The law of the iterated logarithm for wavelet series (Q1815746)

Let \(Q\) be a dyadic cube in \(\mathbb{R}^n\) and \(\psi^\varepsilon_Q\), \(\varepsilon= 1,2,\dots,2^n-1\), be a wavelet base. A wavelet series is a sequence of functions \(\{f_n\}_{n\in\mathbb{N}}\), \(f_n= \sum_Q a_Q\psi^\varepsilon_Q\), \(a_Q\in\mathbb{R}\), where the sum extends over all dyadic cubes with side-length \(\geq 2^{-n}\). The corresponding square function is \(S_n[f](x)=(\sum_Q(\text{vol }Q)^{-1}a^2_Q)^{1/2}\) -- summation is over all dyadic cubes with side-lengths in \([2^{-n},2^{-k_0-3}]\) and such that \(x\in 2^{k_0+3}Q\). It is shown that \[ \limsup_{n\to\infty} {\sup_{k\leq n}|f_n|\over \sqrt{S^2_n[f]\log(\log S_n[f])}}\leq C\tag{\(*\)} \] holds a.e. on the set \(\{x: S_\infty[f](x)=\infty\}\) with an absolute constant \(C\geq 0\). The crucial idea in proving \((*)\) is to split the functions \(f_n\) into a martingale part and remainder terms. The remainder terms are shown to be negligible w.r.t. the limit in \((*)\). For the martingale parts, \((*)\) follows from well-known martingale techniques developed by \textit{R. F. Gundy} [Ann. Math. Stat. 38, 725-734 (1967; Zbl 0153.20901)] and \textit{W. F. Stout} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 15, 279-290 (1970; Zbl 0209.49004)].