Pub Date : 2022-01-01DOI: 10.1215/00127094-2021-0042
Javier Parcet, Éric Ricard, Mikael de la Salle
We establish precise regularity conditions for Lp-boundedness of Fourier multipliers in the group algebra of SLn(R). Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing metric with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle’s rigidity theorem fits here. The proof includes a new relation between Fourier and Schur Lp-multipliers for nonamenable groups. In SLn(R), this holds in terms of Harish-Chandra’s almost L2 matrix coefficients. By transference, matters are reduced to a rather nontrivial RCp-inequality for SLn(R)-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in SLn(R). More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order ∼ | 1 2 − 1 p |(n − 1) for large enough n in terms of p. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of Lp-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our Hörmander-Mikhlin sufficient conditions in a given rank n and failing the rigidity conditions for ranks m >> n. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of K-biinvariant multipliers in the rank 1 groups SO(n, 1).
{"title":"Fourier multipliers in SLn(R)","authors":"Javier Parcet, Éric Ricard, Mikael de la Salle","doi":"10.1215/00127094-2021-0042","DOIUrl":"https://doi.org/10.1215/00127094-2021-0042","url":null,"abstract":"We establish precise regularity conditions for Lp-boundedness of Fourier multipliers in the group algebra of SLn(R). Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing metric with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle’s rigidity theorem fits here. The proof includes a new relation between Fourier and Schur Lp-multipliers for nonamenable groups. In SLn(R), this holds in terms of Harish-Chandra’s almost L2 matrix coefficients. By transference, matters are reduced to a rather nontrivial RCp-inequality for SLn(R)-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in SLn(R). More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order ∼ | 1 2 − 1 p |(n − 1) for large enough n in terms of p. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of Lp-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our Hörmander-Mikhlin sufficient conditions in a given rank n and failing the rigidity conditions for ranks m >> n. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of K-biinvariant multipliers in the rank 1 groups SO(n, 1).","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43652466","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 : 2022-01-01DOI: 10.1215/00127094-2021-0062
Qi-Rui Li
{"title":"An intersection formula for CM cycles on Lubin–Tate spaces","authors":"Qi-Rui Li","doi":"10.1215/00127094-2021-0062","DOIUrl":"https://doi.org/10.1215/00127094-2021-0062","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46116647","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 : 2022-01-01DOI: 10.1215/00127094-2021-0063
M. Kisin, Keerthi Madapusi Pera, S. Shin
A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are non-empty provided the corresponding group G is quasi-split at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda-Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.
{"title":"Honda–Tate theory for Shimura varieties","authors":"M. Kisin, Keerthi Madapusi Pera, S. Shin","doi":"10.1215/00127094-2021-0063","DOIUrl":"https://doi.org/10.1215/00127094-2021-0063","url":null,"abstract":"A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are non-empty provided the corresponding group G is quasi-split at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda-Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47942181","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 : 2022-01-01DOI: 10.1215/00127094-2021-0026
Keaton Naff
{"title":"A planarity estimate for pinched solutions of mean curvature flow","authors":"Keaton Naff","doi":"10.1215/00127094-2021-0026","DOIUrl":"https://doi.org/10.1215/00127094-2021-0026","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47754278","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 : 2022-01-01DOI: 10.1215/00127094-2022-0040
Jan Sbierski
{"title":"On holonomy singularities in general relativity and the Cloc0,1-inextendibility of space-times","authors":"Jan Sbierski","doi":"10.1215/00127094-2022-0040","DOIUrl":"https://doi.org/10.1215/00127094-2022-0040","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66770007","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 : 2022-01-01DOI: 10.1215/00127094-2022-0012
J. Bernstein, Lu Wang
{"title":"Closed hypersurfaces of low entropy in R4 are isotopically trivial","authors":"J. Bernstein, Lu Wang","doi":"10.1215/00127094-2022-0012","DOIUrl":"https://doi.org/10.1215/00127094-2022-0012","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47167732","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 : 2021-11-09DOI: 10.1215/00127094-2021-0014
Xiannan Li
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape x + 2y where y is small.
{"title":"Prime values of a sparse polynomial sequence","authors":"Xiannan Li","doi":"10.1215/00127094-2021-0014","DOIUrl":"https://doi.org/10.1215/00127094-2021-0014","url":null,"abstract":"A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape x + 2y where y is small.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43746543","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 : 2021-10-15DOI: 10.1215/00127094-2021-0052
B. Dodson
{"title":"Scattering for the radial defocusing cubic nonlinear wave equation with initial data in the critical Sobolev space","authors":"B. Dodson","doi":"10.1215/00127094-2021-0052","DOIUrl":"https://doi.org/10.1215/00127094-2021-0052","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45998174","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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