一类与Kakeya极大函数相联系的BMO空间

介绍了一类与Kakeya极大函数相联系的BMO空间BMOδ及其与古典BMO空间类似一些的性质,算例表明BMOδ具有古典BMO不具备的性质.

一类交换子型Kakeya极大函数的加权估计

研究一类乘积形式函数的交换子型Kakeya极大函数的加权估计。

THE KAKEYA MAXIMAL OPERATOR WITH A SPECIAL BASE to the memory of Professor Ruilin Long

Let M f be the Kakeya maximal function in d-dimensional Euclidean space, with same base consisting of cylinders of eccentricity N. The inequality shoum for a base satisfying a direction condition, where ?and c are constants depending only on d.

The Kakeya Problem

This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.

SOME REFINEMENTS OF ENESTROM-KAKEYA THEOREM

In this paper we present certain interesting refinements of a well-known Enestrom- Kakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Aziz and Mohammad, Govil and Rehman and others.

A Generalization of Eneström-Kakeya Theorem and a Zero Free Region of a Polynomial

For the polynomial P (z) = ajzj, aj ≥ aj-1, a0 > 0, j = 1, 2, …, n, an > 0, a classical result of Eneström-Kakeya says that all the zeros of P (z) lie in |z|≤ 1. This result was generalised by A. Joyall and G. Labelle, where they relaxed the non-negativity condition on coefficients. It was further generalized by M.A Shah by relaxing the monotonicity of some coefficients. In this paper, we use some known techniques and provide some more generalizations of the above results by giving more relaxation to the conditions.

随机算子值Enestrm-Kakeya定理的一个注记

关于代数方程和算子情形的Enestr¨om Kakeya定理已被讨论。我们在这篇注记中 ,利用作用在Hilbert空间上的随机算子值的随机数值半径不等式证明了随机算子值Enestr¨om

Kakeya极大算子及其分数次情形的正则性

研究中心Kakeya （Nikodym）极大算子KN（N＞2）及其分数次情形Kα,N（0＜α＜d）的正则性.特别地,建立了中心分数次Kakeya极大算子Kα,N是从W^1,p（R^d）到W^1,q（R^d）上的有界连续算子,其中1＜p＜∞,q=dp/（d-αp）和0≤α＜ d/p.还证明了中心Kakeya极大算子KN是分数次Sobolev空间W^s,p（R^d）,非齐次Triebel-Lizorkin空间Fs^p,q（R^d）以及非齐次Besov空间Fs^p,q（R^d）上的有界连续算子,其中0＜s＜1,1＜p,q＜∞.此外,也考虑分数次Kakeya极大函数的弱导数的两种点态估计以及其离散情形的正则性.

振荡型积分对极大函数的估计

首先将2维KaKeya型的极大函数通过Fourier变换转化成振荡型积分的形式,对此振荡型积分作相应的估计。通过L ittlewood-Paley方法将此振荡型积分分解成不同的算子,在估计极大算子时,通过Newton-Le ibn ig公式,将其转化为相应的积分形式,最后通过经典的振荡型积分的估计,得到最后的结果,本方法强调振荡型积分的方法和以前几何测度论的方法不同。

R^3上一类特殊Besicovitch集的维数估计

本文研究了R^3上一类圆盘型Besicovitch集的Hausdorff维数问题.将Kakeya问题二维情形的其中一种证明方法推广到R^3空间,获得了该类圆盘型Besicovitch集的Hausdorff维数为3.

BOUNDS FOR THE ZEROS OF POLYNOMIALS

Let P(z)= be a polynomial of degree n. In this paper we prove a more general result whichinteralia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrom-Kakeya theorem.

R^3上一类圆盘型Besicovitch集的Hausdorff维数估计

文中研究了R3上一类圆盘型Besicovitch集的Hausdorff维数,将Kakeya问题二维情形的其中一种证明方法推广到R3空间,证明了该类圆盘型Besicovitch集的Hausdorff维数为3.

A Study Guide for the l^2 Decoupling Theorem

This paper contains a detailed, self contained and more streamlined proof of the l^2 decoupling theorem for hypersurfaces from the paper of Bourgain and Demeter in 2015. The authors hope this will serve as a good warm up for the readers interested in understanding the proof of Vinogradov's mean value theorem from the paper of Bourgain,Demeter and Guth in 2015.