The Resolvent and Riesz Transform on Connected Sums of Manifolds with Different Asymptotic Dimensions

Abstract

We consider the class of manifolds obtained by taking the connected sum of a finite number of N-dimensional Riemannian manifolds of the form (R ni , δ) × (Mi , g), where Mi is a closed manifold, equipped with the product metric. The case of greatest interest is when the Euclidean dimensions ni are not all equal. This means that the ends have different ‘asymptotic dimension’, and implies that these connected sums are non-doubling spaces. In this thesis, we take a connected sum of two such product manifolds and assume that one of the Euclidean dimensions, ni , is equal to 2, which is a special case. Our approach is to construct the low energy resolvent and determine the asymptotics of the resolvent kernel as the energy tends to zero. The interesting feature of this case is the logarithmic behaviour of the resolvent kernel on the end with Euclidean dimension 2. We express the Riesz transform in terms of the resolvent to show that it is bounded on L p for 1 < p ≤ 2. This extends results from a paper of Hassell and Sikora [51] which considered connected sums of products of Euclidean spaces with closed manifolds, where each Euclidean space has a dimension of at least 3. We also analyse 1-dimensional models of connected sums, which extends the work of Hassell and Sikora [50]. We consider the real line, which is equipped with a weighted Lebesgue measure with different power behaviour at ±∞. This space mimics higher dimensional connected sums considered above in the sense of the Laplacian being restricted to radial functions. In the case that the negative half-line is weighted like |r| dr and the positive half-line is weighted like r d+−1 dr, d+ ≥ 3, we show the Riesz transform is L p -bounded for the range 1 < p ≤ 2, which agrees with our multidimensional case. Show more