Kato Square Root Problem on Riemannian Symmetric Spaces

The resolution of the Kato Square Root problem by Auscher-Hofmann-Lacey-McIntosh-Tchamitchian in 2002, which took in excess of 30 years, was a major achievement in mathematics. First proposed by Kato in the 1960s, the problem was to identify the domain of the square root of elliptic differential operators with complex and measurable coefficients in divergence form. Such equations arise naturally in partial differential equations and have applications to problems in engineering and physics.; ;Novel methods linking operator theory and real-variable harmonic analysis were developed in its resolution. These methods have become more widely applicable in signal processing through wavelet theory. A generalisation of this problem was studied and resolved in the setting of Lie groups in 2012 by Bandara-ter Elst-McIntosh. This concerned a subelliptic version of the problem, arising from algebraic vector fields of a dimension less than that of the Lie group. At the same time, geometric generalisations were also obtained but in the elliptic setting.; ;This leaves a major research gap in the geometric subelliptic setting.The aim of the proposed project would be to study this problem for subelliptic geometric operators in non-flat Riemannian symmetric spaces,  which are homogeneous spaces of semisimple Lie groups . Its objectives are to identify sufficiently general geometric conditions under which the subelliptic Kato square root problem admits a resolution. This would likely begin with considering the special case of curved Lie groups before moving on to the more general setting of Riemannian symmetric spaces.;;Riemannian symmetric spaces constitute a beautiful and important class of Riemannian manifolds where the rich interplay between the Lie theoretic structures and the Riemannian geometric aspects have historically made them (and indeed, continue to make them) a very attractive setting to analyse various questions of geometric analysis. In particular, the richness of the Lie theoretic and geometric structures often facilitate that calculations can be made in very explicit terms. We think that this would be the case with the proposed problems as well.; ;While one of the investigators (LB) has, in fact, contributed to progress on analogous problems in related geometric contexts, the other one (AP) has worked extensively on various facets of geometry (including complex geometry) and (microlocal, spectral) analysis on symmetric spaces. The complementarity of the areas of our expertise in the context of the proposed problems makes us, in our opinion, very well positioned to guide a PhD student to resolve these important and of current questions of relevance in geometry and analysis.