FY2023 Annual Report

Analysis on Metric Spaces Unit
Assistant Professor Xiaodan Zhou


Abstract

In the past fiscal year,  April 2023 to March 2024, our unit had 3 accepted and 6 submitted papers, 8 talks and 2 poster presentations delivered by unit members in seminars and conferences domestically and internationally. At OIST, we successfully hosted 3 mini-courses, organized 3 seminar talks, and organized 1 workshop.

In research, joint with our collaborators, we study topics including 1) h-convexity preserving (breaking) property for flows, heat kernel asymptotics, weak notions of curvature-dimension condition, maximal functions, and geometric inequalities in the setting of sub-Riemannian manifolds; 2) discontinuous eikonal equations, BV functions and nonlocal functionals in metric measure spaces; 3) Sobolev maps and weak* differentiability and Sobolev spaces on variable domains.

One of our postdocs, Dr. Ye Zhang was awarded JSPS Grant-in-Aid for Early-Career Scientists (April 2024-March 2027) for his research on “Weak notions of curvature-dimension conditions on step-two Carnot groups”. A new member, Dr. Nikita Evseev joined us as a postdoc in 2023 Fall. We also hosted two undergraduate research interns, Jason Lu from University of Pennsylvania and Viktoria Ipatenkova from Moscow Institute of Physics and Technology in the summer of 2023. 

This yearly summary lists the key research projects, accomplishments and events of our unit in fiscal year 2023. 

 

1. Staff

  • Dr. Antoni Kijowski, Postdoctoral Scholar (till October 2023)
  • Dr. Ye Zhang, Postdoctoral Scholar
  • Dr. Nikita Evseev, Postdoctoral Scholar
  • Chiyo Eto, Administrative Assistant

2. Collaborations​

2.1 Horizontal quasiconvexity preserving for horizontal curvature flow on Heisenberg group

  • Type of collaboration: Joint research
  • Researchers:
  1. Qing Liu, OIST
  2. Ye Zhang, OIST
  3. Xiaodan Zhou, OIST

 

2.2 BV functions and nonlocal functionals in metric measure 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

 

2.3 Discontinuous eikonal equations in metric measure spaces

  • Type of collaboration: Joint research
  • Researchers:
  1. Qing Liu, OIST
  2. Nageswari Shanmugalingam, University of Cincinnati
  3. Xiaodan Zhou, OIST

 

2.4 Horizontal semiconcavity for the square of Carnot-Caratheodory distance on Carnot groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Fedrica Dragoni, Cardiff University
  2. Qing Liu, OIST
  3. Ye Zhang, OIST

 

2.5 The measure contraction properties on Sard-regular Carnot groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Ye Zhang, OIST
  2. Sebastiano Nicolussi Golo, University of Jyvaskyla

 

2.6 Loomis–Whitney inequalities on corank 1 Carnot groups

  • Type of collaboration: Joint research
  • Researchers:
  1. Dr. Ye Zhang, OIST

 

2.7 1. Heat kernel asymptotics and heat semigroup estimates on Carnot groups; 2. Weak (1,1) bounds for centered Hardy-Littlewood Maximal Functions on Carnot groups; 3. Heat kernel asymptotics for Grushin operators

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

 

2.8 Sobolev spaces on variable domains

  • Type of collaboration: Joint research
  • Researchers:
  1. Nikita Evseev,, OIST
  2. Malte Kampschulte, Charles University, Prague, Czech Republic
  3. Alexander Menovschikov, Ben-Gurion University of the Negev, Beer Sheva, Israel

3. Activities and Findings

3.1 Horizontal quasiconvexity preserving for horizontal curvature flow on Heisenberg group

By a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator, we discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

 

3.2 Discontinuous eikonal equations in metric measure spaces

We study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely p-integrable in the domain with a finite p. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Hölder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term.

 

3.3 BV functions and nonlocal functionals in metric measure spaces

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincar´e inequality. We show that the limits of these nonlocal functionals are comparable to the variation or the Sobolev semi-norm, which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.

