Weighted integral inequalities in Orlicz martingale classes
Let \(\phi\) be a positive nondecreasing right continuous function on \(\mathbb{R}^+\) with \(\phi(0+)= 0\) and \(\phi(s)\to\infty\) as \(s\to\infty\). Define the Young function \(\Phi\) by \(\Phi(t)= \int^t_0 \phi(s)\,ds\) an let \(q_\Phi= \text{inf}_{t>0} t\phi(t)/\Phi(t)\). A Young function \(\Phi\) is said to satisfy the condition \(\Delta_2)\) if, for some constant \(C\), \(\Phi(2t)\leq C\Phi(t)\), \(t> 0\). If \(\Phi_1\), \(\Phi_2\) are Young's functions. \(\Phi_1\ll \Phi_2\) means that, for some constant \(C\), \[ \sum\Phi_2\circ \Phi^{-1}_1(a_n)\leq C\Phi_2\circ\Phi^{-1}_1(\sum a_n) \] holds for every nonnegative sequence \((a_n)_n\). Let \((\Omega,{\mathcal F},\mu)\) be a complete probability space and let \({\mathcal F}_0,{\mathcal F}_1,\dots\) be a filtration such that \({\mathcal F}= \sigma({\mathcal F}_0\cup{\mathcal F}_1\cup\cdots)\). A weight \(\omega\) is a random variable such that \(\omega>0\) and \(\text{E}[\omega]<\infty\). Let \((f_n)\) \((n\geq 0)\) be a martingale of the form \(f_n= \text{E}[f|F_n]\). Let \(\omega\) be a weight and \(\Phi\) a Young function. Then writing \(f= (f_n)\in L_\Phi(\omega)\) means that \[ \int_\Omega\Phi(|f|)\omega d\mu<\infty. \] Dobb's maximal operator \(M\) is defined by \(Mf= \sup_{n\geq 0}|f_n|\). For any weight \(\omega\) and \(B\in{\mathcal F}\), let \(|B|_\omega:= \int_\Omega 1_B\omega d\mu\). The authors obtain the following result involving two Young functions. Theorem. Let \(u\), \(v\) be weights and suppose \(\Phi_1\ll \Phi_2\), \(\Phi_2\in\Delta_2\) and \(q_{\Phi_1}> 1\). Then the following three statements are equivalent: (1) there exists a positive constant \(C\) such that, for all stopping times \(\tau\), \[ \Phi^{-1}_2\Biggl(\int_{\{\tau<\infty\}} \Phi_2(|\text{E}[fv|{\mathcal F}_\tau]|)\,ud\mu\Biggr)\leq C\Phi^{-1}_1\Biggl(C\int_\Omega \Phi_1(|f|)\,vd\mu\Biggr) \] whenever \(f= (f_n)\in L_{\Phi_1}(v)\); (2) there exists a positive constant \(C\) such that, for all stopping times \(\tau\), \[ \Phi^{-1}_2 \Biggl(\int_{\{\tau< \infty\}} \Phi_2(t\text{E}v|{\mathcal F}_\tau])\,ud\mu\Biggr)\leq C\Phi^{-1}_1 \Biggl(C\int_{\{\tau<\infty\}} \Phi_1(t)\,vd\mu\Biggr),\;t> 0; \] (3) there exists a positive constant \(C\) such that, for all \(\lambda> 0\), \[ \Phi^{-1}_2 (\Phi_2(\lambda)|\{M(fv)> \lambda\}|_u)\leq C\Phi^{-1}_1\Biggl(C\int_\Omega \Phi_1(|f|)\,vd\mu\Biggr), \] whenever \(f= (f_n)\in L_{\Phi_1}(v)\). If \(0< p\leq q<\infty\) and \(\Phi_1(t)= t^p\), \(\Phi_2(t)= t^q\), the inequalities revert to the ones of strong or weak \((p,q)\)-type on the martingale space. From the above theorem, one easily obtains inequalities involving only one Young function.
scientific article; zbMATH DE number 5997898
scientific article