On an inequality of Sagher and Zhou concerning Stein’s lemma

We provide two alternative proofs of the following formulation of Stein’s lemma obtained by Sagher and Zhou [6]: there exists a constant A > 0 such that for any measurable setE⊂ [0, 1], |E| ≠ 0, there is an integerN that depends only onE such that for any square-summable real-valued sequence {ck} k =0/∞ we have: $$A \cdot \sum\limits_{k > N} {\left| {c_k } \right|} ^2 \leqslant \mathop {sup}\limits_I \mathop {inf}\limits_{a \in \mathbb{R}} \frac{1}{{\left| I \right|}} \int_{I \cap E} {\left| {f(t) - a} \right|^2 } dt,$$ (1) where the supremum is taken over all dyadic intervals I and $$f(t) = \sum\limits_{k = 0}^\infty {c_k r_k } (t),$$wherer k denotes thekth Rademacher function. The first proof does not rely on Khintchine’s inequality while the second is succinct and applies to general lacunary Walsh series.