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Project Details

Funding Scheme :

General Research Fund

Project Number :

14313716

Project Title(English) :

Restrictions of pseudodifferential operators of mixed homogeneities

Project Title(Chinese) :

子流形上的偽微分算子

Principal Investigator(English) :

Prof Leung, Chi-wai

Principal Investigator(Chinese) :

Department :

Dept of Mathematics

Institution :

The Chinese University of Hong Kong

E-mail Address :

cwleung@math.cuhk.edu.hk

Tel :

26097982

Co - Investigator(s) :

Prof Yung, Po-lam

Panel :

Physical Sciences

Subject Area :

Mathematics

Exercise Year :

2016 / 17

Fund Approved :

727,647

Project Status :

Completed

Completion Date :

30-6-2020

Project Objectives :

Construct algebras of pseudodifferential operators, that can be used to understand the restrictions of algebras of pseudodifferential operators from an ambient manifold to a submanifold (with particular attention to the algebra of pseudodifferential operators of mixed homogeneities introduced by E.M. Stein and the PI).

Understand the compositions of the above pseudodifferential operators.

Estimate the sizes of the kernels of the above pseudodifferential operators, and study their cancellation properties.

Obtain various mapping properties of the above operators (e.g. on various L^p, Sobolev and Holder spaces).

Abstract as per original application
(English/Chinese):

我們將研究多複變函數中一類帶有 mixed homogeneity 的偽微分算子. 研究將著重於它們在子流形上的特性, 如它們的連續性及卷積核的特性. 這是 PI 跟 E.M. Stein 之前的工作的推廣. 我們希望藉此更理解一些跟 Grushin 的例子相關的偏微分方程.

Realisation of objectives:

The first PI: Po-Lam, Yeung We now understand how pseudodifferential operators restrict to submanifolds of the ambient manifold quite well. They form an algebra, and we understand the sizes and the L^p and Sobolev mapping properties of such operators. This has been achieved for ambient pseudodifferential operators with both a single and mixed homogeneity. We applied it to solve a question from several complex variables and CR geometry, in a joint work with Chin-Yu Hsiao. The second PI: Chi-Wai Leung: 1. Tingley's problem asks whether every metric preserving bijection between the unit spheres of two real Banach spaces extends to a bijective isometry between the two Banach spaces. In this work, We show that every metric preserving bijection between the positive parts of units spheres in certain $L_p$ spaces can be extended (necessarily uniquely) to an isometric order isomorphism. 2. Spectral sets and measures have been studied in the Euclidean setting and locally compact Abelian group. In this work, we are going to generalize spectral sets and measures to locally compact groups (not necessarily Abelian) and certain homogeneous spaces.

Summary of objectives addressed:

	Objectives	Addressed	Percentage achieved
1.	Construct algebras of pseudodifferential operators, that can be used to understand the restrictions of algebras of pseudodifferential operators from an ambient manifold to a submanifold (with particular attention to the algebra of pseudodifferential operators of mixed homogeneities introduced by E.M. Stein and the PI).	Yes	100%
2.	Understand the compositions of the above pseudodifferential operators.	Yes	100%
3.	Estimate the sizes of the kernels of the above pseudodifferential operators, and study their cancellation properties.	Yes	100%
4.	Obtain various mapping properties of the above operators (e.g. on various L^p, Sobolev and Holder spaces).	Yes	100%

Research Outcome

Major findings and research outcome:

