BMO estimates for lacunary series
Let \(0<p<\infty\) and \(f=\sum_{k}c_ k\psi_{n_ k}\) be a lacunary Walsh series or trigonometric series with \(C\equiv \{\sum_{k\neq 0}| c_ k|^ 2\}^{1/2}<\infty.\) It is known that if I is an interval sufficiently large then the local \(L^ p\) norm of f, averaged on I, is equivalent to C. Here the authors show that when \(p=\infty\), the \(L^ p\) norm must be replaced by the BMO norm (dyadic BMO in the Walsh case, classical BMO in the trigonometric case). Namely, they prove there are constants, A, B (which depend only on the lacunarity constant q) such that \(AC\leq \| f\|_{BMO}\leq BC.\) They give an example to show dyadic BMO cannot be replaced by BMO in the Walsh case. They also extend both the \(L^ p\) and the BMO inequalities to multiple Walsh series and multiple trigonometric series. The proofs are clear and well presented, decomposing the sums into managable pieces which are carefully estimated by classical methods.
scientific article; zbMATH DE number 4198746