We show $L^p\to L^q$ estimates for local and global $r$-variation operators associated to the family of spherical means. These can be understood as a strengthening of $L^p$-improving estimates for the spherical maximal function. Our bounds turn out to be sharp up to the endpoints (except for dimension 3) although we also provide positive results in certain endpoints. The results imply associated sparse domination and consequent weighted inequalities.

This is joint work with David Beltran, Richard Oberlin, Andreas Seeger, and Betsy Stovall.