Pub Date : 2020-07-27DOI: 10.1512/iumj.2021.70.9153
Serge Cantat, Julie D'eserti, Junyi Xie
This article is made of three independent parts, the three of them concerning the Cremona group in 2 variables.
本文由三个独立的部分组成，其中三个部分涉及2个变量中的Cremona群。
{"title":"Three chapters on Cremona groups","authors":"Serge Cantat, Julie D'eserti, Junyi Xie","doi":"10.1512/iumj.2021.70.9153","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.9153","url":null,"abstract":"This article is made of three independent parts, the three of them concerning the Cremona group in 2 variables.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49460300","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}
{"title":"On factorization of separating maps on noncommutative L^p-spaces","authors":"C. Merdy, S. Zadeh","doi":"10.1512/iumj.2022.71.9111","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9111","url":null,"abstract":"For any semifinite von Neumann algebra ${mathcal M}$ and any $1leq p<infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({mathcal M};S^1)$. We say that a bounded map $Tcolon L^p({mathcal M})to L^p({mathcal N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $Totimes I_{S^1}$ extends to a bounded (resp. contractive) map $Toverline{otimes} I_{S^1}$ from $ L^p({mathcal M};S^1)$ into $L^p({mathcal N};S^1)$. We show that any completely positive map is $S^1$-bounded, with $Vert Toverline{otimes} I_{S^1}Vert =Vert TVert$. We use the above as a tool to investigate the separating maps $Tcolon L^p({mathcal M})to L^p({mathcal N})$ which admit a direct Yeadon type factorization, that is, maps for which there exist a $w^*$-continuous $*$-homomorphism $Jcolon{mathcal M}to{mathcal N}$, a partial isometry $win{mathcal N}$ and a positive operator $B$ affiliated with ${mathcal N}$ such that $w^*w=J(1)=s(B)$, $B$ commutes with the range of $J$, and $T(x)=wBJ(x)$ for any $xin {mathcal M}cap L^p({mathcal M})$. Given a separating isometry $Tcolon L^p({mathcal M})to L^p({mathcal N})$, we show that $T$ is $S^1$-contractive if and only if it admits a direct Yeadon type factorization. We further show that if $pnot=2$, the above holds true if and only if $T$ is completely contractive.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43752560","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}
Pub Date : 2020-07-08DOI: 10.1512/iumj.2023.72.9202
Hongjie Dong, T. Phan
We study both divergence and non-divergence form parabolic and elliptic equations in the half space ${x_d>0}$ whose coefficients are the product of $x_d^alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $alpha in (-1, infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $alpha =0$
我们研究了半空间${x_d>0}$中发散和非发散形式的抛物型和椭圆型方程，其系数是$x_d^alpha$和一致非退化有界可测矩阵值函数的乘积，其中$alpha in (-1, infty)$。因此，系数在半空间边界附近是奇异的或简并的。对于具有正法边界条件或Neumann边界条件的方程，我们证明了当系数仅在$x_d$方向上可测，且在小柱体中其他方向上有较小的平均振荡时，在加权Sobolev空间和混合范数加权Sobolev空间中解的存在性、唯一性和正则性。即使在系数为常数的特殊情况下，我们的结果也是新的，并且它们被简化为经典结果 $alpha =0$
{"title":"On parabolic and elliptic equations with singular or degenerate coefficients","authors":"Hongjie Dong, T. Phan","doi":"10.1512/iumj.2023.72.9202","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9202","url":null,"abstract":"We study both divergence and non-divergence form parabolic and elliptic equations in the half space ${x_d>0}$ whose coefficients are the product of $x_d^alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $alpha in (-1, infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $alpha =0$","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45382984","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}
Pub Date : 2020-07-05DOI: 10.1512/iumj.2022.71.9150
F. Bernicot, Polona Durcik
We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.
{"title":"Boundedness of some multi-parameter fiber-wise multiplier operators","authors":"F. Bernicot, Polona Durcik","doi":"10.1512/iumj.2022.71.9150","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9150","url":null,"abstract":"We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42461847","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}
Pub Date : 2020-07-05DOI: 10.1512/iumj.2022.71.9174
P. Lions, B. Seeger, P. Souganidis
We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov, Holder, and variation norms, and criteria for the Hamiltonians are presented in terms of both regularity and structure. We also explore various properties of functions that are representable as the difference of convex functions, the largest space of Hamiltonians for which the equation is well-posed for all continuous paths. Finally, we discuss some open problems and conjectures.
