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} ^=0/∞ _k 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.

中文翻译：

---

关于斯坦因引理的萨格和周的不等式

我们提供由Sagher和周获得Stein的引理的以下配方的两种可供选择的证明[6]：存在一个常数A> 0，使得对于任何可测量的集合Ë ⊂[0，1]，| E | ≠0，存在一个整数Ñ仅依赖Ë使得对于任何方累加型实数值序列{ CK } ^{= 0 /∞} _ķ我们有： $$ A \ CDOT \总和\ limits_ {K> N} { \ left | {c_k} \ right |} ^ 2 \ leqslant \ mathop {sup} \ limits_I \ mathop {inf} \ limits_ {a \ in \ mathbb {R}} \ frac {1} {{\ left | I \ right |}} \ int_ {I \ cap E} {\ left | {f（t） -a } \ right | ^ 2} dt，$$ （1）其中，在所有二元间隔I和$$ f（t）= \ sum \ limits_ {k = 0} ^ \ infty {c_k r_k}（t），$$ 其中 r _k表示第k个Rademacher函数。第一个证明不依赖于Khintchine的不等式，而第二个证明是简洁的，适用于一般的凹陷沃尔什级数。