Let $M^{\circ }$ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^{\circ }$, in the sense that $M^{\circ }$ compactifies to a manifold with boundary $M$ so that $g$ becomes a scattering metric on $M$. We study the resolvent kernel ${(P+k^2)}^{-1}$ and Riesz transform $T$ of the operator $P={\Delta }_g+V$, where ${\Delta }_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function smooth on $M$ and vanishing at the boundary.In our first paper we assumed that $P$ has neither zero modes nor a zero-resonance and showed...

We study the Riesz means for the eigenfunction expansions of a class of hypoelliptic differential operators on the Heisenberg group. The operators we consider are homogeneous with respect to dilations and invariant under the action of the unitary group. We obtain convergence results in $L^p$ norm, at Lebesgue points and almost everywhere. We also prove localization results.

We show that the $L^p$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

We study the validity of the $L^p$ inequality for the Riesz transform when $p\>2$ and of its reverse inequality when $1\<p\<2$ on complete riemannian manifolds under the doubling property and some Poincaré inequalities.

We prove $L^p$-bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.

Assume that $M_0$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_0$ satisfies a Sobolev inequality of dimension $\nu \>3$. Let $M$ be a complete Riemannian manifold isometric at infinity to $M_0$ and let $p\in (\nu /(\nu -1),\nu )$. The boundedness of the Riesz transform of $L^p\left(M_0\right)$ then implies the boundedness of the Riesz transform of $L^p\left(M\right)$