Tian-Jun Shen

Riesz transform on manifolds with mixed ends for $1

Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R<\infty$, $$c_i\l...