Some new results related to the Bochner-Riesz problems

Wu, Shukun

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https://hdl.handle.net/2142/113138 Copy

Description

Title

Some new results related to the Bochner-Riesz problems

Author(s)

Wu, Shukun

Issue Date

2021-07-12

Director of Research (if dissertation) or Advisor (if thesis)

Li, Xiaochun

Doctoral Committee Chair(s)

Erdogan, Burak

Committee Member(s)

• Laugesen, Richard
• Tzirakis, Nikolaos

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Fourier analysis, Bochner-Riesz

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."

Graduation Semester

2021-08

Type of Resource

Thesis

Permalink

http://hdl.handle.net/2142/113138

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Copyright 2021 Shukun Wu

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