UCLA

We examine the L^p norm dimensional asymptotics of spherical and ball maximal averaging operators on Cartesian powers of finite graphs. We extend the past techniques that prove that the spherical maximal operators on Cartesian powers of finite cliques satisfy L^p bounds independent of dimension for all p ∈ (1, ∞]. We use these tools to prove computable sufficient conditions for dimension-independent L^2 bounds on the ball and spherical maximal operators on powers of some distance regular graphs. For instance, our techniques prove that the ball maximal operators on powers of the 5-cycle graph and the spherical maximal operators on powers of the Petersen graph each satisfy dimension-independent L^2 bounds.

We go on to discuss some less well-behaved asymptotics. We prove that for a substantial family of graphs, including all trees with more than 2 vertices and even some regular graphs, the spherical maximal averaging operators have exponentially growing L^p norms with respect to dimension for all p ∈ [1, ∞). We prove that for another family, the ball maximal averaging operators have exponentially growing L^p norms for some p > 1. Finally, we identify some graphs whose behavior lies between these two extremes and discuss a few related open questions.