POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES
<jats:p>We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press (CUP)
2021
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| Online Access: | https://hdl.handle.net/1721.1/134734 |
| _version_ | 1811091798740697088 |
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| author | DU, XIUMIN GUTH, LARRY LI, XIAOCHUN ZHANG, RUIXIANG |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics DU, XIUMIN GUTH, LARRY LI, XIAOCHUN ZHANG, RUIXIANG |
| author_sort | DU, XIUMIN |
| collection | MIT |
| description | <jats:p>We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline1" /><jats:tex-math>$n\geqslant 3$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline2" /><jats:tex-math>$\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula><jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline3" /><jats:tex-math>$=f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula> almost everywhere with respect to Lebesgue measure for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline4" /><jats:tex-math>$f\in H^{s}(\mathbb{R}^{n})$</jats:tex-math></jats:alternatives></jats:inline-formula> provided that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline5" /><jats:tex-math>$s>(n+1)/2(n+2)$</jats:tex-math></jats:alternatives></jats:inline-formula>. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.</jats:p> |
| first_indexed | 2024-09-23T15:08:12Z |
| format | Article |
| id | mit-1721.1/134734 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:08:12Z |
| publishDate | 2021 |
| publisher | Cambridge University Press (CUP) |
| record_format | dspace |
| spelling | mit-1721.1/1347342023-03-15T19:28:38Z POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES DU, XIUMIN GUTH, LARRY LI, XIAOCHUN ZHANG, RUIXIANG Massachusetts Institute of Technology. Department of Mathematics <jats:p>We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline1" /><jats:tex-math>$n\geqslant 3$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline2" /><jats:tex-math>$\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula><jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline3" /><jats:tex-math>$=f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula> almost everywhere with respect to Lebesgue measure for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline4" /><jats:tex-math>$f\in H^{s}(\mathbb{R}^{n})$</jats:tex-math></jats:alternatives></jats:inline-formula> provided that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline5" /><jats:tex-math>$s>(n+1)/2(n+2)$</jats:tex-math></jats:alternatives></jats:inline-formula>. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.</jats:p> 2021-10-27T20:08:54Z 2021-10-27T20:08:54Z 2018 2019-11-13T16:38:13Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/134734 en 10.1017/FMS.2018.11 Forum of Mathematics, Sigma Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Cambridge University Press (CUP) Cambridge University Press |
| spellingShingle | DU, XIUMIN GUTH, LARRY LI, XIAOCHUN ZHANG, RUIXIANG POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title | POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title_full | POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title_fullStr | POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title_full_unstemmed | POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title_short | POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES |
| title_sort | pointwise convergence of schrodinger solutions and multilinear refined strichartz estimates |
| url | https://hdl.handle.net/1721.1/134734 |
| work_keys_str_mv | AT duxiumin pointwiseconvergenceofschrodingersolutionsandmultilinearrefinedstrichartzestimates AT guthlarry pointwiseconvergenceofschrodingersolutionsandmultilinearrefinedstrichartzestimates AT lixiaochun pointwiseconvergenceofschrodingersolutionsandmultilinearrefinedstrichartzestimates AT zhangruixiang pointwiseconvergenceofschrodingersolutionsandmultilinearrefinedstrichartzestimates |