Abstract We consider different notions of boundary traces for functions in Sobolev spaces defined on regular trees and show that the almost everywhere existence of these traces is independent of the chosen definition of a trace.
{"title":"Trace Operators on Regular Trees","authors":"P. Koskela, K. N. Nguyen, Zhuang Wang","doi":"10.1515/agms-2020-0117","DOIUrl":"https://doi.org/10.1515/agms-2020-0117","url":null,"abstract":"Abstract We consider different notions of boundary traces for functions in Sobolev spaces defined on regular trees and show that the almost everywhere existence of these traces is independent of the chosen definition of a trace.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0117","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48105395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.
{"title":"Construction of Frames on the Heisenberg Groups","authors":"D. Chang, Yongsheng Han, Xinfeng Wu","doi":"10.1515/agms-2020-0118","DOIUrl":"https://doi.org/10.1515/agms-2020-0118","url":null,"abstract":"Abstract In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0118","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47928216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳) H_*^{p,q}left( mathcal{X} right) with the optimal range p∈(ωω+η,∞) p in left( {{omega over {omega + eta }},infty } right) and q ∈ (0, ∞]. When and p∈(ωω+η,1] p in ({omega over {omega + eta }},1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.
设(f, d， μ)是Coifman和Weiss意义上的齐次型空间，其上维数为ω。设η∈(0,1)是由Auscher和Hytönen构造的小波的光滑指数。本文通过极大函数，引入了Hardy-Lorentz空间H*p,q(∈)H＿*^{p,q}left（ mathcal{X} right)，最优范围p∈(ωω+η，∞)p in left（ {{omega over {omega + eta }}，infty } right)，且q∈(0，∞)。和p∈(ωω+η，1) p in （{omega over {omega + eta }}，1] q∈(0，∞)，分别用径向极大函数、非切向极大函数、原子、分子和各种Littlewood-Paley函数建立了它的实变量刻画。作者还得到了它的有限原子性质。作为应用，作者在Hardy-Lorentz空间上建立了一个实插值定理，并得到了Calderón-Zygmund算子的有界性，包括临界情况。本文的新颖之处在于充分利用了由其并矢参考点和并矢立方体表示的函数的几何性质，消除了μ的反向加倍假设，并且在满足反向加倍条件的情况下，对于q∈(0,1)，本文的结果也是新的。
{"title":"Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators","authors":"Xilin Zhou, Ziyi He, Dachun Yang","doi":"10.1515/agms-2020-0109","DOIUrl":"https://doi.org/10.1515/agms-2020-0109","url":null,"abstract":"Abstract Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳) H_*^{p,q}left( mathcal{X} right) with the optimal range p∈(ωω+η,∞) p in left( {{omega over {omega + eta }},infty } right) and q ∈ (0, ∞]. When and p∈(ωω+η,1] p in ({omega over {omega + eta }},1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49179473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.
{"title":"Ultradiversification of Diversities","authors":"Pouya Haghmaram, K. Nourouzi","doi":"10.1515/agms-2020-0100","DOIUrl":"https://doi.org/10.1515/agms-2020-0100","url":null,"abstract":"Abstract In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43759774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.
{"title":"An Intrinsic Characterization of Five Points in a CAT(0) Space","authors":"T. Toyoda","doi":"10.1515/agms-2020-0111","DOIUrl":"https://doi.org/10.1515/agms-2020-0111","url":null,"abstract":"Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49654740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ruming Gong, Ji Li, Elodie Pozzi, Manasa N. Vempati
Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space Lωp,k(X) L_omega ^{p,k}left( X right) with κ ∈ (0, 1) and ω ∈ Ap(X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ.
{"title":"Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type","authors":"Ruming Gong, Ji Li, Elodie Pozzi, Manasa N. Vempati","doi":"10.1515/agms-2020-0116","DOIUrl":"https://doi.org/10.1515/agms-2020-0116","url":null,"abstract":"Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space Lωp,k(X) L_omega ^{p,k}left( X right) with κ ∈ (0, 1) and ω ∈ Ap(X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0116","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43158160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Admissibility versus Ap-Conditions on Regular Trees","authors":"K. N. Nguyen, Zhuang Wang","doi":"10.1515/agms-2020-0110","DOIUrl":"https://doi.org/10.1515/agms-2020-0110","url":null,"abstract":"Abstract We show that the combination of doubling and (1, p)-Poincaré inequality is equivalent to a version of the Ap-condition on rooted K-ary trees.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0110","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43722350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this note is to prove a representation theorem for left–invariant functionals in Carnot groups. As a direct consequence, we can also provide a Г-convergence result for a smaller class of functionals.
{"title":"Integral Representation of Local Left–Invariant Functionals in Carnot Groups","authors":"Alberto Maione, E. Vecchi","doi":"10.1515/agms-2020-0001","DOIUrl":"https://doi.org/10.1515/agms-2020-0001","url":null,"abstract":"Abstract The aim of this note is to prove a representation theorem for left–invariant functionals in Carnot groups. As a direct consequence, we can also provide a Г-convergence result for a smaller class of functionals.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45275584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov’s book [4]. We obtain a complete answer for this question.
{"title":"Concentration of Product Spaces","authors":"Daisuke Kazukawa","doi":"10.1515/agms-2020-0129","DOIUrl":"https://doi.org/10.1515/agms-2020-0129","url":null,"abstract":"Abstract We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov’s book [4]. We obtain a complete answer for this question.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45798753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For any real number κ kappa and any integer n ≥ 4 nge 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 nge 5 .
摘要:对于任意实数κ kappa和任意整数n≥4 n ge 4, Gromov (CAT(κ)-spaces: construction and concentration, Zap，引入Cycl n (κ) {{rm{Cycl}}}＿n{}left (kappa)条件。午餐。Sem。彼得堡。奥德尔。斯特克洛夫博士。(POMI) 280 (2001)， (Geom)。i Topol. 7)， 100-140, 299-300)是度量空间允许等距嵌入到CAT (κ) {rm{CAT}}left (kappa)空间的必要条件。对于测地线度量空间，满足Cycl 4 (κ) {{rm{Cycl}}}＿4{}left (kappa)条件等价于CAT (κ) {rm{CAT}}left (kappa)。本文证明了满足Cycl 4 (κ) {{rm{Cycl}}}＿4{}left (kappa)条件的(可能是非测地的)度量空间Reshetnyak最大化定理的一个类比。由我们的结果可知，对于一般度量空间，Cycl 4 (κ) {{rm{Cycl}}}＿4{}left (kappa)条件意味着对于所有整数n≥5 n {{rm{Cycl}}}{}ge 5, Cycl n (κ) _nleft (kappa)条件。
{"title":"A non-geodesic analogue of Reshetnyak’s majorization theorem","authors":"T. Toyoda","doi":"10.1515/agms-2022-0151","DOIUrl":"https://doi.org/10.1515/agms-2022-0151","url":null,"abstract":"Abstract For any real number κ kappa and any integer n ≥ 4 nge 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 nge 5 .","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45856277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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