 

3.4 Horizontal semiconcavity for the square of Carnot-Caratheodory distance on Carnot groups

We show that the square of Carnot-Caratheodory distance from the origin, in ideal Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We apply this property to show h-semiconcavity for the solutions of a class of non-coercive evolutive Hamilton-Jacobi equations, by using the associated Hopf-Lax solutions.

 

3.5 The measure contraction properties on Sard-regular Carnot groups

On Sard-regular Carnot groups we characterize the geodesic dimension and give a new lower bound to the curvature exponent. As an application, we give an example of step-two Carnot group on which the curvature exponent is strictly larger than the geodesic dimension: this answers a question posed by Rizzi.

 

3.6 Loomis–Whitney inequalities on corank 1 Carnot groups

Using the equivalence between the Brascamp-Lieb inequality and the subadditivity of the entropy, we provide another way to deduce the Loomis-Whitney inequality on higher dimensional Heisenberg groups based on the one on the first Heisenberg group and the known nonlinear Loomis-Whitney inequality. Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups.

 

3.7 Heat kernel asymptotics and heat semigroup estimates on Carnot groups

We establish the uniform heat kernel asymptotics as well as sharp bounds for its derivatives on the free step-two Carnot group with 3 generators. As a by-product, on this highly non-trivial toy model, we completely solve the Gaveau-Brockett problem, in other words, we obtain the expression of the squared Carnot-Caratheodory distance, as explicitly as one can possibly hope for. Furthermore, the precise estimates of the heat kernel, its small-time asymptotic behaviors, and the gradient estimate for the heat semi-group are also deduced.

3.8 Weak (1,1) bounds for centered Hardy-Littlewood Maximal Functions on Carnot groups

Using the method of stationary phase, we obtain the uniform asymptotic behavior of the Poisson kernel, associated to the sub-Laplacian as well as the full Laplacian, on Heisenberg-type groups. We prove that there exists a constant C > 0, independent of dimension, such that the weak (1,1) bounds for centered Hardy-Littlewood maximal operator defined by the Koranyi norm is bounded by Cn, where n is the dimension of the first layer. While for centered Hardy-Littlewood maximal operator related to the canonical sub-Riemannian and Riemannian distance respectively, we obtain the bound C (3/2)^{m/2} n instead, where m is the dimension of the second layer.

 
 

4. Publications

4.1 Journals

Submitted paper during April 1st, 2023 to March 31st, 2024

  1. BV functions and nonlocal functionals in metric measure spaces, (P. Lahti, A. Pinamonti and X. Zhou), submitted

  2. Discontinuous eikonal equations in metric measure spaces, (Q. Liu, N. Shanmugalingam and X. Zhou), submitted

  3. A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow, (A. Kijowski, Q. Liu, Y. Zhang and X. Zhou), submitted

  4. Loomis–Whitney inequalities on corank 1 Carnot groups, (Y. Zhang), submitted

  5. Heat kernel asymptotics on the free step-two Carnot group with 3 generators, (H.-Q. Li, S.-C. Mao and Y. Zhang), submitted

  6. Curvature exponent and geodesic dimension on Sard-regular Carnot groups, (S. Nicolussi Golo and Y. Zhang), submitted

 

Accepted paper during April 1st, 2023 to March 31st, 2024

  1. Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces, (P. Lahti and X. Zhou), Analysis and Geometry in Metric Spaces (to appear)

  2. Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces, (P. Lahti and X. Zhou), Tohoku Mathematical Journal (to appear)

  3. A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces, (P. Lahti,  A. Pinamonti, and X. Zhou), Nonlinear Analysis (to appear)

4.2 Books and other one-time publications

4.3 Oral and Poster Presentations

  1. Seminar, Shandong University Research Center for Mathematics and Interdisciplinary Sciences (online), X. Zhou, Oct 20th, 2023

