The boundedness of the Riesz transform on a metric cone

Abstract

In this thesis we study the boundedness, on L-p spaces, of the Riesz transform associated to a Schroedinger operator with an inverse square potential on a metric cone of dimension greater or equal to 3. The definition of the Riesz transform involves the Laplacian on the cone. However, the cone is not a manifold at the cone tip, so we initially define the Laplacian away from the cone tip, and then consider its self-adjoint extensions. The Friedrichs extension is adopted as the definition of the Laplacian. Using functional calculus, we can express the Riesz transform in terms of the resolvent kernel of the Schroedinger operator. Therefore we construct and at the same time collect information about this resolvent kernel, and then use the information to study the boundedness of the Riesz transform. The two most interesting parts in the construction of the resolvent kernel are the behaviours of the kernel as both the left and right variables approach the cone tip, and as both the left and right variables approach infinity. To study them, a process called the blow-up is performed on the domain of the kernel. We use the b-calculus to study the kernel near the cone tip, while the scattering calculus is used near infinity. The main result of this thesis provides a necessary and sufficient condition on p for the boundedness of the Riesz transform on the space of L-p functions on the metric cone. When the potential function is positive, we have shown that the lower threshold is 1, and the upper threshold is strictly greater than the dimension d; when the potential function is negative, we have shown that the lower threshold is strictly greater than 1, and the upper threshold is strictly between 2 and d. Our results for p less or equal to 2 are contained in the work of J. Assaad, but we use different methods in this thesis. Our boundedness results for p greater or equal to d over 2 for positive inverse square potentials, and for p greater than 2 for negative inverse square potentials, are new.