Richard L. Baker
We present a general method of constructing an uncountable family of regular Borel measures on certain path spaces of Lipschitz functions having fixed Lipschitz constants. We use this method to give a definition of Lebesgue measure and uniform probability measre on these spaces.
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A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements
本文研究了Borel测度诱导Lipschitz自由空间元素的条件，证明了其与度量空间重量的关系，并指出在ZFC框架下无法证明存在特定测度的例外情况。
We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metr…
On the weak-fragmentability index of some Lipschitz-free spaces
本文构造了具有任意高弱可碎片化指标的 Lipschitz 自由空间，证明纯 1-不可修正空间不能是 Lipschitz 泛通用的，并得到不可数多个两两非同构的例子。
We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely func…
We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group taking values of its Fourier coefficients from this set has natural spectrum. For measures with Fourier coefficients tending to 0 we construct tho open set with this property.
In this paper we prove that the typical Lipschitz function has no directional derivative at any point of a Borel set $E$ if and only if $E$ is contained in a countable union of closed purely unrectifiable sets.
For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. We discuss possible characterizations of those compact spaces $K$ for which $P(K)$ is $\aleph_0$-monolithic. The main result states that under $\diamondsuit$ there exists a nonseparable Corson compact space $K$ such that $P(K)$ is $\aleph_0$-monolithic but $K$ supports a measure of uncountable type.
Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions
本文在有界Lipschitz函数空间中，通过McShane扩展算子构造规范极端函数子集，证明其在单位球极端点中弱星稠密，为Dudley范数的计算开辟了新途径。
We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_σ}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_σ} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$.
For a compact metric space $(K, ρ)$, the predual of $Lip(K, ρ)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of $M(K)$ by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between $Lip(K, ρ)$ and $(lip(K, ρ))_{**}$ [15]. In this work we also show that the pair $(lip(K, ρ), Lip(K, ρ))$ can be framed in the theory of o-O type structures introduced by K. M. Perfekt.
Linear structure of functions with maximal Clarke subdifferential
本文构造性地证明了具有极大 Clarke 次微分的 Lipschitz 函数集合包含不可数维线性子空间，并确立了其线性性与空间性。
A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm or Fortet-Mourier norm to a measure. Moreover, we prove three consequences of this result: the first is equivalence of concepts of equicontinuity in th…
Yet another proof of the density in energy of Lipschitz functions
该研究提出了Lipschitz函数在能量意义下稠密于带重心平面定义的度量Sobolev空间（及Newtonian-Sobolev空间）的新短证明，通过归约到Banach空间框架、结合凸分析工具与正规1-流叠加原理完成，同时得到光滑柱函数稠密性的新证。
We provide a Rademacher theorem for intrinsically Lipschitz functions $ϕ:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a Carnot group, where we require that $\mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.
