Riesz transform on non-compact Riemannian manifolds and heat kernel regularity
speaker: Thierry Coulhon (Université de Cergy-Pontoise)abstract: These lectures are about a joint work with Pascal Auscher, Xuan Thinh Duong, and Steve Hofmann. Its aim is to give a necessary and sufficient condition for the two natural definitions of homogeneous first order $Lp$ Sobolev spaces to coincide on a large class of Riemannian manifolds, for $p$ in an interval $(q0,p0)$, where $2
Let $M$ be a complete, connected, non-compact Riemannian manifold. Denote by $B(x,r)$ the open ball of radius $r>0$ and center $x\in M$, and by $V(x,r)$ its measure $\mu(B(x,r))$. One says that $M$ satisfies the doubling property if for all $x\in M$ and $r>0$ $$ V(x,2r)\le C\,V(x,r).\eqno (D) $$ Denote by $pt(x,y)$, $t>0$, $x,y\in M$, the heat kernel of $M$, that is the kernel of the heat semigroup $e{-t\Delta}$. One says that $M$ satisfies the on-diagonal heat kernel upper estimate if $$pt(x,x)\le \frac{C}{ V(x,\sqrt{t})}, \eqno(DU\!E)$$ for all $x\in M$, $t>0$ and some constant $C>0$, and that it satisfies the Li-Yau estimates if $$ \frac{c}{ V(y,\sqrt{t})}\exp\left(-C \frac{d2(x,y)}{ t}\right)\le pt(x,y)\le \frac{C}{ V( y,\sqrt{t})}\exp\left(-c \frac{d2(x,y)}{ t}\right).\eqno (LY) $$ Assume $M$ satisfies $(LY)$. Let $p0 \in (2,\infty]$. The following assertions are equivalent: \begin{enumerate} \item For all $p \in (2,p0)$, there exists $Cp$ such that $$\|\nabla e{- t\Delta}
\
{p\to p} \le \frac{C}{\sqrt t}, $$ for all $t>0$. \item The Riesz transform $\nabla \Delta{-12}$ is bounded on $Lp$ for $p\in (2,p0)$. \end{enumerate} \end{theo} \begin{theo}\label{maincor} Assume $M$ satisfies $(D)$ and $(DU\!E)$. If there exists $C$ such that, for all $x,y\in M$, $t>0$, $$
\nablax\,pt(x,y)
\le \frac{C}{ \sqrt{t}\left V(y,\sqrt{t})\right}, \eqno(G)$$ then the Riesz transform $\nabla \Delta{-12}$ is bounded on $Lp$ and the equivalence $$cp\norm{\Delta{12} f}p\le \norm{
\nabla f
}p\leq Cp\norm{\Delta{12} f}p. $$ holds for all $p\in(1,\infty)$.
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