Local ergodic theorems for noncommuting semigroups

Abstract:

Let $(X,\mu )$ be a $\sigma$-finite measure space and ${L_p}(\mu ),1 \leqslant p \leqslant \infty$, the usual Banach spaces of complex-valued functions. For $k = 1,2, \ldots ,n$, let $\{ {T_k}(t):t \geqslant 0\}$ be a strongly continuous semigroup of Dunford-Schwartz operators. If \[ f \in {R_{n - 1}} = \left \{ {g:\int _{|g| > t} {|g/t|{{(\log |g/t|)}^{n - 1}}d\mu < \infty {\text {for}}\;{\text {all}}\;t > 0} } \right \},\] then \[ \frac {1}{{{\alpha _1}{\alpha _2} \cdots {\alpha _n}}}\int _0^{{\alpha _n}} \cdots \int _0^{{\alpha _i}} {{T_n}({t_n}) \cdots {T_1}({t_1})f(x)d{t_1} \cdots d{t_n} \to f(x)} \] $\mu$-a.e. as ${\alpha _1} \searrow 0, \ldots ,{\alpha _n} \searrow 0$ independently. If $f \in {L_p}(\mu ),1 < p < \infty$, then the limit exists in norm as well as pointwise.