Pub Date : 2024-08-07DOI: 10.4310/ajm.2024.v28.n1.a2
Xinbao Lu, Ge Xiong, Jiawei Xiong
The existence and uniqueness of solutions to the $L_p$ Minkowski problem for $mathfrak{p}$-capacity for $p gt 1$ and $mathfrak{p} geqslant n$ are proved. For this task, the estimation of $mathfrak{p}$−capacitary measure controlled below by the surface area measure is achieved. This work is a sequel to the results $href{https://doi.org/10.4310/jdg/1606964418}{[45]}$ for $p gt 1$ and $1 lt mathfrak{p} lt n$.
{"title":"The $L_p$ Minkowski problem for the electrostatic $mathfrak{p}$-capacity for $p gt 1$ and $mathfrak{p} geqslant n^ast$","authors":"Xinbao Lu, Ge Xiong, Jiawei Xiong","doi":"10.4310/ajm.2024.v28.n1.a2","DOIUrl":"https://doi.org/10.4310/ajm.2024.v28.n1.a2","url":null,"abstract":"The existence and uniqueness of solutions to the $L_p$ Minkowski problem for $mathfrak{p}$-capacity for $p gt 1$ and $mathfrak{p} geqslant n$ are proved. For this task, the estimation of $mathfrak{p}$−capacitary measure controlled below by the surface area measure is achieved. This work is a sequel to the results $href{https://doi.org/10.4310/jdg/1606964418}{[45]}$ for $p gt 1$ and $1 lt mathfrak{p} lt n$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.4310/ajm.2024.v28.n1.a4
Tohru Ozawa, Durvudkhan Suragan
In this paper, we obtain remainder term representation formulae for the higher-order Steklov inequality for vector fields which imply short and direct proofs of the sharp (classical) Steklov inequalities. The obtained results directly imply sharp Steklov type inequalities for some vector fields satisfying Hörmander’s condition, for example. We also give representation formulae for the $L^{2^m}$-Friedrichs inequalities for vector fields.
{"title":"Representation formulae for the higher-order Steklov and $L^{2^m}$-Friedrichs inequalities","authors":"Tohru Ozawa, Durvudkhan Suragan","doi":"10.4310/ajm.2024.v28.n1.a4","DOIUrl":"https://doi.org/10.4310/ajm.2024.v28.n1.a4","url":null,"abstract":"In this paper, we obtain remainder term representation formulae for the higher-order Steklov inequality for vector fields which imply short and direct proofs of the sharp (classical) Steklov inequalities. The obtained results directly imply sharp Steklov type inequalities for some vector fields satisfying Hörmander’s condition, for example. We also give representation formulae for the $L^{2^m}$-Friedrichs inequalities for vector fields.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.4310/ajm.2024.v28.n1.a1
Guorui Ma, Yang Wang, Stephen S.-T. Yau, Huaiqing Zuo
We introduce the Hodge moduli algebras and Hodge moduli sequence associated with an isolated hypersurface singularity. These are new subtle invariants of singularities. We propose several characterization conjectures by using of these invariants. We investigate structural properties and numerical invariants of Hodge ideals naturally associated with isolated hypersurface singularities.In particular, we establish that the analytic isomorphisms class of an isolated two dimensional rational hypersurface singularities is determined by the Hodge moduli algebras and Hodge moduli sequence. As a result, we prove that Hodge moduli algebra together with the geometric genus give complete characterization of such singularities. In the proof, we concretely compute the Hodge ideals and the associated Hodge moduli algebras of these singularities.
