Summary of Research
The analysis research group focuses on harmonic analysis, complex analysis, and operator theory. One particular focus is on harmonic analysis on weighted Lebesgue spaces, and on variable exponent Lebesgue spaces.
Prof. Bortz’ research is at the interface of harmonic analysis, partial differential equations and geometric measure theory. In particular, he is interested in free boundary problems and properties of harmonic and caloric functions in rough settings. In addition, he studies boundary value problems for elliptic and parabolic differential equations and analysis in metric spaces. The tools in Prof. Bortz’ work can often be traced back to the Calderón-Zygmund school of harmonic analysis.
Professor Cruz-Uribe’s research is in harmonic analysis. He is particularly interested in the study of the classical operators of harmonic analysis—maximal operators, the Hilbert transform and other singular integrals, Riesz potentials/fractional integrals—on weighted and variable exponent Lebesgue spaces. These kinds of spaces occur naturally in situations where there is a lack of homogeneity in the underlying problem being studied. He also works on the theory of Rubio de Francia extrapolation, which allows weighted classical norm inequalities to be extended to a variety of function space settings.
Finally, he is interested in the applications of harmonic analysis to the study of partial differential equations, particularly degenerate elliptic equations.
Find Dr. Cruz-Uribe’s publications on MathSciNet.
Much of Prof. Ferguson’s research is in complex analysis, specifically involving the study of spaces of analytic functions. This has connections to operator theory and harmonic analysis, and he is interested in all of these areas in general. One area of particular interest to him is the study of extremal problems in spaces of analytic functions, especially in Bergman spaces. He is also interested in the connections between geometric complex analysis and spaces of analytic functions, like in his study (with W. Ross) of the range and valences of functions in the Smirnov class with real boundary values. A new topic of interest for him is Hardy and Bergman spaces with variable exponents.
Prof. Moen’s research centers around a change of density in classical problems of harmonic analysis and PDE. This area of study is weighted norm inequalities. He is particularly interested in sharp weighted estimates and two-weight estimates for classical operators such as the Hardy-Littlewood maximal operator and Calderón-Zygmund operators. He is also interested in the applications to PDE and in particular elliptic PDE. Finally, Prof. Moen also studies quasi-regular mappings and mappings of finite distortion.
Prof. Shao’s research has been primarily focusing on partial differential equations and geometric analysis. Currently, he is particularly interested in the study of differential operator theory on non-compact especially singular manifolds as well as its applications to geometry (including the Yamabe flow, the harmonic map heat flow, the mean curvature flow) and image processing (non-local denoising models).
Prof. Shao is also working on non-local diffusion (including the fractional porous medium equation), free boundary problems (including the Stefan problem, two-phase Navier-Stokes equations), and relativistic viscous fluids.