{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113138"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113138","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some new results related to the Bochner-Riesz problems","abstract":"\"We study Fourier analysis problems related to the Bochner-Riesz mean. In particular, it is shown that the R3 Bochner-Riesz operator T is bounded in Lp(R3) when p 3:25, in the optimal range of . The dissertation consists of six chapters. In Chapter 1, we briefly review the background of the Bochner-Riesz problems and state our main result. An outline of our proofs is given along the way. In the end, we briefly discuss the connection between the Bochner-Riesz problems and other well-known problems. Chapter 2 is devoted to some preliminaries in Fourier analysis and several reductions of the Bochner-Riesz operator. There are mainly two reductions. The first one is quite standard: we reduce the Bochner-Riesz operator to a spherical operator, whose Fourier multiplier is supported in a thin neighborhood of the unit sphere. The second one is based on the Bourgain-Guth broad-narrow argument. Basically, we further decompose the spherical operator into a broad part and a narrow part. The narrow part can be handled easily by induction, and the broad part behaves better than the original spherical operator because it contains some additional geometric properties. In Chapter 3, we built some wanted wave packets at different scales. We also discuss some intuitions and key features for the method of wave packet decomposition. Chapter 4 is designed for the polynomial partitioning iteration. We first revisit Guth's polynomial partitioning algorithm. Then we apply it repeatedly to set up the iteration and hence break down the broad part introduced in Section 2. When the iteration stops, we have two situations: \\Small\"\" and \\Tangent\"\". The first situation is the easier one, and in the end of the section, we conclude the proof of the broad part in this situation. Chapter 5 is the heart of our proof. Now we stop at the second situation \\Tangent\"\" in the polynomial partitioning iteration in Section 4. As mentioned above, the iteration helps us break down the broad part into pieces. We build a backward algorithm to sum up the pieces efficiently, and hence prove the broad part in the second situation. This also concludes the proof of our main result.\"","abstract_html":""We study Fourier analysis problems related to the Bochner-Riesz mean. In particular, it is shown that the R3 Bochner-Riesz operator T is bounded in Lp(R3) when p 3:25, in the optimal range of . The dissertation consists of six chapters. In Chapter 1, we briefly review the background of the Bochner-Riesz problems and state our main result. An outline of our proofs is given along the way. In the end, we briefly discuss the connection between the Bochner-Riesz problems and other well-known problems. Chapter 2 is devoted to some preliminaries in Fourier analysis and several reductions of the Bochner-Riesz operator. There are mainly two reductions. The first one is quite standard: we reduce the Bochner-Riesz operator to a spherical operator, whose Fourier multiplier is supported in a thin neighborhood of the unit sphere. The second one is based on the Bourgain-Guth broad-narrow argument. Basically, we further decompose the spherical operator into a broad part and a narrow part. The narrow part can be handled easily by induction, and the broad part behaves better than the original spherical operator because it contains some additional geometric properties. In Chapter 3, we built some wanted wave packets at different scales. We also discuss some intuitions and key features for the method of wave packet decomposition. Chapter 4 is designed for the polynomial partitioning iteration. We first revisit Guth's polynomial partitioning algorithm. Then we apply it repeatedly to set up the iteration and hence break down the broad part introduced in Section 2. When the iteration stops, we have two situations: \\Small"" and \\Tangent"". The first situation is the easier one, and in the end of the section, we conclude the proof of the broad part in this situation. Chapter 5 is the heart of our proof. Now we stop at the second situation \\Tangent"" in the polynomial partitioning iteration in Section 4. As mentioned above, the iteration helps us break down the broad part into pieces. We build a backward algorithm to sum up the pieces efficiently, and hence prove the broad part in the second situation. This also concludes the proof of our main result."","abstract_has_math":false,"creators":["Wu, Shukun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Li, Xiaochun","Erdogan, Burak","Laugesen, Richard","Tzirakis, Nikolaos"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T22:34:51Z","date_published":"2022-01-12T22:34:51Z","updated_at":"2026-07-22T22:24:53Z","subjects":["Fourier analysis, Bochner-Riesz"],"languages":["en"],"rights":["Copyright 2021 Shukun Wu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113138","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Li, Xiaochun","Erdogan, Burak","Laugesen, Richard","Tzirakis, Nikolaos"]},{"key":"dc:creator","label":"Author","values":["Wu, Shukun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T22:34:51Z","2024-01-12T22:35:30Z","2021-07-12","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fourier analysis, Bochner-Riesz"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Shukun Wu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113138"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"We study Fourier analysis problems related to the Bochner-Riesz mean. In particular, it is shown that the R3 Bochner-Riesz operator T is bounded in Lp(R3) when p 3:25, in