{"title":"Hodge moduli algebras and complete invariants of singularities","authors":"Guorui Ma, Yang Wang, Stephen S.-T. Yau, Huaiqing Zuo","doi":"10.4310/ajm.2024.v28.n1.a1","DOIUrl":"https://doi.org/10.4310/ajm.2024.v28.n1.a1","url":null,"abstract":"We introduce the Hodge moduli algebras and Hodge moduli sequence associated with an isolated hypersurface singularity. These are new subtle invariants of singularities. We propose several characterization conjectures by using of these invariants. We investigate structural properties and numerical invariants of Hodge ideals naturally associated with isolated hypersurface singularities.In particular, we establish that the analytic isomorphisms class of an isolated two dimensional rational hypersurface singularities is determined by the Hodge moduli algebras and Hodge moduli sequence. As a result, we prove that Hodge moduli algebra together with the geometric genus give complete characterization of such singularities. In the proof, we concretely compute the Hodge ideals and the associated Hodge moduli algebras of these singularities.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.4310/ajm.2024.v28.n1.a3
Lin Feng Wang
Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation[Delta_p u - lambda_p {lvert u rvert}^{p-2} u]on $G$ with the $mathrm{CD}^psi_p (m,-K)$ condition, where $p geq 2$, $m gt 0$, $K geq 0$, and $Delta_p$ denotes the $ptextrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.
{"title":"Elliptic gradient estimate for the $p$−Laplace operator on the graph","authors":"Lin Feng Wang","doi":"10.4310/ajm.2024.v28.n1.a3","DOIUrl":"https://doi.org/10.4310/ajm.2024.v28.n1.a3","url":null,"abstract":"Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation[Delta_p u - lambda_p {lvert u rvert}^{p-2} u]on $G$ with the $mathrm{CD}^psi_p (m,-K)$ condition, where $p geq 2$, $m gt 0$, $K geq 0$, and $Delta_p$ denotes the $ptextrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.4310/ajm.2024.v28.n1.a5
Dong Uk Lee
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz [Kot90], for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport [LR87] of deriving the Kottwitz’s formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands–Rapoport conjecture), but replaces their Galois gerb theoretic arguments by more standard group-theoretic ones, using Kisin’s geometric work [Kis17]. We also prove a generalization of Honda–Tate theorem in the context of Shimura varieties and fix an error in the Kisin’s work. We do not assume that the derived group is simply connected.
{"title":"Lefschetz number formula for Shimura varieties of Hodge type","authors":"Dong Uk Lee","doi":"10.4310/ajm.2024.v28.n1.a5","DOIUrl":"https://doi.org/10.4310/ajm.2024.v28.n1.a5","url":null,"abstract":"For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz [Kot90], for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport [LR87] of deriving the Kottwitz’s formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands–Rapoport conjecture), but replaces their Galois gerb theoretic arguments by more standard group-theoretic ones, using Kisin’s geometric work [Kis17]. We also prove a generalization of Honda–Tate theorem in the context of Shimura varieties and fix an error in the Kisin’s work. We do not assume that the derived group is simply connected.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-12DOI: 10.4310/ajm.2023.v27.n5.a2
Uta Freiberg, Stefan Kohl
We consider fractals generated by a probabilistic iterated function scheme with open set condition and we interpret the probabilities as weights for every part of the fractal. In the homogeneous case, where the weights are not taken into account, Denker and Sato introduced in 2001 a Markov chain on the word space and proved that the Martin boundary is homeomorphic to the fractal set. Our aim is to redefine the transition probability with respect to the weights and to calculate the Martin boundary. As we will see, the inhomogeneous Martin boundary coincides with the homogeneous case.
