[{"quality_controlled":"1","year":"2017","publication_status":"published","intvolume":"      2017","date_published":"2017-04-01T00:00:00Z","article_processing_charge":"No","fulldoi":"https://doi.org/10.1093/imrn/rnw096","arxiv":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1510.00136"}],"day":"01","oa":1,"external_id":{"arxiv":["1510.00136"]},"month":"04","publisher":"Oxford University Press","volume":2017,"doi":"10.1093/imrn/rnw096","_id":"268","title":"A transference approach to a Roth-type theorem in the squares","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177"},{"full_name":"Prendiville, Sean","last_name":"Prendiville","first_name":"Sean"}],"date_created":"2018-12-11T11:45:31Z","status":"public","article_type":"original","oa_version":"Preprint","type":"journal_article","abstract":[{"text":"We show that any subset of the squares of positive relative upper density contains nontrivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.","lang":"eng"}],"extern":"1","citation":{"chicago":"Browning, Timothy D, and Sean Prendiville. “A Transference Approach to a Roth-Type Theorem in the Squares.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2017. <a href=\"https://doi.org/10.1093/imrn/rnw096\">https://doi.org/10.1093/imrn/rnw096</a>.","apa":"Browning, T. D., &#38; Prendiville, S. (2017). A transference approach to a Roth-type theorem in the squares. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnw096\">https://doi.org/10.1093/imrn/rnw096</a>","mla":"Browning, Timothy D., and Sean Prendiville. “A Transference Approach to a Roth-Type Theorem in the Squares.” <i>International Mathematics Research Notices</i>, vol. 2017, no. 7, Oxford University Press, 2017, pp. 2219–48, doi:<a href=\"https://doi.org/10.1093/imrn/rnw096\">10.1093/imrn/rnw096</a>.","ama":"Browning TD, Prendiville S. A transference approach to a Roth-type theorem in the squares. <i>International Mathematics Research Notices</i>. 2017;2017(7):2219-2248. doi:<a href=\"https://doi.org/10.1093/imrn/rnw096\">10.1093/imrn/rnw096</a>","ieee":"T. D. Browning and S. Prendiville, “A transference approach to a Roth-type theorem in the squares,” <i>International Mathematics Research Notices</i>, vol. 2017, no. 7. Oxford University Press, pp. 2219–2248, 2017.","ista":"Browning TD, Prendiville S. 2017. A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices. 2017(7), 2219–2248.","short":"T.D. Browning, S. Prendiville, International Mathematics Research Notices 2017 (2017) 2219–2248."},"page":"2219 - 2248","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2024-03-05T11:52:36Z","publication":"International Mathematics Research Notices","issue":"7","publication_identifier":{"issn":["1073-7928"]},"acknowledgement":"Whilst working on this paper the authors were supported by ERC grant 306457.","language":[{"iso":"eng"}],"publist_id":"7634"},{"day":"01","oa":1,"external_id":{"arxiv":["1205.5784"]},"publisher":"Oxford University Press","month":"01","volume":2016,"doi":"10.1093/imrn/rnv338","publication_status":"published","year":"2016","quality_controlled":"1","intvolume":"      2016","date_published":"2016-01-01T00:00:00Z","article_processing_charge":"No","das_tickbox":"1","fulldoi":"https://doi.org/10.1093/imrn/rnv338","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1205.5784"}],"arxiv":1,"publication":"International Mathematics Research Notices","scopus_import":"1","issue":"19","publication_identifier":{"issn":["1073-7928"],"eissn":["1687-0247"]},"OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"title":"Riesz transforms outside a convex obstacle","_id":"22037","author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"first_name":"Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"},{"first_name":"Xiaoyi","last_name":"Zhang","full_name":"Zhang, Xiaoyi"}],"status":"public","date_created":"2026-06-19T07:44:28Z","oa_version":"Preprint","type":"journal_article","article_type":"original","abstract":[{"lang":"eng","text":"The goal of this paper was to develop some basic harmonic analysis tools for the Dirichlet–Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions d>=3⁠. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood–Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent p and regularity s⁠. Counterexamples are included to show that these results are indeed sharp. In particular, we precisely settle the question of boundedness of Riesz transforms on L^p⁠, including the endpoint. The utility of such results in the study of nonlinear partial differential equation (PDE) is that they allow us to deduce important results, such as the fractional product and chain rules for the Dirichlet–Laplacian, directly from the classical Euclidean setting. As an application, we discuss the local well-posedness and stability problems for energy-critical NLS. All the results of this paper play an essential role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three-dimensional exterior domains; see [29]."