Evgueni M. Semenov, Stephen J. Montgomery-Smith
Consider an n by n matrix x_ij, and consider the quantity || x_{i,pi(i)} ||_X where X is a symmetric sequence space as a random variable where the permutation pi is chosen randomly. This was considered by Kwapien and Schutt, and we extend their results to rearrangement invariant spaces. We also consider the property of D and D^* convexity for rearrangement invariant spaces.
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We construct a Banach rearrangement invariant norm on the measurable space for which the finiteness of this norm for measurable function (random variable) is equivalent to suitable tail (heavy tail and light tail) behavior. We investigate also a conjugate to offered spaces and obtain some embedding theorems. Possible applications: Functional Analysis (for instance, interpolation of operators), Integral Equations, Probability Theory and Statistics (tail estimations for random variables).
We determine the extrapolation law for rearrangement operators of the Haar system on vector valued Hardy spaces.
We study the extrapolation properties of vector valued rearrangement operators acting on the normalized Haar basis in $L^p_X .$
Let X_1, X_2,..., X_n be a sequence of independent random variables, let M be a rearrangement invariant space on the underlying probability space, and let N be a symmetric sequence space. This paper gives an approximate formula for the quantity || ||(X_i)||_N ||_M whenever L_q embeds into M for some 1 le q < infty. This extends work of Johnson and Schechtman who tackled the case when N = l_p, and recent work of Gordon, Litvak, Schuett and Werner who obtained similar results for Orlicz spaces.
We prove a sharp integral inequality for the dyadic maximal operator and give as an application another proof for the computation of its Bellman function of three variables.
We characterize, in the context of rearrangement invariant spaces, the optimal range space for a class of monotone operators related to the Hardy operator. The connection between optimal range and optimal domain for these operators is carefully analyzed.
We investigate the rearrangement of the Haar system induced by the postorder on the set of dyadic intervals in $[0,1]$ with length greater than or equal to $2^{-N}$. By means of operator norms on $\text{BMO}_N$ we prove that the postorder has maximal distance to the usual lexicographic order.
Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
本文系统研究了加权 Hardy 型算子在重排不变函数空间上的最优行为，刻画了保证有界性的最优空间，探讨了最优范数的简化表达，并分析了迭代算子情形。
We study the behaviour on rearrangement-invariant spaces of such classical operators of interest in harmonic analysis as the Hardy-Littlewood maximal operator (including the fractional version), the Hilbert and Stieltjes transforms, and the Riesz potential. The focus is on sharpness questions, and we present characterisations of the optimal domain (or range) partner spaces when the range (domain) is fixed. When a rearrangement-invariant partner space exists at all, a complete characterisation of the situation is given. We illustrate the results with a variety of examples of sharp particular re…
In this paper general rearrangements of the Haar system in BMO are considered. Several, necessary and suficient, conditions for the boundednes of the induced permutation operator are given. Using analytic families of operators extensions to the case of $L^p$ are obtained .
We determine a class of rearrangements that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension. We show that there exists a large subspace of $L^p$ on which a bounded rearrangement operator acts as an isomorphism. The combinatorial issues of these problems give rise to a two-person game, to be played with colored dyadic intervals. We determine winning strategies for each of the players.
We characterize the rearrangement invariant spaces for which there exists a non-constant fixed point, for the Hardy-Littlewood maximal operator (the case for the spaces $L^p(\mathbb{R}^{n})$ was first considered by Korry in \cite{Ko}). The main result that we prove is that the space $L^{\frac{n}{n-2},\infty}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ is minimal among those having this property
Let $Γ$ be an oriented Jordan smooth curve and $α$ be a diffeomorphism of $Γ$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertiblity of the binomial functional operator \[ A=aI-bW \] where $a$ and $b$ are continuous functions, $I$ is the identity operator, $W$ is the shift operator $Wf=f\circα$, in a reflexive rearrangement-invariant space $X(Γ)$ with Boyd indices $α_X,β_X$ and Zippin indices $p_X,q_X$ satisfying inequalities \[ 0<α_X=p_X\le q_X=q_X<1. \]
In this article we introduce and investigate some new Banach spaces, so - called moment spaces, and consider applications to the Fourier series, singular integral operators, theory of martingales.
