Uniform vector-valued pointwise ergodic theorems for operators

The paper proves exactly the uniform Wiener–Wintner statement the model labels as likely open: if T:L1(X,µ;E)→L1(X,µ;E) is spaCb and f is weakly mixing, then for a.e. x one has lim_{N→∞} sup_{|λ|=1} |(1/N)∑_{n=1}^N λ^n T^n f(x)|=0 (Theorem 1.5). The authors’ proof relies on an ultraproduct/sequence-space framework AU(E), showing that spaCb promotes pointwise orbits (T^n f(x)) into A(E) so the shift SU is well-defined and isometric; then Lemma 3.1 transfers mixing from T to SU; Theorem 3.2 turns weak mixing of f into almost weak mixing of the pointwise orbit in AU(E); and (together with Theorem 3.5) this yields the desired uniform-in-λ decay. This approach avoids the model’s missing step (deriving L1/a.e. control of Cesàro first differences from weak mixing), which indeed is not available under spaCb alone via classical maximal-inequality methods. See the statement of Theorem 1.5 and the proof blueprint given around it, along with the spaCb/A(E)/SU infrastructure and transfer lemmas .