Darko Mitrovic, Nenad Antonic
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.
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On the measure of the boundary of a Hajłasz-Sobolev $(M^{1,p},M^{1,q})$-extension domain
该研究在Q加倍度量测度空间中，证明了满足特定参数条件的有界$(M^{1,p},M^{1,q})$-扩张域的边界测度为零。
In this article, for an optimal range of the smoothness parameter $s$ that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajłasz--Triebel--Lizorkin spaces $M^s_{p,q}$ and Hajłasz--Besov spaces $N^s_{p,q}$ in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results in this article, the…
We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.
Extensions of homogeneous distributions on deformations to the normal cone
本文基于Yves Meyer的弱齐次分布延拓技术，证明了法锥形变上齐次分布的延拓存在性，描述了所有此类延拓，并关联了Van Erp和Yuncken的相关工作。
We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove several implications between the strong BV extension property and extendability of two different non-equivalent versions of Sobolev $W^{1,1}$-spaces and show via examples that the remaining implications fail.
A theorem of Bayer and Teichmann implies that if a finite real multisequence β= β^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence of positive, recursively generated extensions M_(d+1),...,M_(2d-1) is sufficient for a measure. Examples illustrate that all of these extensions may be required to show that βhas a measure. We describe in detail a constructive procedure for determining whether such extensions exist. Under …
In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of algebraic structure (for usual convolution product $\ast$) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of $k$-distribution semigroups to extend previous concepts of distribution semigroups.
In this paper we introduce a nonlinear version of the notion of Anzellotti's pairing between divergence--measure vector fields and functions of bounded variation, motivated by possible applications to evolutionary quasilinear problems. As a consequence of our analysis, we prove a generalized Gauss--Green formula.
Kolmogorov extension theorem for non-probability measures on Cayley trees
本文研究Cayley树上概率与非概率测度向Borel域上σ-可加测度的扩张问题，阐明其在Gibbs测度理论中的基础地位。
Here using some methods of combinatorial set theory, particularly the ones related to the construction of independent families of sets and some modified version of the notion of small sets originally introduced by Riecan, Riecan and Neubrunn, we give abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby relating to nonseparable extension of Lebesgue measure in spaces with transformation groups
We show that a domain is an extension domain for a Hajłasz-Besov or for a Hajłasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case $0<p<1$. The necessity of the measure density condition is derived from embedding theorems; in the case of Hajłasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Hajłasz-Besov spaces ar…
We show that, whenever Gamma is a countable abelian group and Delta is a finitely-generated subgroup of Gamma, a generic measure-preserving action of Delta on a standard atomless probability space (X,mu) extends to a free measure-preserving action of Gamma on (X,mu). This extends a result of Ageev, corresponding to the case when Delta is infinite cyclic.
In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.
Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual
本文利用 Stiefel 流形参数化建立 $k$-平面变换的分布延拓，刻画值域各向同性条件，并构造使对偶反投影算子可逆的 Banach 空间，进而用于逆问题的新型正则化。
Closed BV-extension and $W^{1,1}$-extension sets
本文研究度量测度空间中闭集上$W^{1,1}$与$BV$函数可延拓性的关系，推导了延拓集的相关性质，并证明欧氏平面中紧有限连通$BV$-延拓集必为$W^{1,1}$-延拓集。
The recent theorem by D.Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions.
Class of extensions of real field and their topological properties
本文定义实数域的真类扩张并研究其拓扑性质，区分连通（对二元运算不封闭）和非连通（为线性序域）两类，提出该构造未来可用于构建能感知零勒贝格测度集的测度。
We prove that if $\mathcal{A}$ is a $σ$-complete Boolean algebra in a model $V$ of set theory and $\mathbb{P}\in V$ is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise convergent sequence of measures on $\mathcal{A}$ in a $\mathbb{P}$-generic extension $V[G]$ is weakly convergent, i.e. $\mathcal{A}$ has the Vitali--Hahn--Saks property in $V[G]$. This yields a consistent example of a whole class of infinite Boolean algebras with this property and of cardinality strictly smaller than the dominating number $\mathfrak{d}$. We also obtain a…
Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces
本文提出集对函数的Choquet扩张等概念，构建集合分数阶优化的等价表示，推导测度空间的谱界，建立全局扩张常数的单调性不等式，为多类结构的相关几何与组合量提供新的界。
Approximation by BV-extension sets via perimeter minimization in metric spaces
该研究在度量测度空间中证明有界区域可由内部闭BV-延拓集测度逼近，通过最小化周长与差集测度之和构造延拓集，并给出PI-空间中极小元开代表元非BV-延拓域且非局部John域的例子。
We study a fractional differentiation operator for functions on the conjugate space to an infinite extension of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. In particular, a representation in the form of a hypersingular integral operator is obtained. It is also shown that the corresponding diffusion measure is not absolutely continuous with respect to the Gaussian measure.
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