Let $M$ be a complete non-compact manifold satisfying the volume doubling condition, with doubling index $N$ and reverse doubling index $n$, $n\le N$, both for large balls. Assume a Gaussian upper bound for the heat kernel, and an $L^2$-Poincare inequality outside a compact set. If $2 2$ on manifolds having at least two Euclidean ends of dimension $n$. For $p\in (\max\{N,2\},\infty)$, the fact that $(R_p)$, $(G_p)$ and $(RH_p)$ are equivalent essentially follows from [22]; moreover, if $M$ is non-parabolic, then any of these conditions implies that $M$ has only one end. For the proof, we develop a new criteria for boundedness of the Riesz transform, which was nontrivially adapted from [4], and make an essential application of results from [22]. Our result allows extensions to non-smooth settings.

中文翻译：

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通过非紧流形上的热核和调和函数进行 Riesz 变换

令$M$为满足体积倍增条件的完全非紧流形，倍增指数$N$和反向倍增指数$n$，$n\le N$，均适用于大球。假设热核有一个高斯上限，并且在紧集外有一个 $L^2$-Poincare 不等式。如果在具有至少两个维度为 $n$ 的欧几里得端的流形上 $2 2$。对于$p\in(\max\{N,2\},\infty)$，$(R_p)$、$(G_p)$和$(RH_p)$等价的事实本质上来自[22]；此外，如果 $M$ 是非抛物线的，那么这些条件中的任何一个都意味着 $M$ 只有一端。为了证明，我们为 Riesz 变换的有界性开发了一个新标准，该标准从 [4] 中改编而来，并对 [22] 中的结果进行了必要的应用。我们的结果允许对非平滑设置进行扩展。