Strong mean deviations of Fourier operators (Q1814611)

The authors prove two theorems on the strong summation of Fourier integrals. The first one will be formulated here using the notations of the paper without recalling them because of their great number. Theorem 1. Let \(\varphi\in{\mathfrak A}_{c,\infty}\), assume that for \(t\geq 1\) \(\gamma(t)=\psi^{-1}(\psi(t)/2)-t\geq\alpha_ 0>0\), \(p>1\), \(d\geq 1\), and the function \(\lambda(\sigma)\) is such that the product \(\lambda(\sigma)\psi^ p(\sigma)\) for \(\sigma>1\) does not increase. Then for any \(f\in \hat C^ \psi_ \beta M\), \(\beta\in R\), \[ \| H^ p_ d(f;x;\lambda)\|_ C\leq K(\lambda(d)\psi^ p(d)\gamma(d)E^ p_{d- 1}(f^ \psi_ \beta)+\int^ \infty_ d \lambda(\sigma)\psi^ p(\sigma)E^ p_{\sigma-1}(f^ \psi_ \beta)d\sigma). \]