Research Research

Jie Xiao

Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John's, NL A1C 5S7, Canada

jxiao@mun.ca

My research interests lie in analysis and partial differential equations with connections to Riemannian geometry. In what follows I roughly divide my research contributions into two parts, with results from the second part being used to partially motivate problems attacked in the first part.


Geometric Harmonic Analysis and Partial Differential Equations (PDEs): An interesting problem in this area is the investigation of fractional energy which is mathematically determined by an integral of quotient of symmetric differences with fractional order. I have been interested in the variations of fractional energy under all linear motions in $\Bbb R^n$, and consequently, the study of Besov/Sobolev spaces and their affine-invariant forms, as well as their applications to conformal/convex/differential geometry and PDEs. Several results on these are known as: harmonic extensions, mean oscillations, wavelets and dualities associated with new tent-like spaces, Hausdorff contents, capacitary strong-type inequalities, sharp isoperimetric-type estimates, sharp Sobolev-type inequalities, a prior estimates of geometric Green's functions, and connections to the heat-type equations, the Navier-Stokes equations and other time-dependent equations.


Complex Variables and Functional Analysis: One very fundamental problem in this area is the discovery of the conformal deformations of weighted image areas of holomorphic functions. The process of doing so has led to the study of $Q_p$ space properties including: series expansions, pseudo-holomorphic extensions, representations via $\bar{\partial}$-estimates, composition embeddings and their geometric variants, canonical decomposition, dual spaces, Hausdorff capacity, symbol spaces, and many others.


  • For a list of my publications please see American Mathematical Society - MathSciNet - Mathematical Reviews