The first PL: Po-Lam Yung: The following 3 papers have been published under this grant: 1. Chin-Yu Hsiao and Po-Lam Yung, Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Art. 71, 62 pp. 2. Haim Brezis, Jean Van Schaftingen, and Po-Lam Yung, A surprising formula for Sobolev norms, in Proc. Natl. Acad. Sci. 118 (2021) no. 8, e2025254118. 3. Jean Van Schaftingen and Po-Lam Yung, Limiting Sobolev and Hardy inequalities on stratified homogeneous groups, arXiv:2007.14532 (submitted for publication, still under review) The first paper uses ideas from this project to solve the tangential Kohn Laplacian on a non-compact CR manifold of dimension 3. It has applications towards a positive mass theorem in 3-dimensional CR geometry. The second paper proves a new and surprising formula for the Sobolev norm of a function. It provides an alternative point of view of how one understands Sobolev norms and Sobolev mapping properties. The third paper proves new Sobolev and Hardy inequalities where the inhomogeneity of stratified groups played a key role. It applies ideas from this project to construct fundamental solutions of certain higher order differential operators on such a group. The second PI: Chi-Wai Leung: 1. Leung, Chi-Wai; Ng, Chi-Keung; Wong, Ngai-Ching On a variant of Tingley's problem for some function spaces. J. Math. Anal. Appl. 496 (2021), no. 1. In this paper, we show that every metric preserving bijection between the positive parts of units spheres in certain $L_p$ spaces can be extended (necessarily uniquely) to an isometric order isomorphism. In addition, we show that for compact Hausdorff spaces $X$ and $Y$, if $Phi$ is a metric preserving bijection from the positive part of the unit sphere of $C(X)$ to that of $C(Y)$, then the map $PhI$ is induced by a certain homeomorphism from $Y$ to $X$. 2. Leung, Chi-Wai, Orthogonality relations on certain homogeneous spaces, has been accepted to publish in Proceedings of the American Mathematical Society. In this paper, we generalize the notion of spectral measures to homogeneous spaces. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.

Potential for further development of the research
and the proposed course of action:

The first PI Po-Lam Yung: The ideas in this project will lead to further development of tools for solving various subelliptic partial differential equations. The work on Sobolev norms has already found impact beyond the harmonic analysis community. For instance, the formula we developed was referred to as the Brezis-Van Schaftingen-Yung inequality in a recent paper by Dominguez and Milman, and followed up in the following papers of the PI: Qingsong Gu and Po-Lam Yung, A new formula for the Lp norm, J. Funct. Anal. 281 (2021), no. 4, 109075. Haim Brezis, Jean Van Schaftingen and Po-Lam Yung, Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail, to appear in Calc. Var. Partial Differential Equations.

Layman's Summary of
Completion Report:

The first PI: Po-Lam Yung: This project deepens our understanding of various partial differential equations arising in complex analysis. It provides a new conceptual framework on how one solves such equations. It also led to unexpected discovery of new ways for quantifying how fast a function changes in space. The discoveries are expected to have an impact for the years to come.

Research Output

Peer-reviewed journal publication(s)
arising directly from this research project :
(* denotes the corresponding author)

Year of Publication	Author(s)	Title and Journal/Book	Accessible from Institution Repository
2019	Chin-Yu Hsiao, Po-Lam Yung	Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds, , Calc. Var. Partial Differential Equations 58 (2019), no. 2, Art. 71, 62 pp.	No
2021	Haim Brezis, Jean Van Schaftingen, and Po-Lam Yung,	A surprising formula for Sobolev norms, in Proc. Natl. Acad. Sci. 118 (2021) no. 8, e2025254118.	No
	Jean Van Schaftingen and Po-Lam Yung,	Limiting Sobolev and Hardy inequalities on stratified homogeneous groups,	No
2021	Chi-Wai Leung; Chi-Keung Ng; Ngai-Ching Wong	On a variant of Tingley's problem for some function spaces. J. Math. Anal. Appl. 496 (2021), no. 1, 124800, 16 pp.	No
	Chi-Wai Leung	Orthogonality relations on certain homogeneous spaces, Proceedings of the American Mathematical Society.	No

Recognized international conference(s)
in which paper(s) related to this research
project was/were delivered :

Other impact
(e.g. award of patents or prizes,
collaboration with other research institutions,
technology transfer, etc.):

SCREEN ID: SCRRM00542