Pub Date : 2019-01-01DOI: 10.4310/arkiv.2019.v57.n2.a8
E. Korotyaev, J. S. Møller
We consider the discrete Laplacian Δ on the cubic lattice Zd, and deduce estimates on the group eitΔ and the resolvent (Δ−z)−1, weighted by q(Zd)-weights for suitable q 2. We apply the obtained results to discrete Schrödinger operators in dimension d 3 with potentials from p(Zd) with suitable p 1.
{"title":"Weighted estimates for the Laplacian on the cubic lattice","authors":"E. Korotyaev, J. S. Møller","doi":"10.4310/arkiv.2019.v57.n2.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a8","url":null,"abstract":"We consider the discrete Laplacian Δ on the cubic lattice Zd, and deduce estimates on the group eitΔ and the resolvent (Δ−z)−1, weighted by q(Zd)-weights for suitable q 2. We apply the obtained results to discrete Schrödinger operators in dimension d 3 with potentials from p(Zd) with suitable p 1.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393744","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 : 2018-12-29DOI: 10.4310/arkiv.2019.v57.n2.a10
J. Verdera
We study differentiability properties of a potential of the type $Kstar mu$, where $mu$ is a finite Radon measure in $mathbb{R}^N$ and the kernel $K$ satisfies $|nabla^j K(x)| le C, |x|^{-(N-1+j)}, quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallee Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity "norm" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calderon--Zygmund sense for $1le p < N/(N-1)$. We show that $Kstar mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $Kstar mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures.
我们研究了类型为$Kstar mu$的势的可微性，其中$mu$是$mathbb{R}^N$中的有限Radon测度，核$K$满足$|nabla^j K(x)| le C, |x|^{-(N-1+j)}, quad j=0,1,2.$。我们在容量意义上引入了可微性的概念，其中容量是与核相关的de la Vallee Poussin意义上的经典容量$|x|^{-(N-1)}.$我们要求当用以该点为中心的小半径球的归一化弱容量“范数”测量时，该点的一阶余量很小。这意味着弱$L^{N/(N-1)}$可微性，因此$1le p < N/(N-1)$在Calderon- Zygmund意义上具有$L^{p}$可微性。我们证明$Kstar mu$在容量意义上是a.e.可微的，从而加强了Ambrosio, Ponce和Rodiac最近的结果。我们还提出了作者刚才提到的一个定量定理的另一种证明，给出了$Kstar mu.$的点向Lipschitz估计。作为一个应用，我们研究了有限Radon测度的牛顿势的水平集。
{"title":"Capacitary differentiability of potentials of finite Radon measures","authors":"J. Verdera","doi":"10.4310/arkiv.2019.v57.n2.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a10","url":null,"abstract":"We study differentiability properties of a potential of the type $Kstar mu$, where $mu$ is a finite Radon measure in $mathbb{R}^N$ and the kernel $K$ satisfies $|nabla^j K(x)| le C, |x|^{-(N-1+j)}, quad j=0,1,2.$ \u0000We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallee Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity \"norm\" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calderon--Zygmund sense for $1le p < N/(N-1)$. We show that $Kstar mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $Kstar mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47693692","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 : 2018-12-04DOI: 10.4310/arkiv.2020.v58.n1.a10
P. Muller, K. Riegler
We provide variational estimates for Bloch functions on the unit ball of $mathbb{R}^d$ extending previous work on the Anderson conjecture for conformal maps on the unit disc.
{"title":"Radial variation of Bloch functions on the unit ball of $mathbb{R}^d$","authors":"P. Muller, K. Riegler","doi":"10.4310/arkiv.2020.v58.n1.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n1.a10","url":null,"abstract":"We provide variational estimates for Bloch functions on the unit ball of $mathbb{R}^d$ extending previous work on the Anderson conjecture for conformal maps on the unit disc.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393761","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 : 2018-11-26DOI: 10.4310/ARKIV.2019.v57.n2.a7
Pengjie Jiao
We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.
在区间有限颤振的有限表示范畴中刻画了不可分解的单射对象。
{"title":"Injective objects in the category of finitely presented representations of an interval finite quiver","authors":"Pengjie Jiao","doi":"10.4310/ARKIV.2019.v57.n2.a7","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.v57.n2.a7","url":null,"abstract":"We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393738","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 : 2018-10-01DOI: 10.4310/ARKIV.2018.V56.N2.A9
R. Kloosterman
In this paper, we describe all (2, 3)-torus structures of a highly symmetric 39-cuspidal degree 12 curve. A direct computer-aided determination of these torus structures seems to be out of reach. We use various quotients by automorphisms to find torus structures. We use a height pairing argument to show that there are no further structures.
