# Po Lam Yung

## Contacts

### Research interests

- Lie Groups, Harmonic And Fourier Analysis
- Partial Differential Equations

I work in a branch of mathematics called harmonic analysis, with particular focus on singular integrals, Sobolev embeddings, time-frequency analysis, oscillatory integrals and Fourier decoupling. The goal is to obtain delicate estimates for various operators between function spaces, often motivated by the study of partial differential equations, several complex variables, Cauchy-Riemann (CR) geometry and analytic number theory.

More precisely, I am interested in sharpening some results of Bourgain and Brezis about embeddings of certain critical Sobolev spaces, and some related cancelling / cocancelling *L*^{1} estimates of Van Schaftingen. I am also developing a pseudodifferential calculus, involving operators of mixed homogeneities, with applications to several complex variables and CR geometry. Furthermore, I am interested in problems from time-frequency analysis, particularly those related to Stein-Wainger type oscillatory integrals. Recently I have been working on Fourier decoupling inequalities and related applications.

### Groups

- Researcher, Analysis & geometry

- Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail

(with Haim Brezis and Jean Van Schaftingen),

to appear in Calc. Var. Partial Differential Equations. [article] - Improved discrete restriction for the parabola

(with Shaoming Guo and Zane Kun Li),

submitted. [article] - A new formula for the L
^{p}norm

(with Qingsong Gu),

J. Funct. Anal. 281 (2021), no. 4, 109075. [article] - Limiting Sobolev and Hardy inequalities on stratified homogeneous groups

(with Jean Van Schaftingen),

submitted. [article] - A surprising formula for Sobolev norms

(with Haim Brezis and Jean Van Schaftingen),

Proc. Natl. Acad. Sci. 118 (2021) no. 8, e2025254118. [article] - A short proof of ℓ
^{2}decoupling for the moment curve

(with Shaoming Guo, Zane Kun Li and Pavel Zorin-Kranich),

to appear in Amer. J. Math. [article] - Decoupling for two quadratic forms in three variables: a complete characterization

(with Shaoming Guo, Changkeun Oh, Joris Roos and Pavel Zorin-Kranich),

submitted. [article] - A bilinear proof of decoupling for the cubic moment curve

(with Shaoming Guo and Zane Kun Li),

to appear in Trans. AMS. [article] - Reversing a philosophy: from counting to square functions and decoupling

(with Philip T. Gressman, Shaoming Guo, Lillian B. Pierce and Joris Roos),

J. Geom. Anal. (2021) In "Elias M. Stein: In Memoriam" [article] - Maximal functions associated with families of homogeneous curves:

L^{p}bounds for p ≤ 2

(with Shaoming Guo, Joris Roos and Andreas Seeger),

Proc. Edinb. Math. Soc. 63 (2020), no. 2, 398-412. [article] - A maximal function for families of Hilbert transforms along homogeneous curves

(with Shaoming Guo, Joris Roos and Andreas Seeger),

Math. Ann. 377 (2020), no. 1-2, 69-114. [article] - Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem

(with Shaoming Guo and Joris Roos),

Analysis & PDE 13 (2020), no. 5, 1457-1500. [article] - Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds

(with Chin-Yu Hsiao),

Calc. Var. Partial Differential Equations 58 (2019), no. 2, Art. 71, 62 pp. [article] - Approximation in higher-order Sobolev spaces and Hodge systems

(with Pierre Bousquet, Emmanuel Russ and Yi Wang),

J. Funct. Anal. 276 (2019), no. 5, 1430-1478. [article] - A polynomial Carleson operator along the paraboloid

(with Lillian B. Pierce),

Rev. Mat. Iberoam. 35 (2019), no. 2, 339-422. [article] - Polynomial Carleson operators along monomial curves in the plane

(with Shaoming Guo, Lillian B. Pierce and Joris Roos),

J. Geom. Anal. 27 (2017), no. 4, 2977-3012. [article] - Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type

(with Sagun Chanillo and Jean Van Schaftingen),

J. Funct. Anal. 273 (2017), no. 4, 1504-1547. [article] - The incompressible Navier-Stokes flow in two dimensions with prescribed vorticity

(with Sagun Chanillo and Jean van Schaftingen),

in Sagun Chanillo, Bruno Franchi, Guozhen Lu, Carlos Perez and Eric T. Sawyer (eds.), Harmonic Analysis, Partial Differential Equations and Applications, Birkhauser, Applied and Numerical Harmonic Analysis, 2017, 19--25. [article] - Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem

(with Sagun Chanillo and Jean Van Schaftingen),

Chin. Ann. Math. Ser. B 38 (2017), no. 1, 235-252. [article] - On a trilinear singular integral form with determinantal kernel

