Extensions of Weak-Type Multipliers

Extension of Mikhlin Multiplier Theorem to Fractional Derivatives and Stable Processes

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

Relations between bilinear multipliers on $ \mathbb R^n, \mathbb{T}^n$ and $\mathbb{Z}^n$

In this paper we prove the bilinear analogue of de Leeuw's result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.

弱(1,1)舒尔乘子与一维Hörmander–Mikhlin–Schur定理

Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem

该文刻画了舒尔乘子的有界性，证明了一维Hörmander–Mikhlin–Schur弱(1,1)估计，解决了Parcet ICM 2026综述中的对应一维问题，所用方法涉及二进网格、二叉树分解与Cuculescu投影。

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator…

Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.

A Homomorphism Theorem for Bilinear Multipliers

In this paper we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of K. de Leeuw's theorem for bilinear multipliers of strong and weak type. We also obtain necessary conditions on bilinear multipliers on non-compact LCA groups, yielding boundedness for the corresponding operators on products of rearrangement invariant spaces. Our investigations extend some existing results in Euclidean spaces to the framework of general LCA groups, and y…

A multiplier theorem for sub-Laplacians with drift on Lie groups

We prove a general multiplier theorem for symmetric left-invariant sub-Laplacians with drift on non-compact Lie groups. This considerably improves and extends a result by Hebisch, Mauceri, and Meda. Applications include groups of polynomial growth and solvable extensions of stratified groups.

径向加权Besov空间中通过弱*序列循环性实现的循环性

Cyclicity via weak$^\ast$ sequential cyclicity in Radially weighted Besov spaces

该研究针对径向加权Besov空间中循环性与弱*序列循环性的差异，给出无零点有界全纯函数循环性的对数条件，提出弱*序列循环乘子的比较原理，为非完全Pick空间下的循环性现象提供新认知。

Some remarks on $M_d$-multipliers and approximation properties

We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbolΛ(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the con…

Joint spectral multipliers for mixed systems of operators

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{…

Rigged modules II: multipliers and duality

In a previous paper with Kashyap we generalized the theory of $W^*$-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. The present paper and its contemporaneous predecessor comprise the sequel which we promised at that time would be forthcoming. We give many new results about rigged and weak*-rigged modules and their tensor products, such as an Eilenberg-Watts type theorem.

Weak amenability for dynamical systems

Using the recently developed notion of a Herz--Schur multiplier of a C*-dynamical system we introduce weak amenability of C*- and W*-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra to crossed products and study its multipliers.

Restriction results for multilinear multipliers on weighted settings

We obtain restriction results of K. de Leeuw's type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some application of our development. In particular we obtain periodic weighted results for Coifman-Meyer, Hörmander and Hörmander-Mihlin type multilinear multipliers.

Multiplier theorems via martingale transforms

We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp $L^p$ bounds for second order Riesz transforms by a liming argument.

An extension of compact operators by compact operators with no nontrivial multipliers

We construct an essential extension of $\mathcal K(\ell_2({\mathfrak{c}}))$ by $\mathcal K(\ell_2)$, where ${\mathfrak{c}}$ denotes the cardinality of continuum, i.e., a $C^*$-algebra $\mathcal A\subseteq \mathcal B(\ell_2)$ satisfying the short exact sequence $$0\rightarrow \mathcal K(\ell_2)\xrightarrowι \mathcal A \rightarrow\mathcal K(\ell_2({\mathfrak{c}}))\rightarrow 0,$$ where $ι[\mathcal K(\ell_2)]$ is an essential ideal of $\mathcal A$ such that the algebra of multipliers $\mathcal M(\mathcal A)$ of $\mathcal A$ is equal to the unitization of $\mathcal A$. In particular $\mathcal A$ i…

Heisenberg群及相关群的可解扩张的尖锐乘子定理

A sharp multiplier theorem for solvable extensions of Heisenberg and related groups

本文针对Heisenberg型群及Métivier群与阿贝尔群直积的可解扩张，将谱乘子定理的光滑性条件从齐次维数阈值改进为拓扑维数阈值，达到尖锐最优结果。

Homogeneous Fourier multipliers of Marcinkiewicz type

This 1995 paper contains a sharp version of the classical Marcinkiewicz multiplier theorem for the class of homogeneous Fourier multipliers in two dimensions; here a one-dimensional Marcinkiewicz condition is sufficient. Examples are given to show that the straightforward extension of this statement to higher dimensions does not hold. We provide appropriate versions in higher dimensions which rely on a Fefferman-Stein weighted norm inequality for a relevant Córdoba type square function. The proof of this inequality is given by an early version of an induction on scales argument. For $p<1$ we p…

H-distributions - an extension of the H-measures

We use the continuity of Fourier multiplier operators on $L^p$ to introduce the $H$-distributions --- an extension of $H$-measures in the $L^p$ framework. We apply the $H$-distributions to obtain an $L^p$ version of the localisation principle, and reprove the $L^p$---$L^q$ variant of the Murat--Tartar div-curl lemma.

Semigroups of Herz-Schur Multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the sec…

可测Schur乘子的马尔可夫半群的膨胀

Dilations of markovian semigroups of measurable Schur multipliers

本文利用概率工具证明了自伴保单位完全正可测Schur乘子半群可被更大von Neumann代数上的Markov *-自同构群膨胀，并推导出生成子的泛函演算有界性及插值结果，回答了相关公开问题。

Laplace 变换类型的 Hankel 乘子

Hankel Multipliers of Laplace Transform Type

本文证明了 $(0,1)^n$ 上 Laplace 变换类型的 Hankel 乘子具有弱型 (1,1) 性质，并分析了 Bessel 算子虚幂的 Lp 有界性。

Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

Let $\Box_b$ be the Kohn Laplacian acting on $(0,j)$-forms on the unit sphere in $\mathbb{C}^n$. In a recent paper of Casarino, Cowling, Sikora and the author, a spectral multiplier theorem of Mihlin--Hörmander type for $\Box_b$ is proved in the case $0<j<n-1$. Here we prove an analogous theorem in the exceptional cases $j=0$ and $j=n-1$, including a weak type $(1,1)$ endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstract multivariate multiplier theorem for systems of commuting operators.

Even Fourier multipliers and martingale transforms in infinite dimensions

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in $\mathcal L(L^p(\mathbb R^d; X))$ for $1<p<\infty$ and for a UMD Banach space $X$ in terms of the range of the corresponding symbol. For example, if the range contains $a_1,\ldots,a_N \in \mathbb C$, then the norm of the multiplier exceeds $\|a_1R_1^2 + \cdots + a_NR_N^2\|_{\mathcal L(L^p(\mathbb R^N; X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Bañuelos-Bogdan type multipliers in terms of the range of the functions involved. The main tools th…

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak $L_1$ endpoint for matrix-valued CZ operat…