We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schr"odinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.
{"title":"On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles","authors":"A. Kulikov, Lucas H. Oliveira, João P. G. Ramos","doi":"10.4171/rmi/1426","DOIUrl":"https://doi.org/10.4171/rmi/1426","url":null,"abstract":"We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schr\"odinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41684696","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}
A BSTRACT . We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in R 3 with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2 , for holomorphic null immersions into C n with n ≥ 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any self-dual or anti-self-dual Einstein four-manifold.
{"title":"The Calabi–Yau problem for minimal surfaces with Cantor ends","authors":"F. Forstnerič","doi":"10.4171/rmi/1365","DOIUrl":"https://doi.org/10.4171/rmi/1365","url":null,"abstract":"A BSTRACT . We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in R 3 with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2 , for holomorphic null immersions into C n with n ≥ 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any self-dual or anti-self-dual Einstein four-manifold.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47494438","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}
A. We address Lp(μ) → Lp(λ) bounds for paraproducts in the Bloom setting. We introduce certain “sparse BMO” functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms – essentially, as Haar multipliers – as well as to obtain an equivalence of norms between sparse operators AS and compositions of paraproducts ΠaΠb.
{"title":"Paraproducts, Bloom BMO and sparse BMO functions","authors":"Valentia Fragkiadaki, Irina Holmes Fay","doi":"10.4171/rmi/1400","DOIUrl":"https://doi.org/10.4171/rmi/1400","url":null,"abstract":"A. We address Lp(μ) → Lp(λ) bounds for paraproducts in the Bloom setting. We introduce certain “sparse BMO” functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms – essentially, as Haar multipliers – as well as to obtain an equivalence of norms between sparse operators AS and compositions of paraproducts ΠaΠb.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48806029","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}
We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that is approximable from above and below by continuous functions. We also show that for each f ∈ L 1 ( ∂ Ω) , there is a least gradient function in Ω whose trace agrees with f at points of continuity of f , and so we obtain existence of solutions for boundary data which is continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan, who constructed an L 1 -function on the unit circle which has no least gradient solution in the unit disk in R 2 . Modifying the example of Spradlin and Tamasan, we show that the space of solvable L 1 -functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition.
{"title":"Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces","authors":"Joshua Kline","doi":"10.4171/rmi/1385","DOIUrl":"https://doi.org/10.4171/rmi/1385","url":null,"abstract":"We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that is approximable from above and below by continuous functions. We also show that for each f ∈ L 1 ( ∂ Ω) , there is a least gradient function in Ω whose trace agrees with f at points of continuity of f , and so we obtain existence of solutions for boundary data which is continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan, who constructed an L 1 -function on the unit circle which has no least gradient solution in the unit disk in R 2 . Modifying the example of Spradlin and Tamasan, we show that the space of solvable L 1 -functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44138346","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":"Minimal Mahler measures for generators of some fields","authors":"A. Dubickas","doi":"10.4171/rmi/1331","DOIUrl":"https://doi.org/10.4171/rmi/1331","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44997767","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}
On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that n-connected norbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions we prove slightly better bounds. We obtain sharp results up to dimension 5.
{"title":"How highly connected can an orbifold be?","authors":"Christian Lange, M. Radeschi","doi":"10.4171/rmi/1375","DOIUrl":"https://doi.org/10.4171/rmi/1375","url":null,"abstract":"On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that n-connected norbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions we prove slightly better bounds. We obtain sharp results up to dimension 5.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41814838","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}
We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ partial_t u - partial_xleft(|D|^{alpha}u - u^2 right)=0, $$ in the whole sub-critical range $alpha in]frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0
{"title":"Asymptotic $N$-soliton-like solutions of the fractional Korteweg–de Vries equation","authors":"Arnaud Eychenne","doi":"10.4171/rmi/1396","DOIUrl":"https://doi.org/10.4171/rmi/1396","url":null,"abstract":"We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ partial_t u - partial_xleft(|D|^{alpha}u - u^2 right)=0, $$ in the whole sub-critical range $alpha in]frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0<c_1<cdots<c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$ lim_{ttoinfty} | U(t,cdot) - sum_{j=1}^NQ_{c_j}(x-rho_j(t)) |_{H^{frac{alpha}2}}=0, $$ where $rho'_j(t) sim c_j$ as $t to +infty$. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state $Q_c$ and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Lin'eaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46861832","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 this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we provide a counterexample for higher dimensional tori, which relies on a computer assisted argument. Moreover we prove a theorem on the distribution symmetry of a certain class of trigonometric polynomials that might be of independent interest. eigenfuntions on Riemannian manifolds.
{"title":"Distribution symmetry of toral eigenfunctions","authors":"'Angel D. Mart'inez, Francisco Torres de Lizaur","doi":"10.4171/rmi/1324","DOIUrl":"https://doi.org/10.4171/rmi/1324","url":null,"abstract":". In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we provide a counterexample for higher dimensional tori, which relies on a computer assisted argument. Moreover we prove a theorem on the distribution symmetry of a certain class of trigonometric polynomials that might be of independent interest. eigenfuntions on Riemannian manifolds.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48137908","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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