Pub Date : 2022-01-01DOI: 10.4310/arkiv.2022.v60.n1.a3
Duc Do Tan
{"title":"Fundamental solutions of generalized non-local Schrodinger operators","authors":"Duc Do Tan","doi":"10.4310/arkiv.2022.v60.n1.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n1.a3","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393850","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 : 2021-12-14DOI: 10.4310/arkiv.2022.v60.n2.a5
E. F. Wold, G. Salvo
We present a construction of a proper holomorphic embedding $fcolon Bbb P^1setminus Chookrightarrow Bbb C^2$, where C is a Cantor set obtained by removing smaller and smaller vertical and horizontal strips from a square of side 2, allowing to realize it to have Lebesgue measure arbitrarily close to 4.
{"title":"Proper holomorphic embeddings of complements of large Cantor sets in $mathbb{C}^2$","authors":"E. F. Wold, G. Salvo","doi":"10.4310/arkiv.2022.v60.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n2.a5","url":null,"abstract":"We present a construction of a proper holomorphic embedding $fcolon Bbb P^1setminus Chookrightarrow Bbb C^2$, where C is a Cantor set obtained by removing smaller and smaller vertical and horizontal strips from a square of side 2, allowing to realize it to have Lebesgue measure arbitrarily close to 4.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394126","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 : 2021-11-05DOI: 10.4310/arkiv.2023.v61.n1.a4
T. Rajala, Ugo Bindini
We study conditions on closed sets $C,F subset mathbb{R}$ making the product $C times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.
{"title":"Removability of product sets for Sobolev functions in the plane","authors":"T. Rajala, Ugo Bindini","doi":"10.4310/arkiv.2023.v61.n1.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a4","url":null,"abstract":"We study conditions on closed sets $C,F subset mathbb{R}$ making the product $C times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49612160","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 : 2021-11-04DOI: 10.4310/arkiv.2023.v61.n1.a8
N. L. Lundstrom, J. Singh
Suppose that $p in (1,infty]$, $nu in [1/2,infty)$, $mathcal{S}_nu = left{ (x_1,x_2) in mathbb{R}^2 setminus {(0, 0)}: |phi|0$ and $omega_p(x)$ be the $p$-harmonic measure of $partial B(0,R) cap mathcal{S}_nu$ at $x$ with respect to $B(0, R)cap mathcal{S}_nu$. We prove that there exists a constant $C$ such that begin{align*} C^{-1}left(frac{|x|}{R}right)^{k(nu,p)} , leq omega_p(x) , leq C left(frac{|x|}{R}right)^{k(nu,p)} end{align*} whenever $xin B(0,R) cap mathcal{S}_{2nu}$ and where the exponent $k(nu,p)$ is given explicitly as a function of $nu$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel"of theorem for $p$-subharmonic functions in the unbounded sector $mathcal{S}_nu$. Moreover, if $p = infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $mathbb{R}^n$. Finally, when $nu in (1/2, infty)$ and $pin (1,infty)$ we prove uniqueness (modulo normalization) of positive $p$-harmonic functions in $mathcal{S}_nu$ vanishing on $partialmathcal{S}_nu$.
假设 $p in (1,infty]$， $nu in [1/2,infty)$， $mathcal{S}_nu = left{ (x_1,x_2) in mathbb{R}^2 setminus {(0, 0)}: |phi|0$ 和 $omega_p(x)$ 做一个 $p$的谐波测度 $partial B(0,R) cap mathcal{S}_nu$ 在 $x$ 关于 $B(0, R)cap mathcal{S}_nu$。我们证明存在一个常数 $C$ 这样 begin{align*} C^{-1}left(frac{|x|}{R}right)^{k(nu,p)} , leq omega_p(x) , leq C left(frac{|x|}{R}right)^{k(nu,p)} end{align*} 无论何时 $xin B(0,R) cap mathcal{S}_{2nu}$ 指数在哪里 $k(nu,p)$ 是明确给出的函数 $nu$ 和 $p$。利用这一估计，我们得出了当地的增长估计 $p$-sub- and $p$-局部可被扇形逼近的平面区域上的超调和函数，例如，我们在区域具有向内或向外尖尖的边界附近得出收敛速度的界限。使用 $p$-谐波测度，我们也得到了一个尖锐的Phragmen-Lindelöf定理 $p$-无界扇区中的次谐波函数 $mathcal{S}_nu$。此外，如果 $p = infty$ 然后，上述估计从二维扇形的设置扩展到中锥的设置 $mathbb{R}^n$。最后，当 $nu in (1/2, infty)$ 和 $pin (1,infty)$ 证明了正的唯一性(模归一化) $p$-调和函数 $mathcal{S}_nu$ 消失在 $partialmathcal{S}_nu$．
{"title":"Estimates of $p$-harmonic functions in planar sectors","authors":"N. L. Lundstrom, J. Singh","doi":"10.4310/arkiv.2023.v61.n1.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a8","url":null,"abstract":"Suppose that $p in (1,infty]$, $nu in [1/2,infty)$, $mathcal{S}_nu = left{ (x_1,x_2) in mathbb{R}^2 setminus {(0, 0)}: |phi|<frac{pi}{2nu}right}$, where $phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and $omega_p(x)$ be the $p$-harmonic measure of $partial B(0,R) cap mathcal{S}_nu$ at $x$ with respect to $B(0, R)cap mathcal{S}_nu$. We prove that there exists a constant $C$ such that begin{align*} C^{-1}left(frac{|x|}{R}right)^{k(nu,p)} , leq omega_p(x) , leq C left(frac{|x|}{R}right)^{k(nu,p)} end{align*} whenever $xin B(0,R) cap mathcal{S}_{2nu}$ and where the exponent $k(nu,p)$ is given explicitly as a function of $nu$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel\"of theorem for $p$-subharmonic functions in the unbounded sector $mathcal{S}_nu$. Moreover, if $p = infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $mathbb{R}^n$. Finally, when $nu in (1/2, infty)$ and $pin (1,infty)$ we prove uniqueness (modulo normalization) of positive $p$-harmonic functions in $mathcal{S}_nu$ vanishing on $partialmathcal{S}_nu$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47414445","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}
Let $mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $mathscr{C}$ is locally finite, then $mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.
