一个统一的Pietsch控制定理

Domination Problem for Narrow Orthogonally Additive Operators

The "Up-and-down" theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result of the operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove that for an order narrow positive abstract Uryson operator $T$ from a vector lattice $E$ to a Dedekind complete vector lattice $F$, every abstract Uryson operator $S:E\to F$, such that $0\leq S\leq T$ is also order narrow.

Long--Term Analysis of Positive Operator Semigroups via Asymptotic Domination

We consider positive operator semigroups on ordered Banach spac\-es and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups which are asymptotically dominated by $\mathcal{T}$. Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time…

On pointwise $\ell^r$-sparse domination in a space of homogeneous type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $\ell^1$-sum in the sparse operator is replaced by an $\ell^r$-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $A_2$-theorem for vector-valued Calderón--Zygmund operators in a space of homogeneous type, from which we deduce an…

多尺度稀疏控制

Multi-scale sparse domination

本文证明了一个适用于多尺度算子的双线性稀疏控制定理，并给出必要条件，涵盖傅里叶乘子、极大函数、平方函数和变差范数算子等广泛类别。

A Minimax Lemma and its Applications

We prove an easy version of the minimax theorem with no topological assumption. We deduce from it some domination criteria as well as an application to $p$-summing operators.

The Dodds-Fremlin type theorem for abstract Uryson operators

We continue the investigation of abstract Uryson operators in vector lattices. Using the recently proved Up-and-down theorem for order bounded, orthogonally additive operators, we consider the domination problem for AM-compact abstract Uryson operators. We obtain the Dodds-Fremlin type theorem and prove that for an AM- compact positive abstract Uryson operator T from a Banach lattice E to a order continuous Banach lattice F, every abstract Uryson operator S from E to F included between 0 and T is also AM-compact.

多线性分数次有界平均振荡算子理论 I：稀疏控制、稀疏 $T1$ 定理、非对角外推、定量加权估计——针对广义交换子

The multilinear fractional bounded mean oscillation operator theory I: sparse domination, sparse $T1$ theorem, off-diagonal extrapolation, quantitative weighted estimate -- for generalized commutators

本文在球基测度空间上引入并研究了多线性分数次有界平均振荡算子及其广义交换子，建立了稀疏控制与稀疏 $T1$ 定理，并在多线性分数次权下获得了多种加权不等式与非对角外推框架。