Ergodic averages on spheres

Abstract

LetU ₁,U ₂, …,U _n denoten commuting ergodic invertible measure preserving flows on a probability space (X,Σ,m). LetS _r denote the sphere of radiusr inR ⁿ, and α_r the rotationally invariant unit measure onS _r. WriteU ^tx to denote$U_1^{t_1 } ...U_n^{t_n } x$ x wheret=(t ₁ …,t_n). Define the ergodic averaging operator$S_r f\left( x \right) = \int_{S_r } f \left( {U^t x} \right)d\sigma _r (t)$. This paper shows that these averages converge for eachf ∈L ^p(X), p>n/(n−1), n≥3. This is closely related to the work on differentiation by E. M. Stein, S. Wainger, and others. Because of their work, the necessary maximal inequality transfers quite easily. The difficulty is to show that we have convergence on a dense subspace. This is done with the aid of a maximal variational inequality.