In the framework of tamed Dirichlet space,
I will talk on the $L^p$-boundedness of positive order Riesz transforms under the distributional curvature lower bound of Kato class. Moreover, under the gaugeability condition for the lower bound of curvature, I will mention the $L^p$-boundedness of 0-order Riesz transforms. This is a jointwork with Syota Esaki (Fukuoka University) and Zi Jian Xu (Fukuoka University).