Hardy Spaces and Heat Kernel Regularity
نویسنده
چکیده
In this article, we will be concerned with questions related to the boundedness of the Riesz transform on manifolds. Since the seminal work of Coulhon and Duong [4], who gave sufficient conditions on the heat kernel so that the Riesz transform is bounded on Lp for 1 < p ≤ 2, several authors have investigated both necessary and sufficient conditions for the boundedness of the Riesz transform on manifolds. For p > 2, one of the main achievements is the following result due to Auscher, Coulhon, Duong and Hofmann [2]: if the manifold satisfies the scaled Poincaré inequalities and the Riemannian measure is doubling, then the boundedness of the Riesz transform on Lq for q ∈ (2, p) is equivalent to the following bounds on the gradient of the heat kernel: for every q ∈ (2, p),
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