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»ã±¨ÌáÒª£ºRiesz transform on bounded domains as well as complete manifolds/ metric spaces have been widely studied. However, not much has been known for the case of exterior domain. In this talk, we report some recent progress in this direction. Briefly speaking, we give an almost complete solution to this problem, which also confirms some conjectures from Auscher et al. Math. Ann. 2001 and Hassell et. al. IUMJ 2009.