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.

More information is available on my webpage: https://maths-people.anu.edu.au/~plyung/.