论文标题
Riesz在流形的转换中，其末端增长的末端为$ 1 <p <2 $
Riesz transform on manifolds with ends of different volume growth for $1<p<2$
论文作者
论文摘要
令$ m_1 $，$ \ cdots $，$ m_ \ ell $是完整的，连接的，并且连接和非collapsed的歧管，其中$ 2 \ le \ ell \ in \ mathbb {n} $，并假设每个$ m_i $ $ m_i $都满足双倍的状态，并且满足热量上的高速公路上的热级别。如果每个歧管$ m_i $具有大于两个或等于两个的数量增长，那么我们表明riesz变换$ \ nabla^{ - 1/2} $在$ 1 <p <2 $的$ l^p（m）$上限制在粘合折线上的$ l^p（m）$，gluing perford $ m = m_1 \ m_1 \＃m_2 \＃m_2 \＃m_2 \＃\ cdots \ cdots \ ell $ m_ m_ m_ m_ m_ m_ m_ \ y $ \ y $。
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 $\nabla Ł^{-1/2}$ is bounded on $L^p(M)$ for each $1<p<2$ on the gluing manifold $M=M_1\#M_2\#\cdots \# M_\ell$.
