Pub Date : 2019-05-20DOI: 10.4310/arkiv.2020.v58.n1.a9
Danka Luvci'c, Enrico Pasqualetto, T. Rajala
We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painleve length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painleve length bound $kappa(E) lepi mathcal{H}^1(E)$ is sharp.
{"title":"Sharp estimate on the inner distance in planar domains","authors":"Danka Luvci'c, Enrico Pasqualetto, T. Rajala","doi":"10.4310/arkiv.2020.v58.n1.a9","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n1.a9","url":null,"abstract":"We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painleve length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painleve length bound $kappa(E) lepi mathcal{H}^1(E)$ is sharp.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47263192","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 : 2019-05-18DOI: 10.4310/arkiv.2020.v58.n2.a4
N. Iyudu, S. J. Sierra
Let $mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(mf g)$ is {em just infinite} in the sense that any proper quotient of $U(mf g)$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(mf g)$ by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.
{"title":"Enveloping algebras with just infinite Gelfand–Kirillov dimension","authors":"N. Iyudu, S. J. Sierra","doi":"10.4310/arkiv.2020.v58.n2.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a4","url":null,"abstract":"Let $mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(mf g)$ is {em just infinite} in the sense that any proper quotient of $U(mf g)$ has polynomial growth. \u0000This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. \u0000We also establish the corresponding results for quotients of the symmetric algebra $S(mf g)$ by proper Poisson ideals. \u0000In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393494","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 : 2019-04-01DOI: 10.4310/ARKIV.2019.V57.N1.A2
Judith Brinkschulte, C. Denson Hill
In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in $mathbb{C}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of $mathbb{C}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.
{"title":"Flexible and inflexible $CR$ submanifolds","authors":"Judith Brinkschulte, C. Denson Hill","doi":"10.4310/ARKIV.2019.V57.N1.A2","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A2","url":null,"abstract":"In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in $mathbb{C}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of $mathbb{C}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393173","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 : 2019-04-01DOI: 10.4310/ARKIV.2019.V57.N1.A4
E. Enochs, J. R. G. Rozas, L. Oyonarte, B. Torrecillas
Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects.
{"title":"Gorenstein flat precovers and Gorenstein injective preenvelopes in Grothendieck categories","authors":"E. Enochs, J. R. G. Rozas, L. Oyonarte, B. Torrecillas","doi":"10.4310/ARKIV.2019.V57.N1.A4","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A4","url":null,"abstract":"Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48557318","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 : 2019-03-13DOI: 10.4310/ARKIV.2020.v58.n1.a5
C. Ciliberto, T. Dedieu, C. Galati, A. L. Knutsen
We prove that the locus of Prym curves $(C,eta)$ of genus $g geq 5$ for which the Prym-canonical system $|omega_C(eta)|$ is base point free but the Prym--canonical map is not an embedding is irreducible and unirational of dimension $2g+1$.
{"title":"On the locus of Prym curves where the Prym-canonical map is not an embedding","authors":"C. Ciliberto, T. Dedieu, C. Galati, A. L. Knutsen","doi":"10.4310/ARKIV.2020.v58.n1.a5","DOIUrl":"https://doi.org/10.4310/ARKIV.2020.v58.n1.a5","url":null,"abstract":"We prove that the locus of Prym curves $(C,eta)$ of genus $g geq 5$ for which the Prym-canonical system $|omega_C(eta)|$ is base point free but the Prym--canonical map is not an embedding is irreducible and unirational of dimension $2g+1$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393865","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 : 2019-02-27DOI: 10.4310/arkiv.2019.v57.n2.a3
M. E. Cejas, Irene Drelichman, Javier C. Mart'inez-Perales
Universidad Nacional de La Plata, under grant 11/X805 Universidad de Buenos Aires, under grant 20020120100050BA Agencia Nacional de Promocion Cientifica y Tecnologica, under grant PICT 2014-1771
{"title":"Improved fractional Poincaré type inequalities on John domains","authors":"M. E. Cejas, Irene Drelichman, Javier C. Mart'inez-Perales","doi":"10.4310/arkiv.2019.v57.n2.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a3","url":null,"abstract":"Universidad Nacional de La Plata, under grant 11/X805 \u0000Universidad de Buenos Aires, under grant 20020120100050BA \u0000Agencia Nacional de Promocion Cientifica y Tecnologica, under grant PICT 2014-1771","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47342079","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 : 2019-02-12DOI: 10.4310/arkiv.2020.v58.n1.a4
E. Carlen, R. Frank, E. Lieb
We study $L^p$ inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$.
我们研究了$L^p$不等式，它使$N$函数在$L^p$中的和的三角不等式更加尖锐。
{"title":"Inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$","authors":"E. Carlen, R. Frank, E. Lieb","doi":"10.4310/arkiv.2020.v58.n1.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n1.a4","url":null,"abstract":"We study $L^p$ inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393814","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 : 2019-01-28DOI: 10.4310/arkiv.2020.v58.n1.a8
G. Karagulyan
We prove that $log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.
{"title":"On systems of non-overlapping Haar polynomials","authors":"G. Karagulyan","doi":"10.4310/arkiv.2020.v58.n1.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n1.a8","url":null,"abstract":"We prove that $log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393432","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 : 2019-01-24DOI: 10.4310/ARKIV.2021.V59.N1.A9
Zijian Zhou
We study the existence of Artin-Schreier curves with large $ a$-number. We also give bounds on the $a$-number of trigonal curves of genus $5$ in small characteristic.
{"title":"On the existence of curves with prescribed $a$-number","authors":"Zijian Zhou","doi":"10.4310/ARKIV.2021.V59.N1.A9","DOIUrl":"https://doi.org/10.4310/ARKIV.2021.V59.N1.A9","url":null,"abstract":"We study the existence of Artin-Schreier curves with large $ a$-number. We also give bounds on the $a$-number of trigonal curves of genus $5$ in small characteristic.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49393135","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 : 2019-01-08DOI: 10.4310/ARKIV.2021.V59.N1.A5
E. Donne, Sebastiano Golo
We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, infty)rightarrowmathtt{Aut}(G)$, $lambdamapstodelta_lambda$, so that $ d(delta_lambda x,delta_lambda y) = lambda d(x,y)$, for all $x,yin G$ and all $lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.
{"title":"Metric Lie groups admitting dilations","authors":"E. Donne, Sebastiano Golo","doi":"10.4310/ARKIV.2021.V59.N1.A5","DOIUrl":"https://doi.org/10.4310/ARKIV.2021.V59.N1.A5","url":null,"abstract":"We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, infty)rightarrowmathtt{Aut}(G)$, $lambdamapstodelta_lambda$, so that $ d(delta_lambda x,delta_lambda y) = lambda d(x,y)$, for all $x,yin G$ and all $lambda>0$. \u0000First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. \u0000Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. \u0000Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48854808","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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