the optimal range of . The dissertation consists of six chapters. In Chapter 1, we briefly review the background of the Bochner-Riesz problems and state our main result. An outline of our proofs is given along the way. In the end, we briefly discuss the connection between the Bochner-Riesz problems and other well-known problems. Chapter 2 is devoted to some preliminaries in Fourier analysis and several reductions of the Bochner-Riesz operator. There are mainly two reductions. The first one is quite standard: we reduce the Bochner-Riesz operator to a spherical operator, whose Fourier multiplier is supported in a thin neighborhood of the unit sphere. The second one is based on the Bourgain-Guth broad-narrow argument. Basically, we further decompose the spherical operator into a broad part and a narrow part. The narrow part can be handled easily by induction, and the broad part behaves better than the original spherical operator because it contains some additional geometric properties. In Chapter 3, we built some wanted wave packets at different scales. We also discuss some intuitions and key features for the method of wave packet decomposition. Chapter 4 is designed for the polynomial partitioning iteration. We first revisit Guth's polynomial partitioning algorithm. Then we apply it repeatedly to set up the iteration and hence break down the broad part introduced in Section 2. When the iteration stops, we have two situations: \\Small\"\" and \\Tangent\"\". The first situation is the easier one, and in the end of the section, we conclude the proof of the broad part in this situation. Chapter 5 is the heart of our proof. Now we stop at the second situation \\Tangent\"\" in the polynomial partitioning iteration in Section 4. As mentioned above, the iteration helps us break down the broad part into pieces. We build a backward algorithm to sum up the pieces efficiently, and hence prove the broad part in the second situation. This also concludes the proof of our main result.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Shukun Wu, accepted the attached license on 2021-07-02 at 12:33.","The student, Shukun Wu, submitted this Dissertation for approval on 2021-07-02 at 13:18.","This Dissertation was approved for publication on 2021-07-12 at 16:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16750 on 2022-01-12 at 12:52:51","Made available in DSpace on 2022-01-12T22:34:51Z (GMT). No. of bitstreams: 3 WU-DISSERTATION-2021.pdf: 552250 bytes, checksum: e1b341fb5bab9e59623fe5e668019779 (MD5) LICENSE.txt: 4206 bytes, checksum: 99f14024e5c1c6ed4cc32bd548e50539 (MD5) PROQUEST_LICENSE.txt: 4552 bytes, checksum: 537745f31c9c08a51da3743a60a2d152 (MD5) Previous issue date: 2021-07-12","Embargo set by: Seth Robbins for item 121064 Lift date: 2024-01-12T22:35:30Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Some new results related to the Bochner-Riesz problems"]}]}],"canonical_facts":{"dc:contributor":["Li, Xiaochun","Erdogan, Burak","Laugesen, Richard","Tzirakis, Nikolaos"],"dc:creator":["Wu, Shukun"],"dc:date":["2022-01-12T22:34:51Z","2024-01-12T22:35:30Z","2021-07-12","2021-08"],"dc:description":["\"We study Fourier analysis problems related to the Bochner-Riesz mean. In particular, it is shown that the R3 Bochner-Riesz operator T is bounded in Lp(R3) when p 3:25, in the optimal range of . The dissertation consists of six chapters. In Chapter 1, we briefly review the background of the Bochner-Riesz problems and state our main result. An outline of our proofs is given along the way. In the end, we briefly discuss the connection between the Bochner-Riesz problems and other well-known problems. Chapter 2 is devoted to some preliminaries in Fourier analysis and several reductions of the Bochner-Riesz operator. There are mainly two reductions. The first one is quite standard: we reduce the Bochner-Riesz operator to a spherical operator, whose Fourier multiplier is supported in a thin neighborhood of the unit sphere. The second one is based on the Bourgain-Guth broad-narrow argument. Basically, we further decompose the spherical operator into a broad part and a narrow part. The narrow part can be handled easily by induction, and the broad part behaves better than the original spherical operator because it contains some additional geometric properties. In Chapter 3, we built some wanted wave packets at different scales. We also discuss some intuitions and key features for the method of wave packet decomposition. Chapter 4 is designed for the polynomial partitioning iteration. We first revisit Guth's polynomial partitioning algorithm. Then we apply it repeatedly to set up the iteration and hence break down the broad part introduced in Section 2. When the iteration stops, we have two situations: \\Small\"\" and \\Tangent\"\". The first situation is the easier one, and in the end of the section, we conclude the proof of the broad part in this situation. Chapter 5 is the heart of our proof. Now we stop at the second situation \\Tangent\"\" in the polynomial partitioning iteration in Section 4. As mentioned above, the iteration helps us break down the broad part into pieces. We build a backward algorithm to sum up the pieces efficiently, and hence prove the broad part in the second situation. This also concludes the proof of our main result.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Shukun Wu, accepted the attached license on 2021-07-02 at 12:33.","The student, Shukun Wu, submitted this Dissertation for approval on 2021-07-02 at 13:18.","This Dissertation was approved for publication on 2021-07-12 at 16:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16750 on 2022-01-12 at 12:52:51","Made available in DSpace on 2022-01-12T22:34:51Z (GMT). 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