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Geometry and the Kato square root problem

Bandara, Lashi

Description

The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A...[Show more]

dc.contributor.authorBandara, Lashi
dc.date.accessioned2013-11-05T03:19:39Z
dc.date.available2013-11-05T03:19:39Z
dc.identifier.otherb32355002
dc.identifier.urihttp://hdl.handle.net/1885/10690
dc.description.abstractThe primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem. In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.
dc.language.isoen_AU
dc.subjectKato square root problem
dc.subjectquadratic estimates
dc.subjectelliptic operator
dc.subjectLipschitz estimates
dc.subjectessentially self-adjoint
dc.subjectvector bundle
dc.subjectmeasure metric space
dc.subjectbounded measurable coefficients
dc.subjectHodge-Dirac operator
dc.titleGeometry and the Kato square root problem
dc.typeThesis (PhD)
local.contributor.supervisorMcIntosh, Alan
local.contributor.supervisorcontactalan.mcintosh@anu.edu.au
dcterms.valid2013
local.description.notesSupervisor: Alan McIntosh. Supervisor's Email Address: alan.mcintosh@anu.edu.au
local.description.refereedYes
local.type.degreeDoctor of Philosophy (PhD)
dc.date.issued2013
local.contributor.affiliationCollege of Physical & Mathematical Sciences
local.identifier.doi10.25911/5d77879796b8e
local.mintdoimint
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