. We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [CZ20] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces.
{"title":"The reconstruction theorem in quasinormed spaces","authors":"Pavel Zorin-Kranich","doi":"10.4171/rmi/1355","DOIUrl":"https://doi.org/10.4171/rmi/1355","url":null,"abstract":". We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [CZ20] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44435635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given −∞ < λ < Λ < ∞, E ⊂ R finite, and f : E → [λ,Λ], how can we extend f to a C(R) function F such that λ ≤ F ≤ Λ and ‖F‖Cm(Rn) is within a constant multiple of the least possible, with the constant depending only on m and n? In this paper, we provide the solution to the problem for the case m = 2. Specifically, we construct a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and we provide an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E).
{"title":"$C^2$ interpolation with range restriction","authors":"C. Fefferman, Fushuai Jiang, Garving K. Luli","doi":"10.4171/rmi/1353","DOIUrl":"https://doi.org/10.4171/rmi/1353","url":null,"abstract":"Given −∞ < λ < Λ < ∞, E ⊂ R finite, and f : E → [λ,Λ], how can we extend f to a C(R) function F such that λ ≤ F ≤ Λ and ‖F‖Cm(Rn) is within a constant multiple of the least possible, with the constant depending only on m and n? In this paper, we provide the solution to the problem for the case m = 2. Specifically, we construct a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and we provide an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E).","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48172847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let u be a harmonic function in a C-Dini domain, such that u vanishes on an open set of the boundary. We show that near every point in the open set, u can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that u has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.
{"title":"Expansion of harmonic functions near the boundary of Dini domains","authors":"C. Kenig, Zihui Zhao","doi":"10.4171/rmi/1380","DOIUrl":"https://doi.org/10.4171/rmi/1380","url":null,"abstract":"Let u be a harmonic function in a C-Dini domain, such that u vanishes on an open set of the boundary. We show that near every point in the open set, u can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that u has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70906735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the sub-Riemannian Heisenberg group H. In this paper, we answer the following question: given a compact set K ⊆ R and a continuous map f : K → H, when is there a horizontal C curve F : R → H such that F |K = f? Whitney originally answered this question for real valued mappings [35], and Fefferman provided a complete answer for real valued functions defined on subsets of R [12]. We also prove a finiteness principle for C √ ω horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman [5].
{"title":"Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group","authors":"Scott Zimmerman","doi":"10.4171/rmi/1339","DOIUrl":"https://doi.org/10.4171/rmi/1339","url":null,"abstract":"Consider the sub-Riemannian Heisenberg group H. In this paper, we answer the following question: given a compact set K ⊆ R and a continuous map f : K → H, when is there a horizontal C curve F : R → H such that F |K = f? Whitney originally answered this question for real valued mappings [35], and Fefferman provided a complete answer for real valued functions defined on subsets of R [12]. We also prove a finiteness principle for C √ ω horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman [5].","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45303691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Auscher, Li Chen, J. M. Martell, Cruz Prisuelos-Arribas
We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let Lw be a degenerate elliptic operator with degeneracy given by a fixed weight w ∈ A2(dx) in R, and consider the associated block second order degenerate elliptic problem in the upper-half space R + . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces L(vdw) where v is a Muckenhoupt weight with respect to the underlying natural weighted space (R, wdx). We recover earlier results in the non-degenerate case (when w ≡ 1, and with or without weight v). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L(dx)-solvability of the regularity problem for the block operator Lαu(x, t) = −|x|divx ( |x| A(x)∇xu(x, t) ) − ∂ t u(x, t) for any complex-valued uniformly elliptic matrix A and for all −ǫ < α < 2n n+2 , where ǫ depends just on the dimension and the ellipticity constants of A.
{"title":"The regularity problem for degenerate elliptic operators in weighted spaces","authors":"P. Auscher, Li Chen, J. M. Martell, Cruz Prisuelos-Arribas","doi":"10.4171/rmi/1357","DOIUrl":"https://doi.org/10.4171/rmi/1357","url":null,"abstract":"We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let Lw be a degenerate elliptic operator with degeneracy given by a fixed weight w ∈ A2(dx) in R, and consider the associated block second order degenerate elliptic problem in the upper-half space R + . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces L(vdw) where v is a Muckenhoupt weight with respect to the underlying natural weighted space (R, wdx). We recover earlier results in the non-degenerate case (when w ≡ 1, and with or without weight v). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L(dx)-solvability of the regularity problem for the block operator Lαu(x, t) = −|x|divx ( |x| A(x)∇xu(x, t) ) − ∂ t u(x, t) for any complex-valued uniformly elliptic matrix A and for all −ǫ < α < 2n n+2 , where ǫ depends just on the dimension and the ellipticity constants of A.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49109657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given any non-compact real simple Lie group Go of inner type and even dimension, we prove the existence of an invariant complex structure J and a Hermitian balanced metric with vanishing Chern scalar curvature on Go and on any compact quotient M = Go/Γ, with Γ a cocompact lattice. We also prove that (M, J) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.
{"title":"Real semisimple Lie groups and balanced metrics","authors":"Federico Giusti, F. Podestà","doi":"10.4171/rmi/1391","DOIUrl":"https://doi.org/10.4171/rmi/1391","url":null,"abstract":"Given any non-compact real simple Lie group Go of inner type and even dimension, we prove the existence of an invariant complex structure J and a Hermitian balanced metric with vanishing Chern scalar curvature on Go and on any compact quotient M = Go/Γ, with Γ a cocompact lattice. We also prove that (M, J) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46394112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1962, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich’s result holds for Hölder vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until 2014, when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin’s result to the case of a smooth domain with several internal sources and sinks.
{"title":"Uniqueness of Yudovich’s solutions to the 2D incompressible Euler equation despite the presence of sources and sinks","authors":"Florent Noisette, F. Sueur","doi":"10.4171/rmi/1370","DOIUrl":"https://doi.org/10.4171/rmi/1370","url":null,"abstract":"In 1962, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich’s result holds for Hölder vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until 2014, when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin’s result to the case of a smooth domain with several internal sources and sinks.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43830268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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