{"title":"Interpolation results for pathwise Hamilton-Jacobi equations","authors":"P. Lions, B. Seeger, P. Souganidis","doi":"10.1512/iumj.2022.71.9174","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9174","url":null,"abstract":"We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov, Holder, and variation norms, and criteria for the Hamiltonians are presented in terms of both regularity and structure. We also explore various properties of functions that are representable as the difference of convex functions, the largest space of Hamiltonians for which the equation is well-posed for all continuous paths. Finally, we discuss some open problems and conjectures.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45806737","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}
Pub Date : 2020-06-24DOI: 10.1512/iumj.2022.71.9189
Wei Yan, Qiaoqiao Zhang, Jinqiao Duan, Meihua Yang
In this paper, we consider the pointwise convergence problem of free Ostrovsky equation with rough data and random data. Firstly, we show the almost everywhere pointwise convergence of free Ostrovsky equation in $H^{s}(mathbb{R})$ with $sgeq frac{1}{4}$ with rough data. Secondly, we present counterexamples showing that the maximal function estimate related to the free Ostrovsky equation can fail if $s
{"title":"Convergence problem of Ostrovsky equation with rough data and random data","authors":"Wei Yan, Qiaoqiao Zhang, Jinqiao Duan, Meihua Yang","doi":"10.1512/iumj.2022.71.9189","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9189","url":null,"abstract":"In this paper, we consider the pointwise convergence problem of free Ostrovsky equation with rough data and random data. Firstly, we show the almost everywhere pointwise convergence of free Ostrovsky equation in $H^{s}(mathbb{R})$ with $sgeq frac{1}{4}$ with rough data. Secondly, we present counterexamples showing that the maximal function estimate related to the free Ostrovsky equation can fail if $s<frac{1}{4}$. Finally, for every $xin mathbb{R}$, we show the almost surely pointwise convergence of free Ostrovsky equation in $L^{2}(mathbb{R})$ with random data. The main tools are the density theorem, high-low frequency idea, Wiener decomposition and Lemmas 2.1-2.6 as well as the probabilistic estimates of some random series which are just Lemmas 3.2-3.4 in this paper. The main difficulty is that zero is the singular point of the phase functions of free Ostrovsky equation. We use high-low frequency idea to conquer the difficulties.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45066675","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}
Pub Date : 2020-06-16DOI: 10.1512/iumj.2023.72.9316
Antonio Trusiani
On a compact K"ahler manifold $(X,omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) K"ahler-Einstein metrics with prescribed singularities. Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities. For Monge-Amp`ere equations of Fano type, we also prove an openness result when the singularities decrease. As an application, we deduce a strong stability result for (log-)K"ahler Einstein metrics on semi-K"ahler classes given as modifications of ${omega}$.
{"title":"Continuity method with movable singularities for classical complex Monge-Ampere equations","authors":"Antonio Trusiani","doi":"10.1512/iumj.2023.72.9316","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9316","url":null,"abstract":"On a compact K\"ahler manifold $(X,omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) K\"ahler-Einstein metrics with prescribed singularities. Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities. For Monge-Amp`ere equations of Fano type, we also prove an openness result when the singularities decrease. As an application, we deduce a strong stability result for (log-)K\"ahler Einstein metrics on semi-K\"ahler classes given as modifications of ${omega}$.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43286974","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}
{"title":"On the smoothness of $C^1$-contact maps in $C^infty$-rigid Carnot groups","authors":"Jona Lelmi","doi":"10.1512/iumj.2022.71.9205","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9205","url":null,"abstract":"We show that in any $C^infty$-rigid Carnot group in the sense of Ottazzi - Warhurst, $C^1$-contact maps are automatically smooth.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46814199","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}
Pub Date : 2020-05-31DOI: 10.1512/iumj.2023.72.9051
Baris Coskunuzer
We study the asymptotic Plateau problem in $BHH$ for area minimizing surfaces, and give a fairly complete solution for finite curves.
研究了$BHH$中面积最小化曲面的渐近平台问题，并给出了有限曲线的一个相当完整的解。
{"title":"Asymptotic Plateau Problem in H^2xR: Tall Curves","authors":"Baris Coskunuzer","doi":"10.1512/iumj.2023.72.9051","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9051","url":null,"abstract":"We study the asymptotic Plateau problem in $BHH$ for area minimizing surfaces, and give a fairly complete solution for finite curves.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765917","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}
Pub Date : 2020-05-25DOI: 10.1512/iumj.2022.71.9054
Yumeng Ou, K. Taylor
In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $Esubset mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the finite point configuration set depends on that of $E$. In particular, we show that if a planar set has dimension exceeding $frac{5}{4}$, then there exists a point $xin E$ so that for each integer $kgeq2$, the set of "$k$-chains" has positive Lebesgue measure. The second problem is a continuous analogue of the Erdős unit distance problem, which aims to determine the maximum number of times a point configuration with prescribed gaps can appear in $E$. For instance, given a triangle with prescribed sides and given a sufficiently regular planar set $E$ with Hausdorff dimension no less than $frac{7}{4}$, we show that the dimension of the set of vertices in $E$ forming said triangle does not exceed $3,{rm dim}_H (E)-3$. In addition to the Euclidean norm, we consider more general distances given by functions satisfying the so-called Phong-Stein rotational curvature condition. We also explore a number of examples to demonstrate the extent to which our results are sharp.
{"title":"Finite point configurations and the regular value theorem in a fractal setting","authors":"Yumeng Ou, K. Taylor","doi":"10.1512/iumj.2022.71.9054","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9054","url":null,"abstract":"In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $Esubset mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the finite point configuration set depends on that of $E$. In particular, we show that if a planar set has dimension exceeding $frac{5}{4}$, then there exists a point $xin E$ so that for each integer $kgeq2$, the set of \"$k$-chains\" has positive Lebesgue measure. \u0000The second problem is a continuous analogue of the Erdős unit distance problem, which aims to determine the maximum number of times a point configuration with prescribed gaps can appear in $E$. For instance, given a triangle with prescribed sides and given a sufficiently regular planar set $E$ with Hausdorff dimension no less than $frac{7}{4}$, we show that the dimension of the set of vertices in $E$ forming said triangle does not exceed $3,{rm dim}_H (E)-3$. In addition to the Euclidean norm, we consider more general distances given by functions satisfying the so-called Phong-Stein rotational curvature condition. We also explore a number of examples to demonstrate the extent to which our results are sharp.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46863202","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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