Vector-Valued Singular Integrals on Locally Doubling Spaces
Mattia Calzi, Elena Rizzo
本文研究了在局部双重空间上Calderon-Zygmund算子和极大函数的向量值有界性问题。
我们证明了在连接的局部双重度量空间上，适合的Calderon-Zygmund算子和截断的Hardy-Littlewood极大函数的向量值有界性。
We prove vector-valued boundedness of (suitable) Calderon-Zygmund operators and of the (truncated) Hardy-Littlewood maximal function on a connected locally doubling metric measure space.
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We obtain a complete characterization of the weak-type $(1,1)$ for Haar shift operators in terms of generalized Haar systems adapted to a Borel measure $μ$ in the operator-valued setting. The main technical tool in our method is a noncommutative Calderón-Zygmund decomposition valid for arbitrary Borel measures.
Given a doubling measure $μ$ on $R^d$, it is a classical result of harmonic analysis that Calderon-Zygmund operators which are bounded in $L^2(μ)$ are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on $μ$ by a mild growth condition on $μ$. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderon-Zygmund decomposition adapted to the non doubling situation.
We prove a vector-valued non-homogeneous Tb theorem on certain quasimetric spaces equipped with what we call an upper doubling measure. Essentially, we merge recent techniques from the domain and range side of things, achieving a Tb theorem which is quite general with respect to both of them.
Let $(X,d,μ)$ denotes non-homogeneous metric measure space satisfying geometrically doubling and the upper doubling measure condition. In this paper, the boundedness in Lebesgue spaces for two kinds of commutators, which are iterated commutators and commutators in summation form, generated by multilinear strongly singular integral operators with RBMO$(μ)$ function on non-homogeneous metric measure spaces $(X,d,μ)$ is obtained.
We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered by David, Kenig, and Toro for codimension-1 under a crucial assumption of flatness, and later by Preiss, Tolsa, and Toro in higher codimension. However, their studies leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here answer the question (in an almost classical sense) for codimension-1 Hölder doubling measures in $\RR^4$.
Let $(X,d,μ)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions, which is called non-homogeneous metric measure space. In this paper, via a sharp maximal operator, the boundedness of commutators generated by multilinear singular integral with $RBMO(μ)$ function on non-homogeneous metric measure spaces in $m$-multiple Lebesgue spaces is obtained.
In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis.
The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``$T(1)\in BMO$'' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space …
Let $ϕ: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $μ= \text{det}\,D^2 ϕ$ be the Monge-Ampère measure generated by $ϕ.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n: ϕ(y)<ϕ(x)+\nabla ϕ(x)\cdot(y-x)+t\}$ denote the section. If $μ$ satisfies the doubling property, Caffarelli and Gutiérrez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calderón-Zygmund decomposition and a John-Nirenberg-type inequality associated with sections. Under a stronger uniform continuity condition on $μ$, they also (Amer. J. Math. 119:423--465, 1997) proved an invar…
We study complex interpolation scales obtained by vector valued amalgamation and the derivations they generate. We study their trivial and singular character and obtain examples showing that the hypotheses in the main theorems of [J.M.F. Castillo, V. Ferenczi and M. González, \emph{Singular exact sequences generated by complex interpolation}, Trans. Amer. Math. Soc. 369 (2017) 4671--4708] are not necessary.
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We establish necessary and sufficient conditions for the stability of the finite section method for operators belonging to a certain $C^*$-algebra of operators acting on the Hilbert space $l^2_H(\mathbb{Z})$ of $H$-valued sequences where $H$ is a given Hilbert space. Identifying $l^2_H(\mathbb{Z})$ with the $L^2_H$-space over the unit circle, the $C^*$-algebra in question is the one which contains all singular integral operators with flip and piecewise quasicontinous $\mathcal{L}(H)$-valued generating functions on the unit circle. The result is a generalization of an older result where the sam…
We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norm…
We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued Hörmander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rule…
We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, `vector-valued' means `taking values in a function lattice with the UMD (unconditional martingale differences) property'. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local…
In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the restriction of a Banach-valued Sobolev function defined on the whole space. We investigate the problem for Hajłasz- and Newton-Sobolev spaces, respectively. First, we show that Hajłasz-Sobolev extendability is independent of the target Banach spaces. We also show that every $c_0$-valued Newton-Sobolev extension set is a Banach-valued Newton-Sobolev extension set …
It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend …
This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $α$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by \begin{equation*} A_{n}x= \frac{1}{m(V^{n})} \int _{V^{n}}α_{g}x\,dm(g),\quad x\in L_{p}( \mathcal{M}),n\in \mathbb{N},1\leq p\leq \infty , \end{equation*} are of weak type $(1,1)$ and of strong type $(p,p)$ for …
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Universal Differentiability Sets in Laakso Space
本文在 Laakso 空间中构造一族相互奇异的加倍测度，使实值 Lipschitz 函数几乎处处可微，并由此得到测度为零的通用可微性集合，同时证明这些测度均支持 Poincaré 不等式。
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Maps of bounded variation from PI spaces to metric spaces
该研究针对满足加倍与庞加莱性质的PI空间上取值于度量空间的有界变差映射，通过简单映射逼近构建良态松弛理论，并给出含弱对偶测试计划概念的多种等价刻画。
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