FY2022 Annual Report

Analysis on Metric Spaces Unit
Assistant Professor Xiaodan Zhou

 

Abstract

Our unit had an eventful and remarkable year during the past fiscal year,  April 2022 to March 2023. We welcomed a valuable addition to our team in September 2022, as Dr. Ye Zhang joined us as a postdoctoral scholar. In total, 5  papers have been accepted and 2 were submitted, and 8 talks are delivered by unit members in seminars and conferences domestically and internationally. Furthermore, we successfully hosted 1 mini-course, organized 9 seminar talks, and collaborated in the co-organization of 3 workshops at OIST.

In research, joint with our collaborators, we study topics including h-quasiconvexity,  maximal functions, heat kernel asymptotics, measure contraction properties and sub-Laplacians in sub-Riemannian setting; Characterization of Sobolev and BV function through nonlocal functionals, and functions attaining nonlinear asymptotic mean value formulas on metric measure spaces.

One highlight in the past year is a hybrid workshop on Analysis on metric spaces bringing together a number of top class mathematicians working in several research lines of this area. We also co-organized two onsite workshops with two other units at OIST respectively: Geometric PDEs and Applications workshop with Geometric Partial Differential Equation Unit and Women at the Intersection of Mathematics and Theoretical Physics Meet in Okinawa Workshop with Gravity, Quantum Geometry and Field Theory Unit. 

This yearly summary lists the key research projects, accomplishments and events of our unit in fiscal year 2022.

1. Staff

  • Dr. Antoni Kijowski, Postdoctoral Scholar
  • Dr. Ye Zhang, Postdoctoral Scholar
  • Chiyo Eto, Administrative Assistant

2. Collaborations​

2.1 Collaborators

​

  •  Description: A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

  • Type of collaboration: Joint research
  • Researchers: 
  1. Professor Panu Lahti, Chinese Academy of Sciences
  2. Professor Andrea Pinamonti, University of Trento
  3. Professor Xiaodan Zhou, OIST

 

  • Description:  Heat kernel asymptotics for a class of Metivier groups; Heat kernel asymptotics on the free step-two Carnot group with three generators; Centered Hardy-Littlewood Maximal Functions on H-Type Groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Professor Hong-Quan Li, Fudan University
  2. Dr. Ye Zhang, OIST

 

  • Description: Subelliptic harmonic functions and sub-Laplacian operator in Carnot groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Professor Tomasz Adamowicz, Institute of Mathematics, Polish Academy of Sciences
  2. Dr. Antoni Kijowski, OIST

 

  • Description: Subelliptic harmonic functions and sub-Laplacian operator in Carnot groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Professor Tomasz Adamowicz, Institute of Mathematics, Polish Academy of Sciences
  2. Dr. Antoni Kijowski, OIST
  3. Dr. Sebastiano Nicolussi Golo, University of Jyvaskyla

 

  • Description: Comparison principle for Hamilton-Jacobi equation in metric spaces

  • Type of collaboration: Joint research
  • Researchers:
  1. Professor Przemysław Górka, Warsaw University of Technology
  2. Dr. Antoni Kijowski, OIST

 

  • Description: Properties of functions attaining mean value formulas

  • Type of collaboration: Joint research
  • Researchers:
  1. Professor Julio Rossi, Universidad de Buenos Aires
  2. Dr. Antoni Kijowski, OIST
  3. Dr. Ye Zhang, OIST

 

3. Activities and Findings

3.1 Characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincare inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure.

3.2 Heat kernel asymptotics for a class of Metivier groups

We study the uniform asymptotic behaviour at infinity of the heat kernel, associated to the sub-Laplacian as well as the full Laplacian, on a large class of step-two Carnot groups. Moreover, sharp bounds for its derivatives will be obtained. As an application, precise estimates and small-time asymptotic behaviours for the heat kernel will be provided, as explicit as one can possibly hope for, in the setting of generalized Heisenberg-type groups.

3.3 On the H.-Q. Li inequality on step-two Carnot groups

We show that the gradient estimate of the heat semigroup, or more precisely the H.-Q. Li inequality, is preserved under tensorization, some suitable group epimorphism, and central sum. We also establish the Riemannian counterpart of the H.-Q. Li inequality. As a byproduct, we provide a simpler proof of the fact that the constant in H.-Q. Li inequality is strictly larger than 1.

3.4 Subelliptic harmonic functions and sub-Laplacian operator in Carnot groups

We are interested in studying the following properties of sub-Laplacian on Carnot groups: logconvexity of the area of level set function of solutions to the subelliptic harmonic equation in smooth annuli with constant boundary data; characterize point symmetries of sub-Laplacians in Carnot groups; characterize jet symmetries of sub-Laplacians in Carnot groups.

3.5 Comparison principle for Hamilton-Jacobi equation in metric spaces

The main purpose of our study is to prove comparison principles for fully nonlinear Hamilton-Jacobi equation, where hamiltonian includes time-derivative of unknown function and suffers lack of convexity. We consider the problem in precompact geodesic subspace of a metric space assuming certain conditions on the Hamiltonian.

3.6 Properties of functions attaining mean value formulas

In a recent paper Blanc–Charro–Manfredi–Rossi proved a characterization of viscosity solutions to second order PDEs via a nonlinear asymptotic mean value property in the viscosity sense.The class of their equations cover a large variety of second order operators, e.g. k-Hessian operators which include the Laplacian and Monge–Ampère operator, truncated Laplacians and Pucci operators. With Julio Rossi and Ye Zhang we study functions having such property focusing on proving regularity, comparison principle, Harnack inequality and a suitable version of the Blaschke–Privaloff–Zaremba theorem.