In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.
Variational Principles and Rearrangement Inequalities for asymmetric Operators on Periodic Lattices
该研究针对周期格上非对称二阶差分算子的主特征值，建立变分公式与重排不等式，推导相关校正子性质，在一维情形下得到可最大化倾斜主特征值的钟形循环重排。
Mean and pointwise ergodicity for composition operators on rearrangement-invariant spaces
本文研究重排不变Banach函数空间上复合算子的遍历性，通过对算子符号施加自然条件并结合新引入的局部凸拓扑，在包含多种非自反空间的大类空间上实现了均值与逐点遍历性。
Rearrangements and infimum convolutions
本文发展了与重排相关的 inf 卷积算子的一般比较结果，导出了 Laplace 和 polar 变换在球面对称重排下的比较结果，并简化了泛函 Blaschke Santalo 不等式的证明及导出了抛物型 PDE 的比较结果。
We study moment rearrangement invariant spaces, which contain as particular cases the generalized Grand Lebesgue Spaces, and provide norm estimates for some operators, not necessarily linear, acting between some measurable rearrangement invariant spaces. The estimations are formulated in the terms of fundamental functions for these spaces.
Fourier rearrangements on discrete groups: $L^p$ and operator-norm subsequence convergence
本文针对双曲群或多项式增长群，构造傅里叶项重排使子序列在算子范数及$L^p$范数下收敛，证实Révész性质与多项式增长猜想，否证自由群反例猜想。
Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the $K$-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of $H^\infty$-functional calculus to this general setting.
Factorization Property in Rearrangement Invariant Spaces
本文研究了重排不变 Haar 系统空间中的因子分解性质，证明了具有大对角线的有界算子近似为某对角算子的因子，并在 Haar 系统作为无条件基时确立其因子分解性质。
On optimal endpoints for integral kernel operators
研究单变量函数上积分核算子行为，在重排不变拟范数赋范的巴拿赫空间类中找最优端点空间候选者，刻画非增重排具体逐点估计的存在性并构建理论，以此为一些积分核算子提出最优端点空间。
We give a new proof to show that the complex interpolation for families of Banach spaces is not stable under rearrangement of the given family on the boundary, although, by a result due to Coifman, Cwikel, Rochberg, Sagher and Weiss, it is stable when the latter family takes only 2 values. The non-stability for families taking 3 values was first obtained by Cwikel and Janson. Our method links this problem to the theory of matrix-valued Toeplitz operator and we are able to characterize all the transformations on $\T$ that are invariant for complex interpolation at 0, they are precisely the orig…
Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of $σ$-algebras $({\cal F}_n)$ such that the conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and such that the sequence (g_n) is conditionally independent. Then it is known that $\normo{\sum f_k}_p \le C \normo{\sum g_k}_p$, $1 \le p \le \infty$ where the constant C is independent of p. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant…
For a separable rearrangement invariant space $X$ on $(0,\infty)$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This characterization hinges upon a description of the set of approximate eigenvalues of the doubling operator $x(t)\mapsto x(t/2)$ in $X$. We prove that this set is surprisingly simple: depending on the values of some dilation indices of such a space, it is either an interval or …
{\it We study the class of all rearrangement-invariant (=r.i.) function spaces $E$ on $[0,1]$ such that there exists $0<q<1$ for which $ \Vert \sum_{_{k=1}}^nξ_k\Vert_{E}\leq Cn^{q}$, where $\{ξ_k\}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on $[0,1]$ and $C>0$ does not depend on $n$. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $exp(L_p)$, $p\ge 1$. We further apply our results to the study of Banach-Saks index sets in r.i. spaces.
Rearrangement-invariant norms commuting with dilations
本文研究了具有排列不变性的空间中，范数与扩张算子相容的条件，证明了只有p-齐次范数满足此条件，并探讨了相关嵌入性质。
This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Levy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms.
