The resolvent and Riesz transform on connected sums of manifolds with different asymptotic dimensions. (Abstract of thesis). (English) Zbl 1492.58004
Summary: We consider the class of manifolds obtained by taking the connected sum of a finite number of \(N\)-dimensional Riemannian manifolds of the form \((\mathbb{R}^{n_i}, \delta)\times (\mathcal{M}_i, g)\), where \(\mathcal{M}_i\) is a closed manifold, equipped with the product metric. The case of greatest interest is when the Euclidean dimensions \(n_i\) are not all equal. This means that the ends have different ‘asymptotic dimension’, and it implies that these connected sums are nondoubling spaces.
In this thesis, we take a connected sum of two such product manifolds and assume that one of the Euclidean dimensions, \(n_i\), 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 two. We express the Riesz transform in terms of the resolvent to show that it is bounded on \(L^p\) for \(1 < p \leq 2\). This extends results from a paper of A. Hassell and A. Sikora [Indiana Univ. Math. J. 58, No. 2, 823–852 (2009; Zbl 1187.42008)] which considered connected sums of products of Euclidean spaces with closed manifolds, where each Euclidean space has a dimension of at least three.
We also analyse one-dimensional models of connected sums, which extends the work of A. Hassell and A. Sikora [Commun. Partial Differ. Equations 44, No. 11, 1072–1099 (2019; Zbl 1440.42055)]. We consider the real line, which is equipped with a weighted Lebesgue measure with different power behaviour at \(\pm\infty\). This space mimics higher-dimensional connected sums considered above in the sense of the Laplacian being restricted to radial functions. In the case where the negative half-line is weighted like \(|r|dr\) and the positive half-line is weighted like \(r^{d_+-1}dr\), \(d_+ \geq 3\), we show that the Riesz transform is \(L^p\)-bounded for the range \(1 < p \leq 2\), which agrees with our multidimensional case.
In this thesis, we take a connected sum of two such product manifolds and assume that one of the Euclidean dimensions, \(n_i\), 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 two. We express the Riesz transform in terms of the resolvent to show that it is bounded on \(L^p\) for \(1 < p \leq 2\). This extends results from a paper of A. Hassell and A. Sikora [Indiana Univ. Math. J. 58, No. 2, 823–852 (2009; Zbl 1187.42008)] which considered connected sums of products of Euclidean spaces with closed manifolds, where each Euclidean space has a dimension of at least three.
We also analyse one-dimensional models of connected sums, which extends the work of A. Hassell and A. Sikora [Commun. Partial Differ. Equations 44, No. 11, 1072–1099 (2019; Zbl 1440.42055)]. We consider the real line, which is equipped with a weighted Lebesgue measure with different power behaviour at \(\pm\infty\). This space mimics higher-dimensional connected sums considered above in the sense of the Laplacian being restricted to radial functions. In the case where the negative half-line is weighted like \(|r|dr\) and the positive half-line is weighted like \(r^{d_+-1}dr\), \(d_+ \geq 3\), we show that the Riesz transform is \(L^p\)-bounded for the range \(1 < p \leq 2\), which agrees with our multidimensional case.
MSC:
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
| 47A10 | Spectrum, resolvent |
| 47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |
References:
| [1] | Hassell, A. and Sikora, A., ‘Riesz transforms in one dimension’, Indiana Univ. Math. J.58(2) (2009), 823-852. · Zbl 1187.42008 |
| [2] | Hassell, A. and Sikora, A., ‘Riesz transforms on a class of non-doubling manifolds’, Comm. Partial Differential Equations44(11) (2019), 1072-1099. · Zbl 1440.42055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
