Riesz transform via heat kernel and harmonic functions on non-compact manifolds

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 $L^2$-Poincar\'e inequality outside a compact set. If $2<n$, then we show that for $p\in (2,n)$, $(R_p)$: $L^p$-boundedness of the Riesz transform, $(G_p)$: $L^p$-boundedness of the gradient of the heat semigroup, and $(RH_p)$: reverse $L^p$-H\"older inequality for the gradient of harmonic functions, are equivalent to each other. Our characterization implies that for $p\in (2,n)$, $(R_p)$ has an open ended property and is stable under gluing operations. This substantially extends the well known equivalence of $(R_p)$ and $(G_p)$ from [4] to more general settings, and is optimal in the sense that $(R_p)$ does not hold for any $p\ge n>2$ 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.