Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, I

We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schr\"odinger operators $P=\Delta+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null space and no resonance at 0. In particular, we show that the Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a desingularized version of $M\times M\times [0,1]$. Using this, we show that the Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that this range is optimal if $V$ is not identically zero or if $M$ has more than one end. We also analyze the case V=0 with one end. In a follow-up paper, we shall deal with the same problem in the presence of zero modes and zero-resonances.Comment: 28 pages, 1 figur

A generalization of Strassen's Positivstellensatz

Strassen's Positivstellensatz is a powerful but little known theorem on preordered commutative semirings satisfying a boundedness condition similar to Archimedeanicity. It characterizes the relaxed preorder induced by all monotone homomorphisms to $\mathbb{R}_+$ in terms of a condition involving large powers. Here, we generalize and strengthen Strassen's result. As a generalization, we replace the boundedness condition by a polynomial growth condition; as a strengthening, we prove two further equivalent characterizations of the homomorphism-induced preorder in our generalized setting.Comment: 24 pages. v6: condition (d) in Theorem 2.12 has been correcte