}],"extern":"1","citation":{"short":"R. Killip, M. Vişan, X. Zhang, International Mathematics Research Notices 2016 (2016) 5875–5921.","ieee":"R. Killip, M. Vişan, and X. Zhang, “Riesz transforms outside a convex obstacle,” <i>International Mathematics Research Notices</i>, vol. 2016, no. 19. Oxford University Press, pp. 5875–5921, 2016.","ista":"Killip R, Vişan M, Zhang X. 2016. Riesz transforms outside a convex obstacle. International Mathematics Research Notices. 2016(19), 5875–5921.","ama":"Killip R, Vişan M, Zhang X. Riesz transforms outside a convex obstacle. <i>International Mathematics Research Notices</i>. 2016;2016(19):5875-5921. doi:<a href=\"https://doi.org/10.1093/imrn/rnv338\">10.1093/imrn/rnv338</a>","mla":"Killip, Rowan, et al. “Riesz Transforms Outside a Convex Obstacle.” <i>International Mathematics Research Notices</i>, vol. 2016, no. 19, Oxford University Press, 2016, pp. 5875–921, doi:<a href=\"https://doi.org/10.1093/imrn/rnv338\">10.1093/imrn/rnv338</a>.","apa":"Killip, R., Vişan, M., &#38; Zhang, X. (2016). Riesz transforms outside a convex obstacle. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnv338\">https://doi.org/10.1093/imrn/rnv338</a>","chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Riesz Transforms Outside a Convex Obstacle.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2016. <a href=\"https://doi.org/10.1093/imrn/rnv338\">https://doi.org/10.1093/imrn/rnv338</a>."},"page":"5875-5921","date_updated":"2026-06-22T13:09:52Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1011.1526","open_access":"1"}],"arxiv":1,"das_tickbox":"1","fulldoi":"https://doi.org/10.1093/imrn/rnr051","article_processing_charge":"No","date_published":"2012-05-01T00:00:00Z","intvolume":"      2012","publication_status":"published","quality_controlled":"1","year":"2012","doi":"10.1093/imrn/rnr051","volume":2012,"publisher":"Oxford University Press","month":"05","external_id":{"arxiv":["1011.1526"]},"oa":1,"day":"01","date_updated":"2026-06-30T11:21:51Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Vişan, Monica. “Global Well-Posedness and Scattering for the Defocusing Cubic Nonlinear Schrödinger Equation in Four Dimensions.” <i>International Mathematics Research Notices</i>, vol. 2012, no. 5, Oxford University Press, 2012, pp. 1037–67, doi:<a href=\"https://doi.org/10.1093/imrn/rnr051\">10.1093/imrn/rnr051</a>.","ama":"Vişan M. Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. <i>International Mathematics Research Notices</i>. 2012;2012(5):1037-1067. doi:<a href=\"https://doi.org/10.1093/imrn/rnr051\">10.1093/imrn/rnr051</a>","ista":"Vişan M. 2012. Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. International Mathematics Research Notices. 2012(5), 1037–1067.","ieee":"M. Vişan, “Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions,” <i>International Mathematics Research Notices</i>, vol. 2012, no. 5. Oxford University Press, pp. 1037–1067, 2012.","short":"M. Vişan, International Mathematics Research Notices 2012 (2012) 1037–1067.","chicago":"Vişan, Monica. “Global Well-Posedness and Scattering for the Defocusing Cubic Nonlinear Schrödinger Equation in Four Dimensions.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2012. <a href=\"https://doi.org/10.1093/imrn/rnr051\">https://doi.org/10.1093/imrn/rnr051</a>.","apa":"Vişan, M. (2012). Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnr051\">https://doi.org/10.1093/imrn/rnr051</a>"},"page":"1037-1067","abstract":[{"text":"In this short note, we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical nonlinear Schrödinger equation (NLS) in four space dimensions obtained previously by Ryckman and Visan [“Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in R^1+4⁠.” American Journal of Mathematics 129 (2007): 1–60. MR2288737]. The argument is inspired by the recent work of Dodson [“Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d≥3.” (2009): preprint arXiv:0912.2467.] on the mass-critical NLS.","lang":"eng"}],"extern":"1","oa_version":"Preprint","type":"journal_article","article_type":"original","date_created":"2026-06-19T08:23:13Z","status":"public","author":[{"full_name":"Visan, Monica","first_name":"Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"title":"Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions","_id":"22075","language":[{"iso":"eng"}],"OA_type":"green","OA_place":"repository","publication_identifier":{"issn":["1073-7928"],"eissn":["1687-0247"]},"scopus_import":"1","issue":"5","publication":"International Mathematics Research Notices"}]