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