Orlicz-type integral inequalities for operators (Q2708268)

Let \(f\to Tf\) be an operator and for \(j=1,2,\ldots\) let \(T^j\) be the \(j\)-times iterated operator of \(T\). The author considers two functions \(\Phi(t)=\int_{0}^{t}\varphi(s) ds\), \(\Psi(t)=\int_{0}^{t} \psi(s) ds\), where \(\varphi, \psi: \mathbb R_+\to \mathbb R_+\). The first result is as follows: The following two are equivalent, NEWLINE\[NEWLINE\int_{\mathbb R^n}\Phi(|Tf|)\leq c\int_{\mathbb R^n}\Psi(c|f|)\tag{1}NEWLINE\]NEWLINE holds for every pair \(\varphi,\psi:\mathbb R_+\to \mathbb R_+\) satisfying \(\int_{0}^{s}\varphi(t)t^{-1} dt\leq c'\psi(c''s)\), \(0\leq s<\infty\). NEWLINE\[NEWLINE|\{Tf|>\lambda\}|\leq c_0\lambda^{-1}\int_{\lambda/c_{0}}^{\infty} |\{|f|>s\}|ds. \tag{2}NEWLINE\]NEWLINE And in this case, for \(j=1,2,\ldots\) NEWLINE\[NEWLINE\int_{\mathbb R^n}\Phi(|T^jf|)\leq c_j\int_{\mathbb R^n}\Psi(c_j|f|)\tag{3}NEWLINE\]NEWLINE holds for every pair \(\varphi,\psi:\mathbb R_+\to \mathbb R_+\) satisfying NEWLINE\[NEWLINE\int_{0}^{s}\varphi(t)t^{-1}\log^{j-1}(s/t) dt\leq c'\psi(c''s), \quad 0\leq s<\infty \text{ (corrected form)}.\tag{4}NEWLINE\]NEWLINE The second one is: Assume \(\psi(t)>0\), \(t>0\), and \(\psi(t')\leq c'\psi(c't'')\) for \(0\leq t'\leq t''\). Assume further \(\int_{|f|>\lambda}|f|\leq c\lambda |\{|Tf|>\lambda \}|\). Then, (3) implies (4). The author applies the two results above to the Hardy-Littlewood maximal operator. He also treats \(L\log L\) and \(\exp L\) cases. His results extend a work by \textit{Y. J. Yoo} [Bull. Korean Math. Soc. 36, No. 2, 225-231 (1999; Zbl 0929.42011)].