Pub Date : 2024-05-17DOI: 10.2140/apde.2024.17.1261
Bingxiao Liu
{"title":"On full asymptotics of real analytic torsions for compact locally symmetric orbifolds","authors":"Bingxiao Liu","doi":"10.2140/apde.2024.17.1261","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1261","url":null,"abstract":"","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"65 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140965008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.2140/apde.2024.17.831
Jonathan Luk, Jared Speck
{"title":"The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy","authors":"Jonathan Luk, Jared Speck","doi":"10.2140/apde.2024.17.831","DOIUrl":"https://doi.org/10.2140/apde.2024.17.831","url":null,"abstract":"","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"30 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140661254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-12DOI: 10.2140/apde.2023.16.1205
P. Ganguly, Ramesh Manna, S. Thangavelu
We study an extension problem for the Ornstein–Uhlenbeck operator L = − (cid:49) + 2 x · ∇ + n , and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for L from which Hardy’s inequality for fractional powers of L is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new L p − L q estimates are obtained for the fractional powers of the Hermite operator.
{"title":"An extension problem, trace Hardy and Hardy’s\u0000inequalities for the Ornstein–Uhlenbeck operator","authors":"P. Ganguly, Ramesh Manna, S. Thangavelu","doi":"10.2140/apde.2023.16.1205","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1205","url":null,"abstract":"We study an extension problem for the Ornstein–Uhlenbeck operator L = − (cid:49) + 2 x · ∇ + n , and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for L from which Hardy’s inequality for fractional powers of L is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new L p − L q estimates are obtained for the fractional powers of the Hermite operator.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"600 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116299689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-15DOI: 10.2140/apde.2023.16.1033
O. Safronov
We consider the Schrödinger operator perturbed by a random complex-valued potential. For this operator, we consider its eigenvalues situated in the unit disk. We obtain an estimate on the rate of accumulation of these eigenvalues to the positive half-line.
{"title":"Eigenvalue bounds for Schrödinger operators\u0000with random complex potentials","authors":"O. Safronov","doi":"10.2140/apde.2023.16.1033","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1033","url":null,"abstract":"We consider the Schrödinger operator perturbed by a random complex-valued potential. For this operator, we consider its eigenvalues situated in the unit disk. We obtain an estimate on the rate of accumulation of these eigenvalues to the positive half-line.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133067966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-05DOI: 10.2140/apde.2022.15.1763
Jean-Paul Daniel, Peter Hornung
{"title":"Revisiting the C1,αh-principle for the\u0000Monge–Ampère equation","authors":"Jean-Paul Daniel, Peter Hornung","doi":"10.2140/apde.2022.15.1763","DOIUrl":"https://doi.org/10.2140/apde.2022.15.1763","url":null,"abstract":"","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131475607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-03DOI: 10.2140/apde.2022.15.891
K. Kellay, Thomas Normand, M. Tucsnak
. This paper gives a complete characterization of the reachable space for a system described by the 1 D heat equation with L 2 (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are n -times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.
{"title":"Sharp reachability results for the heat equation in one space dimension","authors":"K. Kellay, Thomas Normand, M. Tucsnak","doi":"10.2140/apde.2022.15.891","DOIUrl":"https://doi.org/10.2140/apde.2022.15.891","url":null,"abstract":". This paper gives a complete characterization of the reachable space for a system described by the 1 D heat equation with L 2 (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are n -times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131533509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-17DOI: 10.2140/apde.2023.16.1433
E. Kakariadis, E. Katsoulis, Marcelo Laca, Xin Li
We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible.
{"title":"Couniversality and controlled maps on product systems over right LCM semigroups","authors":"E. Kakariadis, E. Katsoulis, Marcelo Laca, Xin Li","doi":"10.2140/apde.2023.16.1433","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1433","url":null,"abstract":"We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125040352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-04DOI: 10.2140/apde.2023.16.279
Tyler Bongers, K. Taylor
Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.
{"title":"Transversal families of nonlinear projections and generalizations of Favard length","authors":"Tyler Bongers, K. Taylor","doi":"10.2140/apde.2023.16.279","DOIUrl":"https://doi.org/10.2140/apde.2023.16.279","url":null,"abstract":"Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125829931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-23DOI: 10.2140/apde.2023.16.1061
S. Bortz, J. Hoffman, S. Hofmann, J. García, K. Nystrom
Let $E subset mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$partial_tu- Delta u=0, quad text{in} quad mathbb R^{n+1}setminus E$$ satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for $E$ in terms of regular Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic version which is an improvement of the known results in that setting.
{"title":"Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets","authors":"S. Bortz, J. Hoffman, S. Hofmann, J. García, K. Nystrom","doi":"10.2140/apde.2023.16.1061","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1061","url":null,"abstract":"Let $E subset mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$partial_tu- Delta u=0, quad text{in} quad mathbb R^{n+1}setminus E$$ satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for $E$ in terms of regular Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic version which is an improvement of the known results in that setting.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134537918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-28DOI: 10.2140/apde.2023.16.1245
R. Killip, Maria Ntekoume, M. Visan
We consider the derivative nonlinear Schr"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is whether ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=int |q|^2<4pi$. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well-posed for initial data in $H^{1/6}$ under the same restriction on $M$. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.
{"title":"On the well-posedness problem for the derivative\u0000nonlinear Schrödinger equation","authors":"R. Killip, Maria Ntekoume, M. Visan","doi":"10.2140/apde.2023.16.1245","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1245","url":null,"abstract":"We consider the derivative nonlinear Schr\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is whether ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=int |q|^2<4pi$. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well-posed for initial data in $H^{1/6}$ under the same restriction on $M$. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.","PeriodicalId":203508,"journal":{"name":"Analysis & PDE","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124055544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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