Work of Renjin Jiang
The Riesz transform (it is also called Hilbert transform in one dimension) is the most important example of singular integral, which is a core of harmonic analysis. In 1983, R. Strichartz raised the question that, whether one can extend the Lp boundedness of Riesz transform to general manifold. This question has attracted many researchers from different background.
In these works, we mainly studied the behavior of Riesz transform and its relationship with regularity of heat kernel and harmonic functions. If the heat kernel satisfies two side Gaussian bound, then we established a point-to-point equivalence among the Lp boundedness of Riesz transform, reverse Holder inequality of gradient of harmonic functions, and Lp boundedness of the gradient heat semigroup. If the heat kernel only enjoys a upper Gaussion bound, then we show the equivalence still holds for p between 2 and the dimension. A special case of our result yields a characterization of Yau’s gradient estimate for harmonic functions. Further, we deduce new conditions such that the Riesz transform is stable under metric perturbation and gluing operations. The result also give solutions and new proofs to several open question raised by Carron-Coulhon-Hassell Duke Math. J. 2006.
J. Math. Pures Appl.
J. Funct. Anal.
 Jiang Renjin, Riesz transform via heatkernel and harmonic functions on non-compact manifolds, Adv. Math.
 Jiang, Renjin; Lin, Fanghua, Riesztransform under perturbations via heat kernel regularity. J. Math. Pures Appl.(9) 133 (2020), 39–65.
 Coulhon Thierry, Jiang, Renjin,Koskela, Pekka, Sikora Adam, Gradient estimates for heat kernels and harmonic functions.J. Funct. Anal. 278 (2020), no. 8, 108398, 67 pp.