Volume 11, Issue 1

Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 ( M ; N ) {C}^{0}\left(M;\hspace{0.33em}N) of continuous maps f : M → N f:M\to N . We review two classical topologies, the “ strong ” and the “ weak ” topology. We propose a definition of “ mild topology ” that is coarser than the “ strong ” and finer than the “ weak ” topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M → N f:M\to N , and affine actions when N = R n N={{\mathbb{R}}}^{n} . To define the “ mild topology ” we use “ separation functions ;” these “separation functions” are somewhat similar to the usual “ distance function d ( x , y ) d\left(x,y) ” in metric spaces ( M , d ) \left(M,d) , but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define “ set separation functions ” that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions : one is the aforementioned case of the mild topology on C 0 ( M ; N ) {C}^{0}\left(M;\hspace{0.33em}N) . Other examples are the Sorgenfrey line and the topology of topological manifolds.

For any real number κ \kappa and any integer n ≥ 4 n\ge 4 , the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov ( CAT(κ)-spaces: construction and concentration , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT ( κ ) {\rm{CAT}}\left(\kappa ) space. For geodesic metric spaces, satisfying the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition is equivalent to being CAT ( κ ) {\rm{CAT}}\left(\kappa ) . In this article, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n ≥ 5 n\ge 5 .

This article analyses two schemes: Mann-type and viscosity-type proximal point algorithms. Using these schemes, we establish Δ-convergence and strong convergence theorems for finding a common solution of monotone vector field inclusion problems, a minimization problem, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. We apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also include a numerical example to show the applicability of the schemes. Our findings corroborate some recent findings.

The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

We study the families of measures on Carnot groups that have vanishing p p -module, which we call M p {M}_{p} -exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be M p {M}_{p} -exceptional for p ≥ 1 p\ge 1 . We describe a wide class of M p {M}_{p} -exceptional intrinsic Lipschitz surfaces for p ∈ ( 0 , ∞ ) p\in \left(0,\infty ) .

The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group , we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of analyst’s traveling salesman theorem , which characterizes the subsets of rectifiable curves in R 2 {{\mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem , Invent. Math. 102 (1990), no. 1, 1–15), in R n {{\mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves in R n {{\bf{R}}}^{n} , J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called Jones’ β \beta - numbers . In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n {{\mathbb{R}}}^{n} that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space .

Let ( X , d , μ ) \left({\mathcal{X}},d,\mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space M p u ( μ ) {M}_{p}^{u}\left(\mu ) , where 1 ≤ p < ∞ 1\le p\lt \infty and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) u\left(x,r):{\mathcal{X}}\times \left(0,\infty )\to \left(0,\infty ) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions u 1 , u 2 {u}_{1},{u}_{2} , and u u belong to W τ {{\mathbb{W}}}_{\tau } with τ ∈ ( 0 , 2 ) \tau \in \left(0,2) , we prove that the bilinear θ \theta -type generalized fractional integral T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , where u 1 u 2 = u {u}_{1}{u}_{2}=u , α ∈ ( 0 , 1 ) \alpha \in \left(0,1) , and 1 q = 1 p 1 + 1 p 2 − 2 α \frac{1}{q}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha with p 1 , p 2 ∈ ( 1 , 1 α ) {p}_{1},{p}_{2}\in \left(1,\frac{1}{\alpha }) , and also show that the T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) , where 1 = 1 p 1 + 1 p 2 − 2 α 1=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha . Meanwhile, we prove that the commutator T ˜ θ , α , b 1 , b 2 {\widetilde{T}}_{\theta ,\alpha ,{b}_{1},{b}_{2}} formed by b 1 , b 2 ∈ RBMO ˜ ( μ ) {b}_{1},{b}_{2}\in \widetilde{{\rm{RBMO}}}\left(\mu ) and T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , and it is also bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) .

The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial self-similarity Φ : X → X \Phi :X\to X satisfies the equality d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of ( X , d ) \left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP {\mathcal{ {\mathcal I} P}} -spaces is fully described.

We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analog of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption, the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some structural properties of extremal subsets in GCBA spaces.