Riesz transform on manifolds with ends of different volume growth for $1<p<2$

Abstract: Let $M_1$, $\cdots$, $M_\ell$ be complete, connected and non-collapsed manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$, and suppose that each $M_i$ satisfies a doubling condition and a Gaussian upper bound for the heat kernel. If each manifold $M_i$ has volume growth either bigger than two or equal to two, then we show that the Riesz transform $\nabla \L^{-1/2}$ is bounded on $L^p(M)$ for each $1<p<2$ on the gluing manifold $M=M_1#M_2#\cdots # M_\ell$.