Stephen J. Montgomery-Smith, Michel Talagrand
We show that for any operator $T:l_\infty^N\to Y$, where $Y$ is a Banach space, that its cotype 2 constant, $K_2(T)$, is related to its $(2,1)$-summing norm, $π_{2,1}(T)$, by $K_2(T) \le c \log\log N π_{2,1}(T) $. Thus, we can show that there is an operator $T:C(K)\to Y$ that has cotype 2, but is not 2-summing.
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We will prove an abstract comparision principle which translates gaussian cotype in Rademacher cotype conditions and vice versa. More precisely, let $2\!<\!q\!<\!\infty$ and $T:\,C(K)\,\to\,F$ a linear, continous operator. T is of gaussian cotype q if and only if ( \summ_1^n (\frac{|| Tx_k||_F}{\sqrt{\log(k+1)}})^q )^{1/q} \, \le c || \summ_1^n \varepsilon_k x_k ||_{L_2(C(K))} , for all sequences with $(|| Tx_k ||)_1^n$ decreasing. T is of Rademacher cotype q if and only if (\summ_1^n (|| Tx_k||_F \,\sqrt{\log(k+1)})^q )^{1/q} \, \le c || \summ_1^n g_k x_k ||_{L_2(C(K))} , for all sequences wi…
Lipschitz stable sequence classes: an approach to Rademacher type and cotype of Lipschitz functions
本文通过引入Lipschitz稳定序列类，将序列类方法从线性与多重线性理论扩展到Lipschitz设定，定义了$(Z,Y)$-求和Lipschitz函数空间，并证明其在稳定序列类下构成Banach Lipschitz理想，进而刻画了Lipschitz函数的Rademacher型与余型。
It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m< n^{1+1/q}$ such that for every f:Z_m^n --> X we have $\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x) ||_X^q ] < C m^q \Avg_{\e,x} [ ||f(x+\e)-f(x) ||_X^q ]$, where the expectations are with respect to uniformly chosen x\in Z_m^n and \e\in \{-1,0,1\}^n, and all the implied constants may depend only on q and the Rademacher cotype q constant of X. This improves the bound of m< n^{2+\frac{1}{q}} from [Mendel, Naor 2008]. The proof of the above inequality is based on a …
A Counterexample to Talagrand's Operator Cotype Problem
针对Talagrand关于算子余型控制的问题，构造了$q=2$时的反例，否定其猜想，反例由ChatGPT发现。
Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.
In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $% (2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from $\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$ has cotype $q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem.
This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The L^p-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to sigma-finite measure spaces with filtrations and the L^p-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for L^p-boundedness and a…
We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L_p coarsely or uniformly embeds into L_q. We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a qu…
We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk po…
It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $\ell_p\tp X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes.
Recently an operator space version of type and cotype, namely type $(p,H)$ and cotype $(q,H)$ of operator spaces for $1\leq p \leq 2\leq q \leq \infty$ and a subquadratic and homogeneous Hilbetian operator space $H$ were introduced and investigated by the author. In this paper we define weak type $(2,H)$ (resp. weak cotype $(2,H)$) of operator spaces, which lies strictly between type $(2,H)$ (resp. cotype $(2,H)$) and type $(p,H)$ for all $1\leq p <2$ (resp. cotype $(q,H)$ for all $2<q \leq \infty$). This is an analogue of weak type 2 and weak cotype 2 in the Banach space case, so we develop a…
We consider two operator space versions of type and cotype, namely $S_p$-type, $S_q$-cotype and type $(p,H)$, cotype $(q,H)$ for a homogeneous Hilbertian operator space $H$ and $1\leq p \leq 2 \leq q\leq \infty$, generalizing "$OH$-cotype 2" of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_p$ spaces, and we investigate the relationship between a homogeneous Hilbertian space $H$ and operator spaces with cotype $(2,H)$. As applications we consider operator space versions of generalized little Grothendieck's theorem and Maurey's extension theorem in terms of thes…
The concepts of Riesz type and cotype of a given Banach space are extended to a non-commutative setting. First, the Banach space is replaced by an operator space. The notion of quantized orthonormal system, which plays the role of the orthonormal system in the classical setting, is then defined. The Fourier type and cotype of an operator space with respect to a non-commutative compact group fit in this context. Also, the quantized analogs of Rademacher and Gaussian systems are treated. All this is used to obtain an operator space version of the classical theorem of Kwapień characterizing Hilbe…
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then $B(\ell_{\infty},X^*)=Π_2(\ell_{\infty},X^*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension property for these classe…
Sharp Metric Cotype Inequalities for $L_1$ via Nonlinear Cut Smoothing
本文通过非线性割集平滑技术，确定了 $L_1$ 的 $p$ 矩环面不等式的最优阶数，并回答了 Mendel 和 Naor 关于 $L_1$ 尖锐度量余型的问题。
In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that …
It is well known in Banach space theory that for a finite dimensional space $E$ there exists a constant $c_E$, such that for all sequences $(x_k)_k \subset E$ one has \[ \summ_k \noo x_k \rrm \kl c_E \pl \sup_{\eps_k \pm 1} \noo \summ_k \eps_k x_k \rrm \pl .\] Moreover, if $E$ is of dimension $n$ the constant $c_E$ ranges between $\sqrt{n}$ and $n$. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces $X$, where the constant $c_E \sim \sqrt{n}$ for all finite dimensional subspaces. More generally, we…
