The distribution of vector-valued Rademacher series

Rademacher 类型与 Enflo 类型再探

Rademacher Type and Enflo Type Revisited

本文在离散立方体上为具有 Rademacher 类型的 Banach 空间建立无维数 Poincaré 型不等式，从而证明 Rademacher 类型与 Enflo 类型一致，并将常数改进为 $p$。

Hilbert points in Hilbert space-valued $L^p$ spaces

Let $H$ be a Hilbert space and $(Ω,\mathcal{F},μ)$ a probability space. A Hilbert point in $L^p(Ω; H)$ is a nontrivial function $φ$ such that $\|φ\|_p \leq \|φ+f\|_p$ whenever $\langle f, φ\rangle = 0$. We demonstrate that $φ$ is a Hilbert point in $L^p(Ω; H)$ for some $p\neq2$ if and only if $\|φ(ω)\|_H$ assumes only the two values $0$ and $C>0$. We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.

On vector-valued inequalities for Sidon sets and sets of interpolation

Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$ is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if $E$ is a set of interpolation ($I_0$-set). However for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if $X$ is restricted to be ``natural'' then the result holds for all Sidon sets. We also cons…

Random Series of Trace Class Operators

In this lecture, we present some results on Gaussian (or Rademacher) random series of trace class operators, mainly due jointly with F. Lust-Piquard. We will emphasize the probabilistic reformulation of these results, as well as the open problems suggested by them. We start by a brief survey of what is known about the problem of characterizing a.s. convergent (Gaussian or Rademacher) series of random vectors in a Banach space. The main result presented here is that for certain pairs of Banach spaces $E,F$ that include Hilbert spaces (and type 2 spaces with the analytic UMD property), we have $…

有偏离散立方体上的向量值集中不等式

Vector-valued concentration inequalities on the biased discrete cube

该研究提出离散超立方体有偏测度下对偏差参数和目标巴拿赫空间Rademacher型具最优依赖的向量值集中不等式，推导出泊松乘积测度的相关新不等式，并得到超立方体嵌入非平凡型巴拿赫空间的平均畸变下界及平均不可嵌入结论。

向量值函数在紧群上的谱Barron空间

Spectral Barron spaces of vector-valued functions on compact groups

本文研究了由在紧群上具有特定可和性质的向量值函数构成的谱Barron空间，探讨了这些空间的功能性质以及与其他函数空间（如向量值Sobolev空间和紧群上的有界向量值函数空间）的连续嵌入性质。

Noncommutative Khintchine and Paley inequalities via generic factorization

We reprove an inequality for Rademacher series with coefficients in the Schatten class $S_1$. Our method yields the same estimate for coefficients after suitable gaps in $S_1$-valued trigonometric series; this was known for scalar-valued functions. A very similar method gives a new proof of the extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary coefficients.

Extensions of the vector-valued Hausdorff-Young inequalities

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…

On the distribution of Sidon series

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure $μ$ and dual group $Γ$. Let E be a Sidon subset of $Γ$ with Sidon constant S(E). Let r_n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all $α> 0$: c^{-1}P[| \sum_{n=1}^Na_nr_n| >= c α] <= μ[| \sum_{n=1}^Na_nγ_n| >= α] <= cP [|\sum_{n=1}^Na_nr_n| >= c^{-1} α]

Abscissas of weak convergence of vector valued Dirichlet series

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $σ(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.

Vector-valued L_p convergence of orthogonal series and Lagrange interpolation

We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the vector-valued expansion problem of $L_p$-functions in terms of Jacobi polynomials and consider the question of unconditional convergence. The notion of type $p$ with respect to orthonormal systems leads to some characterizations of Hilbert spaces. It is also shown that various vector-valued Jacobi means are equivalent.

Series representations in spaces of vector-valued functions via Schauder decompositions

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation. This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space over a field $\mathbb{K}$, $\mathcal{F}(Ω)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $Ω$ and $\mathcal{F}(Ω,E)$ be an $E$-valued counterpart of $\mathcal{F}(Ω)$ (where the term $E$-valued counterpa…

Vector-valued general Dirichlet series

Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series $\sum a_n n^{-s}$, and more recently even on general Dirichlet series $\sum a_n e^{-λ_n s}$. This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a …

Henry Helson meets other big shots -- A brief survey

A theorem of Henry Helson shows that for every ordinary Dirichlet series $\sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $\sum a_n χ(n) n^{-s}$, where $χ: \mathbb{N} \to \mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Cesàro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.

