Sharp Orlicz Endpoints for Spatial-Temporal Ergodic Averaging

Abstract

We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by $NΛ_q(N)$ ($q\ge 0$), the sharp Orlicz endpoint is $L\log_{q+1}L$. In particular, the ordinary case $q=0$ yields an $L\log L$ local joint convergence theorem under the Lebesgue differentiation property alone, while $L^1$ fails even on the Euclidean interval, answering two questions of Young. The same $L\log_{q+1}L$ endpoint holds for prime averages for every $q\ge 1$. For arbitrary time sequences, $L\log_qL$ always suffices at the same normalization $NΛ_q(N)$ ($q\ge 1$), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences $k^n$ ($k\ge 2$) and for sequences with polynomial ratio separation, including $n!$. The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an $L^1$ majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local $\infty$-sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.