On the expansiveness of coarse maps between Banach spaces and geometry preservation
该研究提出巴拿赫空间间新的可嵌入性概念，证明其严格弱于粗可嵌入性，借助度量余型技术研究余型保持与ℓ_p空间嵌入，结合Kalton非线性不变量对比，还探讨了ℓ_∞嵌入问题。
In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property. These results generalize, on the one hand Pisier and Xu's result on the variational inequalities for $\mathcal B$-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype $q$ is necessary for the $\mathcal B$-valued $q$-variational inequalities. As applications…
Dimension-free cotype for isotropic log-concave random polytope spaces
研究各向同性对数凹随机多面体空间，通过替换高斯估计的方法证明其具有与 \(n\) 和 \(\mu\) 无关的余型，且余型常数仅依赖于 \(\gamma\)，还得到一个有限余型可分巴拿赫空间，其 \(k\) 维子空间巴拿赫 - 马祖尔直径为 \(k\) 阶。
In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
Using notions from the geometry of Banach spaces we introduce square functions $γ(Ω,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(Ω,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $γ(Ω,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a…
Preservation of primariness under $\ell_1$-, $c_0$-, and $\ell_\infty$-sums of Banach spaces
本文证明Banach空间到向量值序列空间的一致素因子化性质传递原理，在有限余型或自相似假设下得到这些空间是素的，并恢复和获得了若干经典空间的素性。
Cotype of random polytopes
本文研究了随机多面体的型界，证明了在高维空间中，随机多面体对应的规范空间具有与维度无关的型界。
The Bohr-Bohnenblust-Hille Theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series $\sum_n a_n n^{-s}$ converges uniformly but not absolutely is less than or equal to 1/2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space $\mathcal{H}_\infty$ equals 1/2. By a surprising fact of Bayart the same result holds true if $\mathcal{H}_\infty$ is replaced by any Hardy space $\mathcal{H}_p$, $1 \le p < \infty$, of Dirichlet series. For Dirichlet series with coefficients i…
The KKL inequality and Rademacher type 2
本文证明了在所有Rademacher类型2的Banach空间中向量值KKL不等式成立，并给出了涉及类型2常数$T_2(X)$的具体不等式形式。
We show that given a positive integer $m$, a real number $p\in\left[ 2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left( r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$ contains, except for the null vector, a closed subspace of maximal dimension whenever $r<\frac{2ms}{s+2m-ms}$. This result is optimal since for $r\geq\frac{2ms}{s+2m-ms}$ all $m$--linear forms on $\ell_{p}\times \cdots\times\ell_{p}$ are multiple $\left( r;s\right)$--summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botel…
Discrete logarithmic Sobolev inequalities in Banach spaces
本文在有限余型Banach空间上建立了离散超立方体上与维数无关的向量值Lp对数Sobolev不等式，并利用该不等式给出Hamming立方体非线性商的双Lipschitz畸变新下界。
A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < \infty.
Functional calculus for $C_{0}$-groups using (co)type
本文研究了在Banach空间上生成$C_0$-群的生成元的函数calculus性质，通过类型和共类型假设，证明了在特定条件下生成元具有有界$\mathcal{H}^\infty$-calculus，并探讨了有界虚数幂与有界$\mathcal{H}^\infty$-calculus之间的差距。
Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,...,f_n:{-1,1}^n\to X$ satisfy \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n \partial_jf_j(\varepsilon)\Bigg|^pdμ(\varepsilon) \leq \mathfrak{P}^p\int_{{-1,1}^n}\int_{{-1,1}^n}\Bigg\|\sum_{j=1}^n \d_jΔf_j(\varepsilon)\Bigg\|^pdμ(\varepsilon) dμ(δ), where $μ$ is the uniform probability measure on the discrete hypercube ${-1,1}^n$ and ${\partial_j}_{j=1}^n$ and $Δ=\sum_{j=1}^n\partial_j$ are the hypercube partial derivatives and the hypercube Laplacian, r…
A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p<\infty, we study the validity of square functions estimates Norm{(\sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^\infty-calculus on $Lp$-spaces by the…
Square functions associated with Ritt$_E$ operators
本文针对单位圆子集定义的Ritt$_E$算子，引入二次泛函演算，证明在有限余型条件下其等价于有界泛函演算，并建立了相关平方函数的等价性。
A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from $L_1$ into an arbitrary Banach space, namely, the operator from $L_1$ into $\ell_\infty$ defined by $$ T_0 (f) =\left( \int r_n f \, dμ\right)_{n\ge 0} \ , $$ where $r_n$ is the $n^{\text{th}}$ Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbit…
Given a Banach space $X$ and a real number $α\ge 1$, we write: (1) $D(X)\leα$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$ itself admits a bilipschitz embedding into $X$ with distortion $\le α\cdot C$; (2) $D(X)=α^+$ if, for every $\varepsilon>0$, the condition $D(X)\leα+\varepsilon$ holds, while $D(X)\leα$ does not; (3) $D(X)\le α^+$ if $D(X)=α^+$ or $D(X)\le α$. It is known that $D(X)$ is bounded by a universal constant, but the available estimates for this constant are rather large. The…
We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset \ell^p$ for $4 < p < \infty$ and $X \subset c_0$ such that $\mathfrak{A}_X$ contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ s…
For a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space the associated conditionality constants $k_{m}[\mathcal{B}]$ verify the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies $k_{m}[\mathcal{B}]=(\log m)^{1-ε}$ for some $0<ε<1$, and this is optimal. Our first goal in this paper will be to fill the gap in betw…
Absolutely summing Toeplitz operators on weighted Fock spaces
本文研究加权Fock空间上复符号Toeplitz算子的绝对r-求和性，通过IDA分解和Rademacher论证等工具，给出了基于局部均值、复球平均和Berezin变换的等价刻画，并证明了准则的最优性。
Necessary conditions for deterministic and stochastic maximal regularity
该论文研究巴拿赫空间几何在确定性与随机最大正则性中的作用，构造反例表明UMD假设不可省略，证明随机卷积算子的R-有界性条件$(S_p)$的必要性及相关等价性等结论。
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