Some rearrangement inequalities for symmetric norms on matrices are given as well as related results for operator convex functions.
Sobolev inequalities for canceling operators
本文研究了椭圆抵消算子的Sobolev不等式，通过单维Hardy不等式特征化，并展示了其与标准梯度算子的不等式在相同范数下的精确性。
A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant function norms. Optimal target and domain norms in the relevant inequalities are characterized via a reduction principle to one-dimensional inequalities for a Calderón type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of specific families of spaces, including Lebesgue, Lorentz, Lorentz…
This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on $B(L_2(T))$. Additionally, a new framework for further investigation is introduced by consideri…
Fix $b\in (0,\infty)$ and $p\in (1,\infty)$. Let $ϕ$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,ϕ}$, at the measurable function $f:\R+\to\R+$ by $\rph(f):=[\int_0^bf^{**}(t)^pϕ(t)dt]^{\frac1p}$, in which $f^{**}(t):=t^{-1}\int_0^tf^{*}(s)ds$, where $f^*(t):=μ_f^{-1}(t)$, with $μ_f(s):=|\{x\in I_b: |f(x)|>s\}|$. Our aim in this paper is to study the rearrangement-invariant space determined by $\rph$. In particular, we determine its Köthe dual and its Boyd indices. Using the latter a sufficient condition is given for a Caldéron-Zygmund operator to ma…
In this paper we obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.
The Fourier transform in weighted rearrangement invariant spaces
该研究探讨加权重排不变空间上傅里叶变换的有界性，推导得出权函数有界性、空间等价性及指数关系等结论，并应用于证明薛定谔方程相关算子的无界性。
Extension and Integral Representation of the finite Hilbert Transform In Rearrangement Invariant Spaces
本文研究重排不变空间中有限Hilbert变换的延拓与反演，通过引入由该算子诱导的向量测度及其积分算子，给出了该算子的积分表示。
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
We give necessary and sufficient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices.
We introduce a Banach rearrangement invariant (tail) quasy-norm by means of Hardy's (Cesaro) average on the (measurable) functions defined on some measurable space which is a slight generalization of classical Lorentz-Marcinkiewicz norm and find for it an equivalent norm expression.
We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space $X$ we characterize its optimal range partner, that is, the smallest r.i. space $Y$ such that the operator is bounded from $X$ to $Y$. We apply the general results to Lorentz spaces to illustrate their strength.
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
Let (X,d,μ) be a space of homogeneous type and E a UMD Banach space. Under the assumption mu({x})=0 for all x in X, we prove a representation theorem for singular integral operators on (X,d,mu) as a series of simple shifts and rearrangements plus two paraproducts. This gives a T(1) Theorem in this setting.
We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a $δ$-ring. A precise description is given for the case of the minimal Lorentz spaces.
In this paper, equivalence between interpolation properties of linear operators and monotonicity conditions are studied, for a pair $(X_0,X_1)$ of rearrangement invariant quasi Banach spaces, when the extreme spaces of the interpolation are $L^\infty$ and a pair $(A_0,A_1)$ under some assumptions. Weak and restricted weak intermediate spaces fall in our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.
The paper gives the background for Toeplitz $T_a$ and Hankel $H_a$ operators acting between distinct Hardy type spaces over the unit circle $\mathbb{T}$. We characterize possible symbols of such operators and prove general versions of Brown-Halmos and Nehari theorems. The lower bound for measure of noncomactness of Toeplitz operator is also found. Our approach allows Hardy spaces associated with arbitrary rearrangement invariant spaces, but part of the results is new even for the classical case of $H^p$ spaces.
