## Abstract

We consider a class of manifolds M obtained by taking the connected sum of a finite number of *N*-dimensional Riemannian manifolds of the form (R^{ni},δ)×(M_{i},g), where M_{i} is a compact manifold, 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 implies that the Riemannian manifold M is not a doubling space. We completely describe the range of exponents

*p*for which the Riesz transform on M is a bounded operator on L

^{p}(M). Namely, under the assumption that each

*n*is at least 3, we show that Riesz transform is of weak type (1, 1), is continuous on

_{i}*L*for all p∈(1,min

^{p}_{i}n

_{i}), and is unbounded on

*L*otherwise. This generalizes results of the first-named author with Carron and Coulhon devoted to the doubling case of the connected sum of several copies of Euclidean space R

^{p}^{N}, and of Carron concerning the Riesz transform on connected sums.

Original language | English |
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Pages (from-to) | 1072-1099 |

Number of pages | 28 |

Journal | Communications in Partial Differential Equations |

Volume | 44 |

Issue number | 11 |

Early online date | 6 Jun 2019 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- heat kernel bounds
- non-doubling spaces
- resolvent estimates
- Riesz transform