The proposed research project includes solving problems in two main directions: sharp geometric inequalities and multiparameter harmonic analysis. Recent developments in the area of sharp geometric inequalities include best constants for Moser-Trudinger inequalities on the entire Heisenberg group and more general Carnot groups and Adams inequalities on high order Sobolev spaces on unbounded domains in Euclidean spaces. These are circumstances where symmetrization properties do not hold. The PI, in collaboration with his PhD students, have very recently succeeded in developing a rearrangement-free argument. This new method suggests that such sharp geometric inequalities can be established in more general scenarios including Riemannian and sub-Riemannian manifolds. Moreover, the PI will investigate the existence of extremal functions for these sharp geometric inequalities where many challenging problems still remain open. Another main direction of research is to develop multiparameter harmonic analysis function space theory in several complicated but important multiparameter settings. The PI, in collaboration with others, has developed a satisfactory theory of discrete Littlewood-Paley square functions in a number of multiparameter scenarios. However, there are still many other important multiparameter settings where such a discrete Littlewood-Paley theory is yet to be established.
Multi parameter Harmonic analysis and nonlinear partial differential equations are central areas of modern mathematics. Findings and new tools discovered in this project may lead to new development in the area of classical harmonic analysis, partial differential equations as well as other branches of mathematics. They have many applications in sciences and engineering. The solution to the proposed project will have impact on many other disciplines, including mechanics engineering (such as vibration and noise reduction for vehicles), imaging processing and pattern recognitions in medical sciences, stochastic control and optimization, game theory, chemical combustion, human vision and other topics in the life and medical sciences. Moreover, this project has a substantial training and educational component. It finely integrates research together with education. Many graduate students will actively participate in this project by receiving research training under the supervision of the principal investigator.