{"title":"Determining all $(2, 3)$-torus structures of a symmetric plane curve","authors":"R. Kloosterman","doi":"10.4310/ARKIV.2018.V56.N2.A9","DOIUrl":"https://doi.org/10.4310/ARKIV.2018.V56.N2.A9","url":null,"abstract":"In this paper, we describe all (2, 3)-torus structures of a highly symmetric 39-cuspidal degree 12 curve. A direct computer-aided determination of these torus structures seems to be out of reach. We use various quotients by automorphisms to find torus structures. We use a height pairing argument to show that there are no further structures.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47991270","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 : 2018-10-01DOI: 10.4310/arkiv.2018.v56.n2.a5
Erwann Delay
On considere une variete riemannienne (M,g) non compacte, complete, a geometrie bornee et courbure de Ricci parallele. Nous montrons que certains operateurs "affines" en la courbure de Ricci sont localement inversibles, dans des espaces de Sobolev classiques, au voisinage de g.
{"title":"Inversion d'opérateurs de courbures au voisinage d'une métrique Ricci parallèle II: variétés non compactes à géométrie bornée.","authors":"Erwann Delay","doi":"10.4310/arkiv.2018.v56.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2018.v56.n2.a5","url":null,"abstract":"On considere une variete riemannienne (M,g) non compacte, complete, a geometrie bornee et courbure de Ricci parallele. \u0000Nous montrons que certains operateurs \"affines\" en la courbure de Ricci sont localement \u0000inversibles, dans des espaces de Sobolev classiques, au voisinage de g.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393107","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 : 2018-10-01DOI: 10.4310/ARKIV.2018.V56.N2.A13
P. Rabier
We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lj ,σj ∩LN,1 uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and L uloc is a “uniformly local” subspace of L loc larger than L p,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lj ,σj ∩Lquloc with q>N whenever pj
{"title":"Uniformly local spaces and refinements of the classical Sobolev embedding theorems","authors":"P. Rabier","doi":"10.4310/ARKIV.2018.V56.N2.A13","DOIUrl":"https://doi.org/10.4310/ARKIV.2018.V56.N2.A13","url":null,"abstract":"We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lj ,σj ∩LN,1 uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and L uloc is a “uniformly local” subspace of L loc larger than L p,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lj ,σj ∩Lquloc with q>N whenever pj<N and that, if so, the limit of f at infinity is uniform if the pj are suitably distributed. Only a few special cases have been considered in the literature, under much more restrictive assumptions that do not involve uniformly local spaces (pj=N and f vanishing at infinity, or ∂jf∈L∩L with p<N<q). Various similar results hold under integrability conditions on the higher order derivatives of f. All of them are applicable to g∗f with g∈L1 and f as above, or under weaker assumptions on f and stronger ones on g. When g is a Bessel kernel, the results are provably optimal in some cases.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393096","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 : 2018-09-26DOI: 10.4310/ARKIV.2019.V57.N1.A9
R. Laterveer
We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).
{"title":"Algebraic cycles and triple $K3$ burgers","authors":"R. Laterveer","doi":"10.4310/ARKIV.2019.V57.N1.A9","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A9","url":null,"abstract":"We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393593","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 : 2018-08-06DOI: 10.4310/arkiv.2019.v57.n2.a9
S. Miihkinen, J. Virtanen
The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1
用分段连续符号对Toeplitz算子的(本质)谱的几何描述是Hardy空间$H^p$与$1
{"title":"Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$","authors":"S. Miihkinen, J. Virtanen","doi":"10.4310/arkiv.2019.v57.n2.a9","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a9","url":null,"abstract":"The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1<p<infty$. In the Hardy space $H^1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C+H^infty$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H^1$. We answer this question in negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a jump discontinuity.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393750","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 : 2018-07-30DOI: 10.4310/ARKIV.2021.V59.N1.A6
Simon Muller
Let $R$ be a not necessarily commutative ring with $1.$ In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on $R$. We proceed by uniformly defining a coarsening relation $leq$ on the set $mathcal{Q}(R)$ of all quasi-orderings on $R.$ One of our main results states that $(mathcal{Q}(R),leq')$ is a rooted tree for some slight modification $leq'$ of $leq,$ i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that $(mathcal{Q}(R),leq')$ is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with $1.$ We conclude this paper by studying $mathcal{Q}(R)$ as a topological space.
{"title":"On the tree structure of orderings and valuations on rings","authors":"Simon Muller","doi":"10.4310/ARKIV.2021.V59.N1.A6","DOIUrl":"https://doi.org/10.4310/ARKIV.2021.V59.N1.A6","url":null,"abstract":"Let $R$ be a not necessarily commutative ring with $1.$ In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on $R$. We proceed by uniformly defining a coarsening relation $leq$ on the set $mathcal{Q}(R)$ of all quasi-orderings on $R.$ One of our main results states that $(mathcal{Q}(R),leq')$ is a rooted tree for some slight modification $leq'$ of $leq,$ i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that $(mathcal{Q}(R),leq')$ is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with $1.$ We conclude this paper by studying $mathcal{Q}(R)$ as a topological space.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393710","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：