We analyze the relationship between Borel measures and continuous linear functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast$ continuous functionals, i.e. members of the Lipschitz-free space $\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting a…
Diversity of Lipschitz-free spaces over countable complete discrete metric spaces
本文证明可数完备离散度量空间上的Lipschitz自由空间具有不可数多样性，且存在无法嵌入一致离散空间自由空间的例子，并揭示了dentability指数的非二进制行为。
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space…
Lipschitz-free spaces over properly metrisable spaces and approximation properties
该论文研究固有可度量化空间上的 Lipschitz-free 空间的逼近性质，证明具有度量逼近性质的度量集合稠密（零维时剩余），且不满足逼近性质的度量集合稠密，并得出其稠密性或空性的结论。
Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces
本文证明了度量Sobolev空间中Lipschitz函数子代数在能量意义下稠密的一般准则，并将其应用于Wasserstein Sobolev空间，证明了其Hilbert性质并给出了Wasserstein梯度的显式特征与微积分法则。
Lipschitz solvability of prescribed Jacobian and divergence for singular measures
本文研究了在奇异测度下，给定散度和雅可比行列式的利普希茨解的存在性，证明了在特定紧凑集上解的近似性及解的 Lipschitz 界限。
Characterization of AC and Sobolev curves via Lipschitz post-compositions
该研究在度量空间框架下，通过Lipschitz函数后复合的性质，分别刻画了p-绝对连续曲线与完备可分度量空间中Sobolev曲线的等价特征，丰富了几何测度与分析理论。
Non-absolute integrable function spaces on metric measure spaces
该研究在有限直径、有限正Borel测度的可分度量测度空间上引入Kuelbs-Steadman空间及Lipschitz型变体，提出欧氏情形下一致的HK-Sobolev空间，并分析Hardy-Littlewood极大算子在两类空间上的有界性。
We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $Ψ$-space and Isbell--Mrówka space. We construct an uncountable, almost disjoint family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded lin…
Let $Q$ denote the space of signed measures on the Borel $σ$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\|q\|=\sup|\intϕdq|$, where the supremum is taken over all Lipschitz with constant 1 functions whose module does not exceed unity. This normed space is incomplete provided $X$ is infinite and has at least one limit point. We call its completion the space of hypermeasures. Necessary and sufficient conditions for precompactness (=relative compactness) of a set of hypermeasures are found. They are similar to those of Prokhorov's and Fernique's theorems for measures.
Square Root Normal Fields for Lipschitz surfaces and the Wasserstein Fisher Rao metric
本文证明 Lipschitz 曲面上的 SRNF 距离等价于球面 Borel 测度的 WFR 度量，并刻画 SRNF 变换的性质，进而推广 WFR 度量至一般测度空间。
Extension of monotone operators and Lipschitz maps invariant for a group of isometries
本文研究对等距同构不变的单调算子与 Lipschitz 映射，证明其保持不变性的极大延拓存在，并推广到保测 $L^p$ 情形及律不变相关对象。
We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated …
Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative vanishing at infinity function; and it produces a signed deficient topological measure if we use a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In parti…
Consider the partition function S(ε) associated in theory of Renyi dimension to a finite Borel measure μon Euclidean d-space. This partion function S(ε) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width ε. We further Guerin's investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of μagainst an approximate identity of Gaussians. We prove a Lipschitz-type esimate on the partition function. This bound on the partition function leads to results regarding the computation of Renyi dimensi…
Borel complexity of sets of ideal limit points
本文研究不可数波兰空间中理想极限点集族的博雷尔复杂度，通过组合刻画揭示其与理想拓扑性质的关系，证明特定博雷尔类不可达并给出实例与开放问题。
A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps
本文在均匀局部加倍且满足弱局部Poincaré不等式的度量测度空间上，证明了Borel映射属于Ambrosio--Reshetnyak--Sobolev类的充要条件是其与任意1-Lipschitz函数的复合属于Sobolev空间，从而提供了该映射类的新刻画。
Metric Sobolev spaces I: equivalence of definitions
本文系统研究完备可分度量测度空间上一阶p-Sobolev空间的三类定义，证明其在无加倍或Poincaré假设、p∈[1,∞]时完全等价，补充了p=1情形的新结果。
The Ces`aro-like operator on some analytic function spaces
本文定义了基于有限正Borel测度矩的类Cesàro算子，研究其在Hardy空间、Morrey空间等多种解析函数子空间对上的作用特性。
On the normal trace space of extended divergence-measure fields
本文通过Arens-Eells空间刻画了扩展散度测度场在集合边界上的法向迹空间，并证明了在Lipschitz域等温和正则条件下迹算子的满射性，同时给出了Anzellotti配对的新逐点描述，并应用于散度测度场和无散度测度的延拓定理。
Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces
本文通过构造与函数差分相关的临界值ε(f)，建立了Lipschitz空间中函数到Sobolev、Besov-Triebel-Lizorkin等子空间距离的等价刻画，并推广到拟范数格框架。
Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces
本文利用分数热半群定义坏点集与容许集函数的临界指数，刻画 Lipschitz 空间中函数到多类函数子空间的距离。
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