Random unconditional convergence of vector-valued Dirichlet series

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it follows that $(x_n n^{-s})_n$ is Random unconditionally convergent (respectively, divergent) in $\mathcal H_2(X)$. The analogous question on $\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.

On the range of $\sum_{n=1}^\infty\pm c_n$

Let $\{c_n\}_{n=1}^\infty$ be a sequence of complex numbers. In this paper we answer when the range of $\sum_{n=1}^\infty\pm c_n$ is dense or equal to the complex plane. Some examples are given to explain our results. As its application, we calculate the Hausdorff dimension of the level sets of a Rademacher series with complex coefficients.

向量值 Dirichlet 级数的 Hardy 空间之间的 Volterra 算子

Volterra operators between Hardy spaces of vector-valued Dirichlet series

本文研究向量值 Dirichlet 级数 Hardy 空间间的有界 Volterra 算子，证明在 $p$-一致 PL-凸空间条件下仅存在平凡算子，并借助新建立的 Littlewood–Paley 不等式完成论证。

Vector valued $q$-variation for differential operators and semigroups I

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…

Hausdorff-Young type inequalities for vector-valued Dirichlet series

We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean …

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give the completeness criteria of these spaces with $c_0(X)$-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through $M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for every $c_0(X)$-multi…

Mean Convergence of Vector--valued Walsh Series

Given any Banach space $X$, let $L_2^X$ denote the Banach space of all measurable functions $f:[0,1]\to X$ for which ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2} is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only if \lim_n||f-S_n(f)||_2=0 for all $f\in L_2^X$, where S_n(f):=sum_{i=0}^{n-1} (f,w_i)w_i is the $n$--th partial sum associated with the Walsh system $(w_i)$.

A note on abscissas of Dirichlet series

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

On the growth of vector-valued Fourier series

We prove the 'little Carleson theorem' on the growth of Fourier series for functions taking values in a UMD Banach space.

Vector-valued Lagrange interpolation and mean convergence of Hermite series

Let X be a Banach space and $1\le p<\infty$. We prove interpolation inequalities of Marcinkiewicz-Zygmund type for X-valued polynomials g of degree $\le n$ on $R$, \[c_p (\sum\limits_{i=1}^{n+1} μ_i \| g(t_i)e^{-t_i^2 /2} \|^p)^{1/p} \le (\int\limits_{\RR}^{} \|g(t)e^{-t^2 /2} \|^p dt)^{1/p} \le d_p (\sum\limits_{i=1}^{n+1} μ_i \|g(t_i)e^{-t_i^2 /2} \|^p)^{1/p}\;\;,\] where $(t_i)_1^{n+1}$ are the zeros of the Hermite polynomial $H_{n+1}$ and $(μ_i)_1^{n+1}$ are suitable weights. The validity of the right inequality requires $1<p<4$ and X being a UMD-space. This implies a mean convergence theo…

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.

Operator-Valued Moment Series of the Generating Operator of L(F_2) Over the Commutator Group von Neumann algebra L(K)

In this paper, we will consider the generating operator of the free group factor L(F_2). Then we can construct the group von Neumann algebra L(K), where K is the commutator group of F_2 and the conditional expectation E. Then (L(F_2), E) is the W*-probability space with amalgamation over L(K). In this paper, we will compute the trivial operator-valued moment series of the generating operator of L(F_2) over L(K). This computation is the good example for studying the operator-valued distribution, since the operator-valued moment series of operator-valued random variables contain algebraic and co…

Hardy spaces of vector-valued Dirichlet series

Given a Banach space $X$ and $1 \leq p \leq \infty$, it is well known that the two Hardy spaces $H_p(\mathbb{T},X)$ ($\mathbb{T}$ the torus) and $H_p(\mathbb{D},X)$ ($\mathbb{D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces $\mathcal{H}_p(X)$ and $\mathcal{H}^+_p(X)$ of Dirichlet series $\sum_n a_n n^{-s}$ with coefficients in $X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the analytic Radon-Nikodým Prop…

On a ${\mathbb C}^2$-valued integral index transform and bilateral hypergeometric series

We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.