混合端流形上的Riesz变换（$1<q<2$）
Riesz transform on manifolds with mixed ends for $1<q<2$
中文总结 AI 辅助
本文证明在满足双侧高斯界和RCA条件的混合端流形粘合体上，Riesz变换在$L^q$空间（$1<q<2$）有界。
中文摘要 AI 辅助
设 $M_1$, $\cdots$, $M_\ell$ 为同维数的完备流形，其中 $2\le \ell\in\mathbb{N}$。假设每个 $M_i$ 满足双侧高斯界。若其中一些流形是抛物型的，且存在常数 $1\le n_i\le 2$，使得对某个 $x_i\in M_i$ 及 $1 \le r \le R < \infty$，有 $$c_i\left(\frac{R}{r}\right)^{n_i}\le \frac{Vol_{M_i}(B(x_i,R))}{Vol_{M_i}(B(x_i,r))}\le C_i\left(\frac{R}{r}\right)^{n_i},$$ 在假设所有流形满足 Grigor'yan 和 Saloff-Coste 引入的环带相对连通性（$RCA$）条件的前提下，我们证明在粘合流形 $M=M_1 \\# M_2 \\# \cdots \\# M_\ell$ 上，对于每个 $1<q<2$，Riesz 变换 $\nabla \mathcal L^{-1/2}$ 在 $L^q(M)$ 上有界。
英文摘要
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 $1 \le r \le R < \infty$, $$c_i\left(\frac{R}{r}\right)^{n_i}\le \frac{Vol_{M_i}(B(x_i,R))}{Vol_{M_i}(B(x_i,r))}\le C_i\left(\frac{R}{r}\right)^{n_i},$$ 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 $\nabla \mathcal L^{-1/2}$ is bounded on $L^q(M)$ for each $1<q<2$ on the gluing manifold $M=M_1 \# M_2 \# \cdots \# M_\ell$.