  2. OIST Workshop Geometric Aspects of Partial Differential Equations, X. Zhou, January 15-18, 2024

  3. OIST Analysis and Partial Differential Equations Summer School, X. Zhou, June 12 – 17, 2023 

  4. The 3rd Global Youth Forum, University of Science and Technology of China, Y. Zhang, January 6, 2024

  5. The Mathematical Society of Japan Spring Meeting 2024, Osaka Metropolitan University, Y. Zhang, March 19, 2024

  6. The Mathematical Society of Japan Autumn Meeting 2023, Tohoku University, Y. Zhang, September 20, 2023

  7. (Poster) Processes and heat kernels with symmetries, The Henri Lebesgue Center, Y. Zhang, June 26-June 30, 2023

  8. (Poster) Sub-Riemannian geometry and beyond, III, The Centro di Ricerca Matematica Ennio De Giorgi, Y. Zhang, June 19-June 23, 2023

  9. Potential theory and random walks in metric spaces, OIST, Y. Zhang, June 2, 2023

  10. The Mathematical Society of Japan Spring Meeting 2024, Osaka Metropolitan University, N. Evseev, March 19, 2024

5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

6. Meetings and Events

6.1 【Workshop】Analysis on Metric Spaces Workshop 2023

  • Date: May 30, 2023 - June 2, 2023
  • Venue: OIST Campus B250 and online
  • Organizers: Sylvester Eriksson-Bique (University of Jyväskylä), Panu Lahti (Chinese Academy of Sciences)

6.2【Seminar】Critical Sobolev Spaces and Subspaces of BMO

  • Date: June 19, 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Daniel Spector (National Taiwan Normal University)

6.3.1【Mini - Course I】Lectures on Capacities | Riemann and Lebesgue Integration

  • Date: June 20, 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Daniel Spector (National Taiwan Normal University)

6.3.2【Mini - Course I】Lectures on Capacities | Capacitary Integration

  • Date: June 21, 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Daniel Spector (National Taiwan Normal University)

6.3.3【Mini - Course I】Lectures on Capacities | Capacitary Sobolev Inequalities and Applications

  • Date: June 22, 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Daniel Spector (National Taiwan Normal University)

6.4.1【Mini - Course II】Metric geometry on the configuration space Part 1

  • Date: Auguest 29 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Kohei Suzuki (Durham University)

6.4.2【Mini - Course II】Metric geometry on the configuration space Part 2

  • Date: Auguest 30 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Kohei Suzuki (Durham University)

6.4.3【Mini - Course II】Metric geometry on the configuration space Part 3

  • Date: Auguest 31 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Kohei Suzuki (Durham University)

6.4.4【Mini - Course II】Metric geometry on the configuration space Part 4

  • Date: September 1 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Kohei Suzuki (Durham University)

6.5【Seminar】Thompson's groups, several generalizations and divergence property　

  • Date: December 13, 2023
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Xiaobing Sheng,(OIST)

6.6 【Seminar】Different definitions of conformal dimension are (essentially) equal!

  • Date: January 22, 2024
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Sylvester Eriksson-Bique, (University of Jyväskylä and OIST TSVP Visiting Scholar)

6.7.1【Mini - Course III】Metric Embeddings - What, How and Why? Part 1

  • Date: February 21, 2024
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Sylvester Eriksson-Bique (University of Jyväskylä)

6.7.2【Mini - Course III】Metric Embeddings - What, How and Why? Part 2

  • Date: February 26, 2024
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Sylvester Eriksson-Bique (University of Jyväskylä)

6.7.3【Mini - Course III】Metric Embeddings - What, How and Why? Part 3

  • Date: February 28, 2024
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Sylvester Eriksson-Bique (University of Jyväskylä)

6.7.4【Mini - Course III】Metric Embeddings - What, How and Why? Part 4

  • Date: March 1, 2024
  • Venue: OIST Campus Lab 4 and online
  • Speaker: Sylvester Eriksson-Bique (University of Jyväskylä)

 

7. Other

Grant: JSPS Grant-in-Aid for Early-Career Scientists (April 2024-March 2027)

  • PI: Ye Zhang, OIST
  • Title: Weak notions of curvature-dimension conditions on step-two Carnot groups