Riesz transform on manifolds with mixed ends for $1<q<2$

Let $M_1$, $\cdots$, $M_{\mathrm{\ell }}$ be complete manifolds of the same dimension, where $2\le \mathrm{\ell }\in \mathbb{N}$. 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 $1\le r\le R<\mathrm{\infty }$, $$c_i{\left(\frac{R}{r}\right)}^{n_i}\le \frac{Vo{l}_{M_i}(B(x_i,R))}{Vo{l}_{M_i}(B(x_i,r))}\le C_i{\left(\frac{R}{r}\right)}^{n_i},$$ by assuming that all the manifolds satisfy the relative connectedness of the annuli ($RCA$) condition introduced by Grigor’yan and Saloff-Coste, we show that the Riesz transform $\mathrm{\nabla }{\mathcal{L}}^{-1\mathrm{/}2}$ is bounded on $L^q(M)$ for each $1<q<2$ on the gluing manifold $M=M_1\mathrm{\#}{M}_2\mathrm{\#}\cdots \mathrm{\#}{M}_{\mathrm{\ell }}$.