Gagliardo-Nirenberg inequality via a new pointwise estimate
该研究提出一类用稀疏平均算子替代极大算子的新型逐点估计，将Gagliardo-Nirenberg插值不等式推广至所有重排不变Banach函数空间，无需上Boyd指数假设，还去除了Orlicz与Lorentz空间框架下该不等式的不必要假设。
In this paper, we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on [0,1] every nonzero element of which does not belong to any included rearrangement invariant space of the same class such that the inclusion operator is disjointly strictly singular. We consider Lorentz, Marcinkiewicz and Orlicz spaces. The answer is affirmative for Marcinkiewicz spaces and negative for Lorentz and Orlicz spaces. Also, the same problem is studied for Nakano spaces assuming different hypothesis.
The aim of this paper is to obtain boundedness conditions for the maximal function Mf and to prove the necessary and sufficient conditions for the fractional maximal oparator Ma in the Lorentz Morrey spaces which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner Riesz operator and the Schrodinger type operators in the Lorentz Morrey spaces, where the nonnegative potential V belongs to the reverse Holder class.
Rearrangement Invariant Orthogonal Sums in Krein Spaces. II
本文研究了Krein空间中伪正则子空间的无穷正交和定义及其条件，通过无条件和Moore-Smith和进行分析。
IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces
本文以 IDA 函数的不增重排刻画加权 Bergman 空间中 Hankel 算子奇异值的渐近行为，并推广至共轭符号联合刻画及加权 Fock 空间中的 Berger-Coburn 现象。
Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $ε> 0$ there exists $x \in X$ with $x^2= χ_A, \int x d μ= 0$ and $\| Tx \| < ε$. In particular all compact operators are narrow. We prove that if $X$ is a Lorentz function space $L_{w,p}$ on [0,1] with $p>2$, then there exists a constant $k_X>1$ so that for every narrow projection $P$ on $L_{w,p}$ $\| Id - P \| \geq k_X. $ This generalizes earlier results on $L_p$ and partially answers a question of E. M. Semenov. Moreover we prove that every rearrangement-invariant func…
Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a $p$-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various suff…
On multiplier analogues of the algebra $C+H^\infty$ on weighted rearrangement-invariant sequence spaces
研究加权重排不变序列空间上代数 $C+H^\infty$ 的乘子类似物，证明由生成有界 Laurent 算子的周期分布构成的集合是 Banach 代数，且其特定闭包与子集之和构成闭子代数。
We prove weighted estimates for singular integral operators which operate on function spaces on a half-line. The class of admissible weights includes Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal $L^p$ regularity of first order, second order and fractional order Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson,…
Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize surjective isometries of $X(H).$ We prove that if $T$ is such an isometry then there exist Borel maps $a:\Om\to\bbK$ and $σ:\Om\lra\Om$ and a strongly measurable operator map $S$ of $\Om$ into $\calB(H)$ so that for almost all $\om$ $S(\om)$ is a surjective isometry of $H$ and for any $f\in X(H)$ $$Tf(\om)=a(\om)S(\om)(f(σ(\om))) \text{ a.e.}$$ As a consequence we …
We consider a bounded linear operator $T$ on a complex Banach space $X$ and show that its spectral radius $r(T)$ satisfies $r(T) < 1$ if all sequences $(< x',T^nx>)_{n \in \mathbb{N}_0}$ ($x \in X$, $x' \in X'$) are, up to a certain rearrangement, contained in a principal ideal of the space $c_0$ of sequences which converge to $0$. From this result we obtain generalizations of theorems of G. Weiss and J. van Neerven. We also prove a related result on $C_0$-semigroups.
Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces
本文完整刻画了离散分数阶极大算子在加权Lorentz序列空间间的有界性，通过尖锐重排估计将其归结为上确界算子的有界性，并利用近期结果得到最终判定。
A reverse entropy power inequality for i.i.d. log-concave random variables
研究独立同分布对数凹随机变量的反向熵幂不等式，通过递减重排、控制和测度变换等工具，证明\(h_\infty(X + Y) \leq h_\infty(Z + W)\)，并得到整数值单调对数凹随机变量类似结果。
We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $R^{n}$. We show that there exists $s_{0}\in(0,1),$ such that for all $0<s<s_{0},$ and for all r.i. spaces $X(Q_{0}),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_{0}}^{\#}$ is the Strömberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows th…
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