Eugene Tokarev
It is proved that there exists a separable reflexive Banach space W that contains an isomorphic image of every separable superreflexive Banach space. This gives the affirmative answer on one J. Bourgain's question
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p-adic Grothendieck Inequality, p-adic Johnson-Lindenstrauss Flattening and p-adic Bourgain-Tzafriri Restricted Invertibility Problems
本文提出了p-adic版本的Grothendieck不等式、Johnson-Lindenstrauss扁平化引理和Bourgain-Tzafriri受限可逆性定理。
Noncommutative Bourgain's circular maximal theorem and a local smoothing estimate on the generalized Moyal planes
本文在二维量子欧几里得空间上建立了Mockenhaupt–Seeger–Sogge局部光滑估计的非交换类比，并由此得到Bourgain圆极大定理的非交换版本，解决了Hong之后的一个问题。
We prove that Bourgain's hyperplane conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.
Expository article: "Bounded orthogonal systems and the $Λ(p)$-set problem'' by Jean Bourgain
本文阐述Jean Bourgain解决Rudin关于$Λ(p)$-集存在性猜想的工作。
Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-Spaces
本文解决马祖尔问题，证明满足特定条件的L_∞空间存在c_0商，通过构造性方法和反例验证假设的必要性。
A Bourgain-Gromov problem on non-compact Sobolev-Lorentz embeddings
本文研究非紧 Sobolev-Lorentz 嵌入的有限严格奇异性，通过约化到 Besov 空间与序列空间嵌入，给出 Bernstein 数衰减率的上界，并简化了 Bourgain-Gromov 与 Lang-Mihula 的原有方法。
The real interpolation spaces between $L^{p}({\mathbb{R}}^{n})$ and $\dot {H}^{t,p}({\mathbb{R}}^{n})$ (resp. $H^{t,p}({\mathbb{R}}^{n})$), $t>0,$ are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be " the correct" fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limit formulae, as well as the Bourgain-Brezis-Mironescu convergence theorem, to …
Motivated by a problem stated by S.A.Argyros and Th. Raikoftsalis, we introduce a new class of Banach spaces. Namely, for a sequence of separable Banach spaces $(X_n,\|\cdot\|_n)_{n\in\mathbb{N}}$, we define the Bourgain Delbaen $\mathcal{L}^{\infty}$-sum of the sequence $(X_n,\|\cdot\|_n)_{n\in\mathbb{N}}$ which is a Banach space $\mathcal{Z}$ constructed with the Bourgain-Delbaen method. In particular, for every $1\leq p<\infty$, taking $X_n=\ell_p$ for every $n\in\mathbb{N}$ the aforementioned space $\mathcal{Z}_p$ is strictly quasi prime and admits $\ell_p$ as a complemented subspace. We s…
We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $δ> 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $δ$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $\ell_1$ dual. The first class of spaces are HI and possess few, but not very few …
Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $…
Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound
结合Milman的M-椭球体理论、Guan界的随机定位及Shannon-Stam不等式稳定性估计，肯定解决了Bourgain切片问题，证明任何单位体积凸体均存在截面大于普适常数的超平面。
The trace-free Beurling--Ahlfors transform and the Bourgain--Brezis problem for Hodge systems
该研究引入无迹Beurling--Ahlfors变换，利用其符号的反射奇性得到端点抵消估计，证明了任意维度与形式次数下Bourgain--Brezis猜想的希尔伯特情形，还得到多类新的临界估计及端点Hodge分解，并证明多数情形下有界选择算子非线性。
Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces
该研究将Bourgain等人1981年针对L_p空间的补子空间构造推广到重排不变空间与Hardy空间，建立了相关二分性并引入序数指标，同时给出特定空间中这些子空间非补的充分条件。
Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality
本文研究了离散Bourgain-Morrey序列空间的结构性质、与经典ℓ^p空间的关系及对偶性，揭示了其可分性、嵌入关系及对偶空间的构造。
In his study of the Radon Nikodým property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set $A$ that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set $A$ (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of $A$-valued quasimartingales with sharply divergent behavior.
In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} μ= μ(N_n,B_n) =& δ_{\frac{1}{N_1}B_1}\astδ_{\frac{1}{N_1N_2}B_2}\ast δ_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& μ_n\astμ_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(Λ) = \{e^{2πi λx}:λ\inΛ\}$ for $L^2(μ)$, where $Λ$ is obtained from suitably modifying $L_n$. Here, $μ_n$ is the convolution of the first $n$ Dirac measures …
We show that the non-separable Banach space $SL^\infty$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL^\infty$. In particular, we bypass Bourgain's localization method.
Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
该论文通过结合无迹 Beurling--Ahlfors 变换与非线性对偶方法，在 Besov 和 Triebel--Lizorkin 空间中扩展了 Hodge 系统的 Bourgain--Brezis 选择映射的适用范围，显著拓宽了已知解范围。
Operator Algebras of Bourgain Delbaen Spaces: Realization, Rigidity, and Ideal Structure
本文研究了Calkin代数及其作为紧算子商的结构，通过保持几何和拓扑信息实现交换C*代数。证明了对于每个紧度量空间K，存在反射Banach空间X_C(K)，其Calkin代数同构于C(K)。同时建立了稳定性、局部化原理、Hölder连续性、刚性定理及理想结构分类等贡献。
We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves a problem of J. Bourgain. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block $q$-Hilbertian and/or a block $p$-Besselian finite dimensional decomposition.
Isoperimetric inequalities in high-dimensional convex sets
本讲义笔记围绕Bourgain切片问题及Kannan、Lovasz与Simonovits提出的KLS等周猜想的近期研究进展展开阐述。
Limitedly L-weakly compact operators
本文引入从Banach空间到Banach格的limitedly L-weakly compact算子新类，证明其介于Bourgain-Diestel算子与L-weakly compact算子之间，并给出基于序列的有效刻画，研究控制问题及完备性。
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if $k>3$. The problem is still open for $k=2,3$. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space $\mathbb{H}^n$.
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = \sum x_j \otimes x_j, rather than on the coordinate one. The John's decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. This is applied to embeddings of l_\infty^k into finite dimensional spaces.
Following results of Bourgain and Gorelik we show that the spaces $\ell_p$, $1<p<\infty$, as well as some related spaces have the following uniqueness property: If $X$ is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a $C(K)$ space is uniformly homeomorphic to $c_0$, then it is isomorphic to $c_0$. We show also that there are Banach spaces which are uniformly homeomorphic to exactly $2$ isomorphically distinct spaces.
Unconditional structure of Banach spaces with few operators
本文通过p-凸化Gowers空间，证明其具有唯一无条件基，推翻了关于无条件基空间等距同构的猜想，并揭示了少数算子空间的无条件结构唯一性。
From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
本文研究Bourgain--Brezis--Mironescu空间的$L^q$变体，刻画了函数与特征函数的端点行为，并建立了精确周长公式的收敛性及其维度限制。
We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in $L^1$ appearing naturally in these contexts.
A Banach space with a Schauder basis is said to be $α$-minimal for some countable ordinal $α$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $α$. We prove a dichotomy that characterises when a Banach space has an $α$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.
Presenting a proof of the Kannan-Lovasz-Simonovits conjecture
本文给出了高维几何中KLS猜想的证明，关键利用Song和Zhang关于倾斜平均高阶导数的准则。
We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric.
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in …
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