Weak type maximal inequality and the rate of growth of integral means (Q2702986)

Let \(I=\bigtimes^n_{k=1}(a_k,b_k)\) be an \(n\)-dimensional interval in \(\mathbb{R}^n\), whose Lebesgue measure and diameter are denoted by \(|I|\) and \(\delta(I)\), respectively. Set \((I)^k= (a_k, b_k)\) and let \(k_1,\dots, k_n\) be a permutation of the natural numbers \(1,2,\dots, n\) such that \(|(I)^{k_1}|\geq\cdots\geq|(I)^{k_n}|\). Set \(M_i(I)= |(I)^{k_i}|\) and \(r(I)= M_n(I)/M_1(I)\).NEWLINENEWLINENEWLINEThe author estimates the growth order of integral means \(|I|^{-1}\int f\), where \(f\in L(\mathbb{R}^n)\). He proves four theorems and two corollaries. The following Theorem 1 is typical: If \(\nu:[1, \infty[\to ]0,\infty[\) is an increasing function such that \(\nu(2t)< \beta\nu(t)\) for all \(t> 0\) with some \(\beta> 0\) and \(\sum^\infty_{k= 1}1/\nu(2^k)< \infty\), then NEWLINE\[NEWLINE\lim_{\substack{ x\in I\\ \delta(I),r(I)\to 0}} \prod^{n-1}_{i=1} \Biggl\{\nu\Biggl({M_i(I)\over M_n(I)}\Biggr)\Biggr\}^{-1} {1\over|I|} \int_I f= 0\quad\text{a.e.}NEWLINE\]NEWLINE and the operator NEWLINE\[NEWLINE\sup_{\substack{ x\in I\\ \delta(I)< 1}} \prod^{n-1}_{i=1} \Biggl\{\nu\Biggl({M_i(I)\over M_n(I)}\Biggr)\Biggr\}^{- 1} {1\over|I|} \int_I fNEWLINE\]NEWLINE is of weak type \((1,1)\).NEWLINENEWLINENEWLINEDivergence theorems for multiple Fourier series are obtained as corollaries.