Pub Date : 2020-12-28DOI: 10.2140/apde.2020.13.2375
Hart F. Smith
We demonstrate a parametrix construction, together with associated pseudodifferential operator calculus, for an operator of sum-of-squares type with semiclassical parameter. The form of operator we consider includes the generator of kinetic Brownian motion on the cosphere bundle of a Riemannian manifold. Regularity estimates in semiclassical Sobolev spaces are proven by establishing mapping properties for the parametrix.
{"title":"Parametrix for a semiclassical subelliptic operator","authors":"Hart F. Smith","doi":"10.2140/apde.2020.13.2375","DOIUrl":"https://doi.org/10.2140/apde.2020.13.2375","url":null,"abstract":"We demonstrate a parametrix construction, together with associated pseudodifferential operator calculus, for an operator of sum-of-squares type with semiclassical parameter. The form of operator we consider includes the generator of kinetic Brownian motion on the cosphere bundle of a Riemannian manifold. Regularity estimates in semiclassical Sobolev spaces are proven by establishing mapping properties for the parametrix.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44794717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-27DOI: 10.2140/apde.2020.13.1269
Sun-Sig Byun, Jehan Oh
We consider a wide class of functionals with the property of changing their growth and ellipticity properties according to the modulating coefficients in the framework of Musielak–Orlicz spaces. In particular, we provide an optimal condition on the modulating coefficient to establish the Holder regularity and Harnack inequality for quasiminimizers of the generalized double phase functional with (G,H)-growth for two Young functions G and H.
{"title":"Regularity results for generalized double phase functionals","authors":"Sun-Sig Byun, Jehan Oh","doi":"10.2140/apde.2020.13.1269","DOIUrl":"https://doi.org/10.2140/apde.2020.13.1269","url":null,"abstract":"We consider a wide class of functionals with the property of changing their growth and ellipticity properties according to the modulating coefficients in the framework of Musielak–Orlicz spaces. In particular, we provide an optimal condition on the modulating coefficient to establish the Holder regularity and Harnack inequality for quasiminimizers of the generalized double phase functional with (G,H)-growth for two Young functions G and H.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/apde.2020.13.1269","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48862701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-15DOI: 10.2140/apde.2020.13.865
D. Bucur, I. Fragalà, A. Giacomini
{"title":"Local minimality results for the Mumford–Shah\u0000functional via monotonicity","authors":"D. Bucur, I. Fragalà, A. Giacomini","doi":"10.2140/apde.2020.13.865","DOIUrl":"https://doi.org/10.2140/apde.2020.13.865","url":null,"abstract":"","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/apde.2020.13.865","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48744556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-30DOI: 10.2140/apde.2021.14.1725
Gelu Popescu
We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${bf D_n^m}(H)$, ${bf n,m}in {bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $$ kappa_{bf m}(z,w):=prod_{i=1}^k frac{1}{(1-bar z_i w_i)^{m_i}},qquad z,win {bf D}^k, $$ where $m_igeq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.
{"title":"Brown–Halmos characterization of multi-Toeplitz\u0000operators associated with noncommutative polyhyperballs","authors":"Gelu Popescu","doi":"10.2140/apde.2021.14.1725","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1725","url":null,"abstract":"We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${bf D_n^m}(H)$, ${bf n,m}in {bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel \u0000$$ \u0000kappa_{bf m}(z,w):=prod_{i=1}^k frac{1}{(1-bar z_i w_i)^{m_i}},qquad z,win {bf D}^k, \u0000$$ \u0000where $m_igeq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44696304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2140/APDE.2020.13.1173
Yunfeng Zhang
Every connected compact Lie group is the quotient by a finite central subgroup of a product of circles and compact simply connected simple Lie groups. Let each of the circles be equipped with a constant metric, and each of the simple groups be equipped a metric that is a constant multiple of the negative of the Cartan-Killing form, under the requirement that the periods of the Schrodinger flow on each component are rational multiples of each other. This metric on the covering group is pushed down to a metric on the original group, which is called a rational metric. This paper establishes scaling invariant Strichartz estimates for the Schrodinger flow on compact Lie groups equipped with rational metrics. The highlights of this paper include a Weyl differencing technique for rational lattices, the different decompositions of the Schrodinger kernel that accommodate different positions of the variable inside the maximal torus relative to the Weyl chamber walls, and the application of the BGG-Demazure operators to the estimate of the difference between characters.
