In this note we show that the gradient estimate of the heat semigroup, or more precisely the H.-Q. Li inequality, is preserved under tensorization, some suitable group epimorphism, and central sum. We also establish the Riemannian counterpart of the H.-Q. Li inequality. As a byproduct, we provide a simpler proof of the fact that the constant in H.-Q. Li inequality is strictly larger than 1.
{"title":"On the H.-Q. Li inequality on step-two Carnot groups","authors":"Ye Zhang","doi":"10.5802/crmath.475","DOIUrl":"https://doi.org/10.5802/crmath.475","url":null,"abstract":"In this note we show that the gradient estimate of the heat semigroup, or more precisely the H.-Q. Li inequality, is preserved under tensorization, some suitable group epimorphism, and central sum. We also establish the Riemannian counterpart of the H.-Q. Li inequality. As a byproduct, we provide a simpler proof of the fact that the constant in H.-Q. Li inequality is strictly larger than 1.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267550","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}
We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order 0
建立了0阶
{"title":"The Caffarelli–Kohn–Nirenberg inequalities for radial functions","authors":"Arka Mallick, Hoai-Minh Nguyen","doi":"10.5802/crmath.503","DOIUrl":"https://doi.org/10.5802/crmath.503","url":null,"abstract":"We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order 0<s≤1. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case s=1. The known proofs used the Riesz potential and inequalities for fractional integrations. Our proof is new, elementary, and is based on one-dimensional case. Applications on compact embeddings are also mentioned.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220205","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}
We obtain improved fractional differentiability of solutions to the Banach-space valued Finsler γ-Laplacian defined on a σ-convex, τ-smooth Banach space. The operators we consider are non-linear and very degenerately elliptic. Our results are new already in the ℝ-valued setting.
{"title":"A note on improved differentiability for the Banach-space valued Finsler γ-Laplacian","authors":"Max Goering, Lukas Koch","doi":"10.5802/crmath.474","DOIUrl":"https://doi.org/10.5802/crmath.474","url":null,"abstract":"We obtain improved fractional differentiability of solutions to the Banach-space valued Finsler γ-Laplacian defined on a σ-convex, τ-smooth Banach space. The operators we consider are non-linear and very degenerately elliptic. Our results are new already in the ℝ-valued setting.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220269","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}
We exhibit counterexamples to F. Morel’s conjecture on the 𝔸 1 -invariance of the sheaves of connected components of 𝔸 1 -local spaces.
我们给出了F. Morel关于局部空间中连通分量束的t_1不变量的猜想的反例。
{"title":"Counterexamples to F. Morel’s conjecture on π 0 𝔸 1 ","authors":"Joseph Ayoub","doi":"10.5802/crmath.472","DOIUrl":"https://doi.org/10.5802/crmath.472","url":null,"abstract":"We exhibit counterexamples to F. Morel’s conjecture on the 𝔸 1 -invariance of the sheaves of connected components of 𝔸 1 -local spaces.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268396","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}
The L p convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for p≠2. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba regarding the L p convergence of the Bessel–Fourier series in the mixed norm space L rad p (L ang 2 ) on the disk for the range 4 3
平面域上拉普拉斯特征函数展开式的lp收敛性对于p≠2是未知的。在讨论了2环面上的经典傅立叶级数之后，我们转向圆盘，其特征函数作为三角函数和贝塞尔函数的乘积显式可计算。我们总结了Balodis和Córdoba关于盘上混合范数空间lrad p (lang 2)中贝塞尔-傅里叶级数在范围为43 ，rdrdt)范数收敛。
{"title":"Remarks on the L p convergence of Bessel–Fourier series on the disc","authors":"Ryan Luis Acosta Babb","doi":"10.5802/crmath.464","DOIUrl":"https://doi.org/10.5802/crmath.464","url":null,"abstract":"The L p convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for p≠2. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba regarding the L p convergence of the Bessel–Fourier series in the mixed norm space L rad p (L ang 2 ) on the disk for the range 4 3<p<4. We then describe how to modify their result to obtain L p (𝔻,rdrdt) norm convergence in the subspace L rad p (L ang q ) (1 p+1 q=1) for the restricted range 2≤p<4.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267748","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}
In this short article, we determine the bigness of the tangent bundle T X of the projective bundle X=ℙ C (E) associated to a vector bundle E on a smooth projective curve C.