(with Philip T. Gressman, Danqing He, Vjekoslav Kovač, Brian Street, Christoph Thiele),

Proc. Amer. Math. Soc. 144 (2016), no. 8, 3465-3477. [article] - Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

(with Sagun Chanillo and Jean van Schaftingen),

C. R. Math. Acad. Sci. Paris, Vol. 354 (2016), no. 1, 51-55. [article] - Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

(with Chin-Yu Hsiao),

Adv. Math. 281 (2015), 734-822. [article] - A sharp subelliptic Sobolev embedding theorem with weights,

Bull. London Math. Soc. 47 (2015), no. 3, 396-406. [article] - Absence of Self-Similar Blow-up and Local Well-posedness for the Constant Mean-Curvature Wave Equation

(with Sagun Chanillo),

J. Funct. Anal. 269 (2015), no. 4, 1180-1202. [article] - A subelliptic Bourgain-Brezis inequality

(with Yi Wang),

J. Eur. Math. Soc 16 (2014), 649-693. [article] - Pseudodifferential operators of mixed type adapted to distributions of k-planes

(with Elias M. Stein),

Math. Res. Lett. 20 (2013), no. 6, 1183-1208. [article] - The tangential Cauchy-Riemann complex on the Heisenberg group Via Conformal Invariance

(with Chin-Yu Hsiao),

Bulletin of the Institute of Mathematics, Academia Sinica (New Series), Vol. 8 (2013), no. 3, 359-375. [article] - Wave Equations Associated to Liouville Systems and Constant Mean Curvature equations

(with Sagun Chanillo),

Adv. Math, Vol 235 (2013), 187-207. [article] - An improved Strichartz estimate for systems with divergence free data

(with Sagun Chanillo),

Comm. PDE., Vol 37 (2012), no. 2, 225-233. [article] - Sobolev inequality for (0,q) forms on CR manifolds of finite type,

Math. Res. Lett. 17 (2010), no. 1, 177-196. [article] - Doubling properties of self-similar measures,

Indiana Univ. Math. J. 56 No. 2 (2007), 965-990. [article] - Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

(with Jonathan Needleman, Robert Strichartz and Alexander Teplyaev),

J. Funct. Anal. 215 (2004), 290-340. [article]

Teaching at Australian National University

- Introduction to Geometry: Curves and Surfaces, Term 2 2020
- Analysis 1 Advanced Extension, Term 1 2020
- Topics course on Fourier Analysis, Term 1 2020

Teaching at the Chinese University of Hong Kong

- Math 1010A, University Mathematics, Fall 2018
- Math 4060, Complex Analysis, Fall 2018
- Math 6081A, Topics in Harmonic Analysis I, Fall 2017
- Math 1010B, University Mathematics, Fall 2017
- Math 1010A, University Mathematics, Fall 2017
- Math 4900F, Capstone Seminar, Fall 2016
- Math 4060, Complex Analysis, Fall 2016
- Math 1010C, University Mathematics, Fall 2016
- Math 4900E, Capstone Seminar, Fall 2015
- Math 4060, Complex Analysis, Fall 2015
- Math 1010C, University Mathematics, Fall 2015
- Math 2060B, Elementary Analysis II, Spring 2015
- Math 1010C, University Mathematics, Fall 2014

Teaching at Oxford (as a Teaching Assistant)

- 2 sets of B4b. Hilbert spaces, Hilary term, 2014
- 2 sets of B5b. Applied PDEs, Hilary term, 2014
- 2 sets of B4a. Banach spaces, Michaelmas term, 2013
- 2 sets of C2.1a. Lie Algebras, Michaelmas term, 2013

Teaching at Rutgers, the State University of New Jersey

- Math 494, Independent Study in Complex Analysis, Spring 2013
- Math 135, Calculus I, Sections 09-11, Fall 2012
- Math 244, Differential Equations for Engineering and Physics, Sections 01-03, Fall 2012
- Math 252, Elementary Differential Equations, Section 02, Spring 2012
- Math 350, Linear Algebra, Sections 01-02, Fall 2011
- Math 403, Introduction to Theory of Functions of a Complex Variable, Section 02, Spring 2011
- Math 152, Calculus II for the Mathematical and Physical Sciences, Sections 01-03 and 01-09, Fall 2010

Teaching at Princeton (as a Teaching Assistant)

- Princeton Summer School in Analysis and Geometry '09, Several Complex Variables, Summer 2009
- MAT 330, Fourier Analysis, Spring 2009
- MAT 218, Analysis in Several Variables, Fall 2008
- MAT 102, Calculus with Analytic Geometry II, Spring 2008
- Princeton Summer School in Analysis and Geometry '07, Harmonic Analysis, Summer 2007