Maximal Subellipticity

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Elliptic partial differential equations (PDE) play a central role in many areas of mathematics and science. A canonical example of an elliptic PDE is Laplace’s equation, which governs steady-state temperature distributions. One reason that elliptic equations are so useful is that precise results are known for very general elliptic PDE, even in the notoriously difficult setting of fully nonlinear equations. Outside of the elliptic setting, current techniques usually require the use of special properties of the equation under consideration, and abstract general results are rare. The theory of maximally subelliptic equations, a far-reaching generalization of elliptic equations, originated in the 1960s and 1970s. In the intervening years, many mathematicians have adapted results from the elliptic theory to various special cases of linear maximally subelliptic equations. This project will develop the regularity theory of linear maximally subelliptic PDE in full generality, and moreover will address the general situation of fully nonlinear maximally subelliptic PDE. This will provide a toolbox, more general than the usual one from the elliptic theory, for mathematicians and scientists who encounter such partial differential equations in their work. The project will provide research opportunities for graduate students.This project will develop the theory of maximally subelliptic partial differential equations in three main steps. The first is a study of general linear maximally subelliptic partial differential operators with smooth coefficients. Special cases have previously been considered, but this will be the first such theory of these operators in full generality. A key tool which will be used is the underlying Carnot-Caratheodory geometry along with associated scaling maps. The next stage of the project will be a development of the theory of Besov and Triebel-Lizorkin function spaces adapted to maximally subelliptic operators. An important property of elliptic operators is that, modulo smooth functions, they are left invertible on many classical function spaces. The aforementioned Besov and Triebel-Lizorkin spaces will generalize this fact to the maximally subelliptic setting. Special cases of these function spaces include both Sobolev and Zygmund-Holder spaces adapted to a maximally subelliptic operator. The third stage of the project will involve a study of fully nonlinear partial differential equations. The theory of function spaces and linear operators as described above will be used to understand the interior regularity of fully nonlinear maximally subelliptic equations. Outside of the elliptic setting, fully nonlinear equations are often difficult to study. The results of this project will provide a framework for future study of fully nonlinear equations in the maximally subelliptic setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.