在这篇短文中，我们确定了光滑投影曲线C上与向量束E相关联的投影束X= C (E)的切束tx的大小。
{"title":"Bigness of the tangent bundles of projective bundles over curves","authors":"Jeong-Seop Kim","doi":"10.5802/crmath.476","DOIUrl":"https://doi.org/10.5802/crmath.476","url":null,"abstract":"In this short article, we determine the bigness of the tangent bundle T X of the projective bundle X=ℙ C (E) associated to a vector bundle E on a smooth projective curve C.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268405","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}
where the component measure μ is the Lebesgue measure supported on [t,t+1] for t∈ℚ∖{-1 2} and δ 0 is the Dirac measure at 0. We prove that ρ is a spectral measure if and only if t∈1 2ℤ. In this case, L 2 (ρ) has a unique orthonormal basis of the form
{"title":"The spectrality of symmetric additive measures","authors":"Wen-Hui Ai, Zheng-Yi Lu, Ting Zhou","doi":"10.5802/crmath.435","DOIUrl":"https://doi.org/10.5802/crmath.435","url":null,"abstract":"where the component measure μ is the Lebesgue measure supported on [t,t+1] for t∈ℚ∖{-1 2} and δ 0 is the Dirac measure at 0. We prove that ρ is a spectral measure if and only if t∈1 2ℤ. In this case, L 2 (ρ) has a unique orthonormal basis of the form","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135526910","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}
We have developed a new simple iterative algorithm to determine entries of a normalized matrix given its marginal probabilities. Our method has been successfully used to obtain two different solutions by maximizing the entropy of a desired matrix and by minimizing its Kullback–Leibler divergence from the initial probability distribution. The latter is fully equivalent to the well-known RAS balancing algorithm. The presented method has been evaluated using a traffic matrix of the GÉANT pan-European network and randomly generated matrices of various sparsities. It turns out to be computationally faster than RAS. We have shown that our approach is suitable for efficient balancing both dense and sparse matrices.
{"title":"An Entropy Optimizing RAS-Equivalent Algorithm for Iterative Matrix Balancing","authors":"Edward Chlebus, Viswatej Kasapu","doi":"10.5802/crmath.398","DOIUrl":"https://doi.org/10.5802/crmath.398","url":null,"abstract":"We have developed a new simple iterative algorithm to determine entries of a normalized matrix given its marginal probabilities. Our method has been successfully used to obtain two different solutions by maximizing the entropy of a desired matrix and by minimizing its Kullback–Leibler divergence from the initial probability distribution. The latter is fully equivalent to the well-known RAS balancing algorithm. The presented method has been evaluated using a traffic matrix of the GÉANT pan-European network and randomly generated matrices of various sparsities. It turns out to be computationally faster than RAS. We have shown that our approach is suitable for efficient balancing both dense and sparse matrices.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135526912","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}
We show that a compact Lorentzian locally symmetric space is geodesically complete if the Lorentzian factor in the local de Rham–Wu decomposition is of Cahen–Wallach type or if the maximal flat factor is one-dimensional and time-like. Our proof uses a recent result by Mehidi and Zeghib and an earlier result by Romero and Sánchez.
{"title":"Completeness of certain compact Lorentzian locally symmetric spaces","authors":"Thomas Leistner, Thomas Munn","doi":"10.5802/crmath.449","DOIUrl":"https://doi.org/10.5802/crmath.449","url":null,"abstract":"We show that a compact Lorentzian locally symmetric space is geodesically complete if the Lorentzian factor in the local de Rham–Wu decomposition is of Cahen–Wallach type or if the maximal flat factor is one-dimensional and time-like. Our proof uses a recent result by Mehidi and Zeghib and an earlier result by Romero and Sánchez.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135526897","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}
We consider a p-Laplace problem in a strip with two-constant boundary Dirichlet conditions. We show that if the width of the strip is smaller than some d 0 ∈(0,+∞], then the problem admits a unique bounded solution, which is strictly monotone. Hence this unique solution is one-dimensional symmetric and belongs to the C 2 class. We also show that the problem has no bounded solution in the case that d 0 <+∞ and the width of the strip is larger than or equal to d 0 . An analogous rigidity result in the whole space was obtained recently by Esposito et al. [8]
{"title":"Uniqueness of bounded solutions to p-Laplace problems in strips","authors":"Phuong Le","doi":"10.5802/crmath.442","DOIUrl":"https://doi.org/10.5802/crmath.442","url":null,"abstract":"We consider a p-Laplace problem in a strip with two-constant boundary Dirichlet conditions. We show that if the width of the strip is smaller than some d 0 ∈(0,+∞], then the problem admits a unique bounded solution, which is strictly monotone. Hence this unique solution is one-dimensional symmetric and belongs to the C 2 class. We also show that the problem has no bounded solution in the case that d 0 <+∞ and the width of the strip is larger than or equal to d 0 . An analogous rigidity result in the whole space was obtained recently by Esposito et al. [8]","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135473935","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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