{"title":"Strichartz estimates for the Schrödinger flow on compact Lie groups","authors":"Yunfeng Zhang","doi":"10.2140/APDE.2020.13.1173","DOIUrl":"https://doi.org/10.2140/APDE.2020.13.1173","url":null,"abstract":"Every connected compact Lie group is the quotient by a finite central subgroup of a product of circles and compact simply connected simple Lie groups. Let each of the circles be equipped with a constant metric, and each of the simple groups be equipped a metric that is a constant multiple of the negative of the Cartan-Killing form, under the requirement that the periods of the Schrodinger flow on each component are rational multiples of each other. This metric on the covering group is pushed down to a metric on the original group, which is called a rational metric. This paper establishes scaling invariant Strichartz estimates for the Schrodinger flow on compact Lie groups equipped with rational metrics. The highlights of this paper include a Weyl differencing technique for rational lattices, the different decompositions of the Schrodinger kernel that accommodate different positions of the variable inside the maximal torus relative to the Weyl chamber walls, and the application of the BGG-Demazure operators to the estimate of the difference between characters.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/APDE.2020.13.1173","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68005282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-26DOI: 10.2140/apde.2021.14.2307
Kewei Zhang
Relying on the recent work of Liu-Szekelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kahler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kahler-Ricci flow on Fano manifolds.
{"title":"Some refinements of the partial C0\u0000estimate","authors":"Kewei Zhang","doi":"10.2140/apde.2021.14.2307","DOIUrl":"https://doi.org/10.2140/apde.2021.14.2307","url":null,"abstract":"Relying on the recent work of Liu-Szekelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kahler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kahler-Ricci flow on Fano manifolds.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2019-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48289627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-23DOI: 10.2140/apde.2021.14.1925
Q. Ding
Let $Sigma$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$mathrm{acute{e}}$ inequality. We show that any positive minimal graphic function on $Sigma$ is a constant.
{"title":"Liouville-type theorems for minimal graphs over manifolds","authors":"Q. Ding","doi":"10.2140/apde.2021.14.1925","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1925","url":null,"abstract":"Let $Sigma$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$mathrm{acute{e}}$ inequality. We show that any positive minimal graphic function on $Sigma$ is a constant.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2019-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42526934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-12DOI: 10.2140/apde.2021.14.2497
A. Gloria, S. Neukamm, F. Otto
This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $beta=2$; we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration --- the results would fail if correlation decay were encoded in terms of an $alpha$-mixing condition. Among other ensembles popular in modelling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. The optimal growth of the corrector $phi$ is derived from bounding the size of spatial averages $F=int gcdotnablaphi $ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $frac{partial F}{partial a}$ of $F$ w.~r.~t.~the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-nablacdot anabla$. The treatment allows for non-symmetric $a$ and for systems like linear elasticity.
{"title":"Quantitative estimates in stochastic homogenization for correlated coefficient fields","authors":"A. Gloria, S. Neukamm, F. Otto","doi":"10.2140/apde.2021.14.2497","DOIUrl":"https://doi.org/10.2140/apde.2021.14.2497","url":null,"abstract":"This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $beta=2$; we capture the correct power of logarithms coming from these two sources of criticality. \u0000The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration --- the results would fail if correlation decay were encoded in terms of an $alpha$-mixing condition. Among other ensembles popular in modelling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. \u0000The optimal growth of the corrector $phi$ is derived from bounding the size of spatial averages $F=int gcdotnablaphi $ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $frac{partial F}{partial a}$ of $F$ w.~r.~t.~the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-nablacdot anabla$. The treatment allows for non-symmetric $a$ and for systems like linear elasticity.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2019-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44380515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-27DOI: 10.2140/apde.2021.14.1905
Raphael Clouatre, Michael Hartz
In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $mathcal H$, we characterize all multipliers of the weak product space $mathcal H odot mathcal H$. In particular, we show that if $mathcal H$ has the so-called column-row property, then the multipliers of $mathcal H$ and of $mathcal H odot mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $mathcal H odot mathcal H$, which enables the use of dilations of completely bounded maps.
{"title":"Multipliers and operator space structure of weak product spaces","authors":"Raphael Clouatre, Michael Hartz","doi":"10.2140/apde.2021.14.1905","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1905","url":null,"abstract":"In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $mathcal H$, we characterize all multipliers of the weak product space $mathcal H odot mathcal H$. In particular, we show that if $mathcal H$ has the so-called column-row property, then the multipliers of $mathcal H$ and of $mathcal H odot mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $mathcal H odot mathcal H$, which enables the use of dilations of completely bounded maps.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44254165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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