 

4. Publications

4.1 Journals

Submitted paper during April 1st, 2022 to March 31st, 2023

1.     Heat kernel asymptotics for a class of Metivier groups, (H.-Q. Li and Y. Zhang), submitted

2.    A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces, (P. Lahti, A. Pinamonti and X. Zhou), submitted, arXiv.

Accepted paper during April 1st, 2022 to March 31st, 2023

1.    Horizontally quasiconvex envelope in the Heisenberg group, A. Kijowski, Q. Liu and X. Zhou, accepted by Rev. Mat. Iberoam, (2023), https://doi.org/10.4171/rmi/1417.

2.    Absolutely continuous mappings on doubling metric measure spaces, P. Lahti and X. Zhou, accepted by Manuscripta Mathematica.

3.    Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces Measure Spaces, A. Kijowski, T. Adamowicz and E. Soultanis, Anal. Geom. Metr. Spaces 10 (2022), no. 1, 344–372.

4.    Asymptotically Mean Value Harmonic Functions in Subriemannian and RCD Settings, A. Kijowski, T. Adamowicz and E. Soultanis,, J Geom Anal 33, 80 (2023). 

5.    On the H.-Q. Li inequality on step-two Carnot groups, Y. Zhang, To appear in Comptes Rendus Mathematique.

4.2 Books and other one-time publications

 

4.3 Oral and Poster Presentations 

 

External conferences and seminars

1.    MATRIX-RIMS Tandem Workshop on “Geometric Analysis in Harmonic Analysis and PDE”,  RIMS, Kyoto University, X. Zhou, March 27-31, 2023

2.    PDE and Analysis Seminar, University of Pittsburgh, X. Zhou, Feb 27th, 2023

3.    Probability and Geometry,  Osaka University, X. Zhou, February 14-16, 2023

4.    RIMS共同研究（公開型）Women in Mathematics, Kyoto, X. Zhou, September 7-9, 2022

5.    The 47th Sapporo Symposium on Partial Differential Equations,  Sapporo, X. Zhou, August 8-10, 2022

6.     Geometry and Probability 2022, Osaka University, Y. Zhang, February 14, 2023

7.    The Mathematical Society of Japan Spring Meeting 2023 (Geometry Section), Chuo University, Y. Zhang, March 17, 2023

8.    The Mathematical Society of Japan Spring Meeting 2023 (Geometry Section), Chuo University, A. Kijowski, March 15, 2023

Conferences and seminars at OIST

  1. Analysis on Metric Spaces Workshop at OIST, H-convex hull and h-quasiconvex envelope in the Heisenberg group, A. Kijowski, May 25, 2022

  2. OIST internal seminar, A. Kijowski, March 3, 2023

5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

6. Meetings and Events

6.1 【Seminar】Sub-Gaussian heat kernel bounds and singularity of energy measures for symmetric diffusions

  • Date: April 22, 2022 
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Naotaka Kajino (Kyoto University)

6.2 【Seminar】Variational problems with gradient constraint　

  • Date: May 10, 2022 
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Xiao Zhong (University of Helsinki)

6.3 【Workshop】Analysis on Metric Spaces Workshop 2022　

  • Date: May 23, 2022 - May 27, 2022
  • Venue: OIST Campus Conference Center and online
  • Organizers: Luca Capogna (Smith College), Qing Liu (OIST),  Shin-ichi Ohta (Osaka University), Nageswari Shanmugalingam (University of Cincinnati), Xiaodan Zhou (OIST)

6.4 【Seminar】Sobolev embeddings of fractional spaces – continuity and compactness

  • Date: June 2, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Artur Słabuszewski (Warsaw University of Technology)

6.5 【Seminar】Conservation law for harmonic mappings in higher dimensions

  • Date: July 1, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Changyu Guo (Shandong University)

6.6 【Mini - Course 】BV functions in Carnot groups

  • Date: Auguest 22 - Auguest 24, 2022
  • Venue: OIST Campus L4E48
  • Speaker: Sebastiano Nicolussi Golo (University of Jyväskylä)

6.7 【Seminar】Cantor Julia sets

  • Date: October 7, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Vyron Vellis (University of Tennessee)

6.8【Seminar】Korevaar-Schoen-Sobolev spaces and critical exponents on metric measure spaces   

  • Date: October 21, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Fabrice Baudoin (University of Connecticut)

6.9【Seminar】Whitney Extension and Lusin Approximation for Horizontal Curves in the Heisenberg Group

  • Date: November 4, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Andrea Pinamonti (University of Trento)

6.10 【Seminar】Poincare inequalities on the Vicsek set

  • Date: November 18, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Li Chen (Louisiana State University)

6.11 【Seminar】Sharp Uncertainty Principles and their stability

  • Date: December 2, 2022
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Nguyen Lam (Memorial University of Newfoundland)

6.12 【Conference】Geometric PDEs and Applications

  • Date: January 16- January 18, 2023
  • Venue: OIST Campus C209 and online
  • Organizers: Qing Liu (OIST), Hiroyoshi Mitake (University of Tokyo), Xiaodan Zhou (OIST)

6.13 【Workshop】 Women at the Intersection of Mathematics and Theoretical Physics Meet in Okinawa

  • Date: March 20 - March 24, 2023
  • Venue: OIST Campus L4E48
  • Organizers: Shihoko Ishii (University of Tokyo), Sylvie Paycha (University of Potsdam), Susanne Reffert (University of Bern), Kasia Rejzner (University of York), Reiko Toriumi(OIST), Xiaodan Zhou (OIST)

7. Other

Nothing to report.