian are given.
This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form $A(u) + |u|^{q-1}u \leq A(v) + |v|^{q-1}v$ on ${\Bbb R}^n$, where $n\geq 1$, $q$ is positive, and the operator $A(w)$ belongs to a class of the so-called $α$-monotone operators. Typical examples of such operators are the $p$-Laplacian and its well-known modifications. The results improve and supplement those in [1].
Hardy inequalities and uncertainty principles in the presence of a black hole
该研究针对Schwarzschild黑洞外部场景，建立了适配几何结构的Hardy型与Heisenberg型不确定性不等式，推导了其与事件视界黎曼距离的关联，并验证了远场极限下可回归经典欧氏不等式，拓展了黑洞几何下的分析不等式理论。
A symmetrization techique, introduced by Cianchi, Fusco, Maggi and Pratelli concerning the Sobolev inequality, is adapted to the Gagliardo-Nirenberg-Sobolev inequality (GNS) to obtain a reduction step of the problem of showing its quantitative version. More precisely we prove a stability result for the GNS inequality under the hypothesis that it holds, in turn, in the smaller class of radial symmetric decreasing functions.
We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.
Logarithmic Sobolev inequalities: a review on stability and instability results; logarithmic uncertainty principle
本文综述对数索伯列夫不等式在强范数下的稳定性与不稳定性近期结果，涉及欧几里得空间上勒贝格及高斯测度的版本差异，给出新直接证明、示例并探讨对数不确定性原理稳定性，旨在确立该领域最新研究现状。
Improved Caffarelli-Kohn-Nirenberg Inequalities and Uncertainty Principle
该研究证明改进的 CKN 不等式与不确定性原理，并引入向量值函数的“相位导数”，将超强不确定性原理推广到球面 $\mathbb S^n$ 上的复值和向量值函数。
The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequali…
A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking
本研究给出加权Hardy-Rellich不等式成立的权重充要条件，提出用Bessel对建立最优不等式的新方法，改进了已有结果，并部分解答了相关公开问题，明确了不等式对所有函数成立等价于对径向函数成立的权重条件。
Higher-order Sobolev and Rellich inequalities via iterated Talenti's principle
本文通过迭代Talenti原理，在非紧致Riemann流形上建立了高阶Sobolev和Rellich不等式，适用于具有非负Ricci曲率或非正截面曲率的流形。
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are define…
We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as w…
The article builds on several recent advances in the Monge-Kantorovich theory of mass transport which have -- among other things -- led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated by Brunn-Minkowski, Sobolev, Gagliardo-Nirenberg, Beckner, Gross, Talagrand, Otto-Villani and their extensions by many others. While this paper continues in this spirit, we however propose here a basic framework to which all of these inequalities belong, and a general unifying principle from which many of them follow. This basic inequality relates the relativ…
Scale-Dependent Poincaré inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space
本文在双曲空间上建立尺度依赖的Poincaré-Hardy型恒等式，推导并改进多类不等式，获得海森堡不确定性原理的L2稳定性，并填补高斯测度对数Sobolev不等式的空白。
We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.
Stability of Gaussian Poincaré inequalities and Heisenberg Uncertainty Principle with monimial weights
该研究利用Bakry-Émery准则与Γ-演算建立单项式高斯测度庞加莱不等式，通过对偶方法研究其改进与梯度稳定性，构建尺度依赖不等式并用于推导带单项式权重的海森堡不确定性原理的稳定性，还得到经典情形下的相关稳定性结果。
Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents
本文建立了一族新的临界高阶Sobolev插值不等式，覆盖径向与非径向函数，利用尖锐Hardy-Rellich不等式的最优二次型，并通过Fourier分析和高阶Talenti比较原理克服高阶情形下的困难，推广了Dietze和Nam的一阶结果。
Renormalized variational principles and Hardy-type inequalities
本文在Hardy不等式成立的有界区域上建立包含Hardy与Trudinger不等式的指数可积性结果，并用重整化泛函给出Liouville方程极大解的变分刻画及整体H1界。
A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities
本文在黎曼流形上提出加性与乘性泛函不等式，引入Riccati对扩展Bessel对，通过求解Riccati型ODE简洁证明多种Hardy型不等式，并在Cartan-Hadamard流形上证明精确不确定性原理。
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times \mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb S$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ …
Differential Harnack inequalities and maximum principles of Morel-Oswald type for elliptic PDE in divergence form
本文为散度型一致椭圆方程正解建立Morel-Oswald型Hopf-Oleinik引理推广及新的微分Harnack估计，并给出最优全局结果。
Variational Principles and Rearrangement Inequalities for asymmetric Operators on Periodic Lattices
该研究针对周期格上非对称二阶差分算子的主特征值，建立变分公式与重排不等式，推导相关校正子性质，在一维情形下得到可最大化倾斜主特征值的钟形循环重排。
Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite …
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in the whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_…
A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators
本文提出多点最大值原理，直接证明黎曼流形上薛定谔方程正解的全局Harnack不等式，避免高阶导数，并给出二次势情形下的更优常数及Ornstein-Uhlenbeck算子的应用。
Shifted Composition I: Harnack and Reverse Transport Inequalities
提出移位复合规则这一信息论新原理，通过辅助移位界定随机过程分布间的散度，借此证明扩散过程的反向传输不等式并蕴含维度无关的 Harnack 不等式，同时连接了连续与离散时间方法。
We prove an improvement for the sharp Adams inequality in $W^{m,\frac nm}_0(Ω)$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our method gives an alternative approach to a Concentration--Compactness principle in $W^{m,\frac nm}_0(Ω)$ recently established by do Ó and Macedo. Moreover, when $m$ is odd, we obtain an improvement for their result by finding the best exponent in this principle. Our approach also is successfully applied to whole space $\mathbb{R}^n$ to establish an improvement for the …
In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequalities which contains the sharp second order uncertainty principle recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case. We also show the sharpness of our inequalities for several classes of parameters. Finally, we prove two stability versions of the sharp second order uncertainty principle of Cazacu, Flynn and Lam by showing that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ norm of gradient of functions.
Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities
本文针对螺线管场的海森堡不确定性原理和二阶CKN不等式，建立了最优稳定性层级，给出显式稳定性常数，并通过Fourier--Hankel变换新方法简化证明。
Pitt Inequalities and Logarithmic-type Uncertainty Principles for Metaplectic Operators
本文完整刻画了 Metaplectic 算子的 Pitt 不等式，提出各向同性与各向异性两种形式，并推导了对数型和熵型不确定性原理，修正了经典熵不等式在奇异方向存在时的失效，应用于二次 Schrödinger 演化。
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
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。