PDE Seminar Abstracts

Let $\left(X,d,\mu \right)$ be a doubling metric measure space endowed with a Dirichlet form satisfying a scale-invariant ${L}^{2}$-PoincarÃ© inequality. We show that, for $p\in \left(2,\infty \right)$, the following conditions are equivalent:

(i) $\left({G}_{p}\right)$: ${L}^{p}$-estimate for the gradient of the associated heat semigroup;

(ii) $\left(R{H}_{p}\right)$: ${L}^{p}$-reverse HÃ¶lder inequality for the gradients of harmonic functions;

(iii) $\left({R}_{p}\right)$: ${L}^{p}$-boundedness of the Riesz transform ($p<\infty $).

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

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