我们考虑的分形是由具有开放集条件的概率迭代函数方案生成的，我们将概率解释为分形每个部分的权重。在不考虑权重的同质情况下，Denker 和 Sato 于 2001 年引入了词空间上的马尔可夫链，并证明了马丁边界与分形集同构。我们的目的是重新定义与权重相关的过渡概率，并计算出马丁边界。我们将看到，非均质马丁边界与均质情况相吻合。
{"title":"Martin boundary theory on inhomogeneous fractals","authors":"Uta Freiberg, Stefan Kohl","doi":"10.4310/ajm.2023.v27.n5.a2","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n5.a2","url":null,"abstract":"We consider fractals generated by a probabilistic iterated function scheme with open set condition and we interpret the probabilities as weights for every part of the fractal. In the homogeneous case, where the weights are not taken into account, Denker and Sato introduced in 2001 a Markov chain on the word space and proved that the Martin boundary is homeomorphic to the fractal set. Our aim is to redefine the transition probability with respect to the weights and to calculate the Martin boundary. As we will see, the inhomogeneous Martin boundary coincides with the homogeneous case.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-12DOI: 10.4310/ajm.2023.v27.n5.a1
De-Jun Feng, Zhou Feng
Let $T_1, dotsc , T_m$ be a family of $d times d$ invertible real matrices with $rVert T_i rvert lt 1/2$ for $1 leq i leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $lbrace T_i x + a_i rbrace$ on $mathbb{R}^d$ has non-empty interior for almost all $(a_1 , dotsc , a_m) in mathbb{R}^{md}$.
让 $T_1, dotsc , T_m$ 是一个 $d times d$ 可逆实矩阵族，其中 $rVert T_i rvert lt 1/2$ for $1 leq i leq m$。我们在这些矩阵上提供了一些充分条件，使得迭代函数系统 $lbrace T_i x + a_i rbrace$ 在 $mathbb{R}^d$ 上产生的自链集对于几乎所有 $(a_1 , dotsc , a_m) in mathbb{R}^{md}$ 都具有非空的内部。
{"title":"Typical self-affine sets with non-empty interior","authors":"De-Jun Feng, Zhou Feng","doi":"10.4310/ajm.2023.v27.n5.a1","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n5.a1","url":null,"abstract":"Let $T_1, dotsc , T_m$ be a family of $d times d$ invertible real matrices with $rVert T_i rvert lt 1/2$ for $1 leq i leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $lbrace T_i x + a_i rbrace$ on $mathbb{R}^d$ has non-empty interior for almost all $(a_1 , dotsc , a_m) in mathbb{R}^{md}$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-12DOI: 10.4310/ajm.2023.v27.n5.a4
Jiaxin Hu, Zhenyu Yu
In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.
{"title":"The weak elliptic Harnack inequality revisited","authors":"Jiaxin Hu, Zhenyu Yu","doi":"10.4310/ajm.2023.v27.n5.a4","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n5.a4","url":null,"abstract":"In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-12DOI: 10.4310/ajm.2023.v27.n5.a3
Alexander Grigor’yan, Eryan Hu, Jiaxin Hu
We study the heat kernel of a regular symmetric Dirichlet form on a metric space with doubling measure, in particular, a connection between the properties of the jump measure and the long time behaviour of the heat kernel. Under appropriate optimal hypotheses, we obtain the Hölder regularity and lower estimates of the heat kernel.
{"title":"Off-diagonal lower estimates and Hölder regularity of the heat kernel","authors":"Alexander Grigor’yan, Eryan Hu, Jiaxin Hu","doi":"10.4310/ajm.2023.v27.n5.a3","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n5.a3","url":null,"abstract":"We study the heat kernel of a regular symmetric Dirichlet form on a metric space with doubling measure, in particular, a connection between the properties of the jump measure and the long time behaviour of the heat kernel. Under appropriate optimal hypotheses, we obtain the Hölder regularity and lower estimates of the heat kernel.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-27DOI: 10.4310/ajm.2022.v26.n6.a2
Alex Casella
Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realization in Cauchy–Riemann (CR) space. By introducing a new type of 3‑cell, we construct a different cell decomposition $mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is contained in the union of all edges of $mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.
{"title":"Branched Cauchy–Riemann structures on once-punctured torus bundles","authors":"Alex Casella","doi":"10.4310/ajm.2022.v26.n6.a2","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n6.a2","url":null,"abstract":"Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realization in Cauchy–Riemann (CR) space. By introducing a new type of 3‑cell, we construct a different cell decomposition $mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is contained in the union of all edges of $mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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