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 con...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.展开更多
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...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.展开更多
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 a...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.展开更多
For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup...For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup></em>, <em>a</em><sub><em>j </em></sub>≥ <em>a</em><sub><em>j</em>-1</sub>, <em>a</em><sub>0</sub> > 0, <em>j</em> = 1, 2, …, <em>n</em>, <em>a<sub>n</sub></em> > 0, a classical result of Enestr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>m-Kakeya says that all the zeros of <em>P</em> (<em>z</em>) lie in |<em>z</em>|≤ 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.展开更多
适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领...适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.展开更多
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,...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。展开更多
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 generali...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.展开更多
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...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.展开更多
基金The author is partly supported by the Grants-in-Aid for Scientific Reseach,The Ministry of Educa-ion,Science and Culture,Japan.
文摘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.
文摘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.
文摘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.
文摘For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup></em>, <em>a</em><sub><em>j </em></sub>≥ <em>a</em><sub><em>j</em>-1</sub>, <em>a</em><sub>0</sub> > 0, <em>j</em> = 1, 2, …, <em>n</em>, <em>a<sub>n</sub></em> > 0, a classical result of Enestr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>m-Kakeya says that all the zeros of <em>P</em> (<em>z</em>) lie in |<em>z</em>|≤ 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.
文摘适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.
文摘首先将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。
文摘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.
基金supported by the NSF Grant DMS-1301619,the NSF Grant DMS-1161752
文摘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.
