The boundedness of the Riesz transform on a metric cone

Abstract

In this thesis we study the boundedness, on L(M), of the Riesz transform T associated to a Schrodinger operator with an inverse square potential V = V0 r2 on a metric cone M defined by T = ∇ ( ∆ + V0(y) r2 )− 1 2 . Here M = Y × [0,∞)r has dimension d ≥ 3, and the smooth function V0 on Y is restricted to satisfy the condition ∆Y + V0(y) + ( d−2 2 ) > 0, where ∆Y is the Laplacian on the compact Riemannian manifold Y . The definition of T involves the Laplacian ∆ on the cone M . 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, T can be written as an integral involving the expression (∆ + V0(y) r2 + λ2)−1. Therefore if we understand the resolvent kernel of the Schrodinger operator ∆ + V0(y) r2 , we have information about T . We construct and at the same time collect information about this resolvent kernel, and then use the information to study the boundedness of T . The two most interesting parts in the construction of the resolvent kernel are the behaviours of the kernel as r, r′ → 0 and r, r′ → ∞. 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 as r, r′ → 0, while the scattering calculus is used as r, r′ →∞. The main result of this thesis provides a necessary and sufficient condition on p for the boundedness of T on L(M). The interval of boundedness depends on V0 through the first and second eigenvalues of ∆Y + V0(y) + ( d−2 2 ). • When the potential function V is positive, we have shown that the lower