{"title":"On the existence of Auslander–Reiten $n$-exangles in $n$-exangulated categories","authors":"Jian He, Jiangsheng Hu, Dongdong Zhang, Panyue Zhou","doi":"10.4310/arkiv.2022.v60.n2.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n2.a8","url":null,"abstract":"Let $mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $mathscr{C}$ is locally finite, then $mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394412","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 : 2021-09-23DOI: 10.4310/arkiv.2023.v61.n1.a3
N. Berestycki, Vittoria Silvestri
We consider a constrained version of the HL$(0)$ Hastings--Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $sqrt{tlog n}$, uniformly in $tin [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL$(0)$.
{"title":"Explosive growth for a constrained Hastings–Levitov aggregation model","authors":"N. Berestycki, Vittoria Silvestri","doi":"10.4310/arkiv.2023.v61.n1.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a3","url":null,"abstract":"We consider a constrained version of the HL$(0)$ Hastings--Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $sqrt{tlog n}$, uniformly in $tin [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL$(0)$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394530","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 : 2021-08-31DOI: 10.4310/arkiv.2022.v60.n2.a1
N. Andruskiewitsch, Franccois Dumas, H'ector Mart'in Pena Pollastri
We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in $texttt{arXiv:2002.02514}$ assuming that the characteristic is zero.
{"title":"On the double of the Jordan plane","authors":"N. Andruskiewitsch, Franccois Dumas, H'ector Mart'in Pena Pollastri","doi":"10.4310/arkiv.2022.v60.n2.a1","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n2.a1","url":null,"abstract":"We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in $texttt{arXiv:2002.02514}$ assuming that the characteristic is zero.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393474","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 : 2021-08-25DOI: 10.4310/ARKIV.2020.v58.n1.a12
Nguyen Thi Thanh Tam, Hoang Le Truong
In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.
{"title":"A note on Chern coefficients and Cohen–Macaulay rings","authors":"Nguyen Thi Thanh Tam, Hoang Le Truong","doi":"10.4310/ARKIV.2020.v58.n1.a12","DOIUrl":"https://doi.org/10.4310/ARKIV.2020.v58.n1.a12","url":null,"abstract":"In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"58 1","pages":"197-212"},"PeriodicalIF":0.7,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393800","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 : 2021-07-13DOI: 10.4310/arkiv.2022.v60.n2.a3
Germán Benitez, Luis Enrique Ram'irez
Relation Gelfand–Tsetlin gln-modules were introduced in [FRZ19], and are determined by some special directed graphs and Gelfand-Tsetlin characters. In this work we constructed polyhedra associated with the class of relation modules, which includes as a particular case, any classical Gelfand– Tsetlin polytope. Following the ideas presented in [LM04] we give a characterization of d-faces of the associated polyhedra in terms of a matrix related to the corresponding graph.
{"title":"Faces of polyhedra associated with relation modules","authors":"Germán Benitez, Luis Enrique Ram'irez","doi":"10.4310/arkiv.2022.v60.n2.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n2.a3","url":null,"abstract":"Relation Gelfand–Tsetlin gln-modules were introduced in [FRZ19], and are determined by some special directed graphs and Gelfand-Tsetlin characters. In this work we constructed polyhedra associated with the class of relation modules, which includes as a particular case, any classical Gelfand– Tsetlin polytope. Following the ideas presented in [LM04] we give a characterization of d-faces of the associated polyhedra in terms of a matrix related to the corresponding graph.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48776826","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 : 2021-07-09DOI: 10.4310/arkiv.2022.v60.n2.a10
S. Thangavelu
We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.
{"title":"Beurling’s theorem on the Heisenberg group","authors":"S. Thangavelu","doi":"10.4310/arkiv.2022.v60.n2.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2022.v60.n2.a10","url":null,"abstract":"We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393517","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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