Variational inequalities for generalized spherical means

In this paper, we establish the $L^p({\mathbb{R}}^d)$-boundedness of the variation operator and the $\delta$-jump operator for generalized spherical means, and we also show the necessary conditions for the $L^p({\mathbb{R}}^d)$-boundedness of these operators. These results are almost optimal when $d=2$.