Convergence of weighted ergodic averages

The paper’s Theorem 1 (under Condition (H1)) states a.e. convergence and a strong L2 maximal inequality with the sharp thresholds H>β+1/2 when α≠1/2 and H>β+3/2 when α=1/2, and proves them via the spectral lemma and a careful block-oscillation control using Móricz’s lemma (including a separate α=1/2 case) . The candidate solution replaces this step by an unsubstantiated “block maximal spectral inequality” that would bound an L2-norm of a block supremum by supθ|VM,N(θ)|·||f||2; this is not available from the spectral lemma (which only controls each fixed polynomial, not the supremum) . Consequently, the candidate’s proof does not justify the required L2-maximal control inside dyadic blocks, nor the α=1/2 threshold, and it also conflates supN||AN(f)||2 with ||supN|AN(f)||2, which is strictly stronger and is what the paper proves.