A Sharp Multidimensional Bergh Type Inequality

A sharp multidimensional integral inequality for functions satisfying two quasi-mono-tonicity conditions is proved. This result generalizes Bergh's inequality valid for quasi-concave functions.     

Note on Sharp Hardy-Type Inequality

We prove Hardy-type inequality with weight singular at ${0 \in \Omega}$ in the class of functions which are not zero on the boundary ${\partial \Omega}$ . The Hardy constant is optimal and the inequality is sharp due to the additional boundary term.     

A Sharp Inequality for a Trigonometric Sum

We prove that the inequality $$-\frac{1}{2}\leq {\sum\limits_{k=1}^{n}} \left( \frac{{\rm cos}(2kx)}{2k - 1}+\frac{{\rm sin}((2k - 1)x)}{2k} \right)$$ holds for all natural numbers n and real numbers x with ${x \in [0, \pi]}$ . The sign of equality is valid if and only if n =  1 and x =  π /2.     

A Sharp Remez Inequality for Trigonometric Polynomials

We obtain a sharp Remez inequality for the trigonometric polynomial T _n of degree n on [0,2π): $$\|T_n \|_{L_\infty([0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T _n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty([0,2\pi ))}}{ \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated.     

球面上的次临界最佳Sobolev不等式

本文在球面SN上建立了一类最佳Sobolev不等式:‖f‖2Lq(SN)≤(q-2)Γ(N-d)2+1/dΓ(N+d/2))(∫SNf(ξ)Adf(ξ)dξ-Γ(N+d/2)Γ/(N-d/2)∫SN |f丨2dξ)+∫SN丨f丨2dξ,其中Ad(0＜d＜N)是SN上的高阶保形算子,dξ是SN的归一化曲面测度,2≤q＜2...     

Jackson-Type Inequality for Doubling Weights on the Sphere

In the one-dimensional case, Jackson's inequality and its converse for weighted algebraic polynomial approximation, as well as many important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumption on the weights. In most cases this minimal assumption is the doubling condition. In this paper, we establish Jackson's theorem and its Stechkin-type converse for spherical polynomial approximation with respect to doubling weights on the unit sphere.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.     

Sobolev-Lorentz范数约束下的次临界型Adams不等式

本文在Sobolev-Lorentz空间W2L2,q(R4)的范数约束下得到了一个最佳的二阶次临界型Adams不等式.进一步,当次临界指标逼近最佳常数时,得到了Adams泛函的上、下界的估计.本文主要采用了Lam和Lu[A new approach to sharp Moser-Trudinger and Adams ...     

A Sharp Version of Mahler's Inequality for Products of Polynomials

In this note we give some sharp estimates for norms of polynomialsvia the products of norms of their linear terms. Different convexnorms on the unit disc are considered. 1991 Mathematics SubjectClassification 30C10, 11C08.     

关于非负独立同分布随机变量停止和的精确不等式的注记(英语)

Hitczenko[2]证明了不等式 E(sum from i=1 to γ(ζ_i))~γ≤2(γ-1)E(sum from i=1 to γ~2(ζ_i))~γ,1≤γ〈∞,其中(ζ_i)为非负独立随机变量,γ为停时,γ′为停时γ的一个复制品,且与(ζ_i)独立,2(γ-1)是最佳常数,我们证明了,对于非负独立同分布的(ζ_i),2(γ-1)也是最佳常数,从而解决了Hitczenko[2]提出的问题。     

On a Property of Functions on the Sphere

According to the Knaster conjecture, for any continuous function $f:S^{n - 1} \to \mathbb{R}$ and any $n$ -point subset of the sphere $S^{n - 1}$ , there exists a rotation mapping all the points of this subset to a level surface of the function $f$ . In the present paper, this conjecture is proved for the case in which ${n = p^1 }$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.     

Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L^p space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of R^d.     

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.     

A Sharp Lp Inequality for Dyadic A1 Weights in Rn

The exact best possible range of p is determined such that anydyadic A₁ weight w on Rⁿ satisfies a reverse Hölder inequalityfor p, which depends on the dimension n and the correspondingA₁ constant of w. The proof is based on an effective linearizationof the dyadic maximal operator applied to dyadic step functions.2000 Mathematics Subject Classification 42B25.     

A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.     

亚纯函数的一个基本不等式——纪念李国平院士吴新谋教授诞辰100周年

该文, 欲求在含有原点的区域D内亚纯函数之囿界, 其中使用熊庆来教授, 李国平教授的方法. 所得结果较熊庆来教授的结果更一般些, 因此较Valiron教授, Mllilloux教授的结果更一般些与精确些.     

与单形外接球心有关的一个不等式

若n维单形A＝A1A2…An r的外接球心O位于其内部，单形A与单形Ai＝A1A2…Ai-1OAi 1…An 1的外接球半径分别为R与Ri(i=1,2,…,n 1），本文证明了：n 1↑П↑i=1Ri≥(nR/2)^n 1，而等号成立的充分必要条件为单形A正则。     

A Hardy Inequality for Ultraspherical Expansions with an Application to the Sphere

We prove a Hardy inequality for ultraspherical expansions by using a proper ground state representation. From this result we deduce some uncertainty principles for this kind of expansions. Our result also implies a Hardy inequality on spheres with a potential having a double singularity.     

Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.