Abstract
Let be a
doubling metric measure space endowed with a Dirichlet form satisfying a scale-invariant
-Poincaré inequality.
We show that, for ,
the following conditions are equivalent:
(i) :
-estimate
for the gradient of the associated heat semigroup;
(ii) :
-reverse
Hölder inequality for the gradients of harmonic functions;
(iii) :
-boundedness of the
Riesz transform ().
This is joint work with Thierry Coulhon, Renjin Jiang and Pekka Koskela.