PDE Seminar Abstracts

Riesz transform and and harmonic functions

Adam Sikora
Macquarie University
Mon 16th Oct 2017, 2-3pm, Carslaw Room 829 (AGR)


Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form satisfying a scale-invariant L2-Poincaré inequality. We show that, for p (2,), the following conditions are equivalent:

(i) (Gp): Lp-estimate for the gradient of the associated heat semigroup;

(ii) (RHp): Lp-reverse Hölder inequality for the gradients of harmonic functions;

(iii) (Rp): Lp-boundedness of the Riesz transform (p < ).

This is joint work with Thierry Coulhon, Renjin Jiang and Pekka Koskela.