The minimal operator and the John-Nirenberg theorem for weighted grand Lebesgue spaces (Q2787134)

Let \((\Omega, \mathcal{A},\mu)\) be a finite measure space, let \(1<p<\infty\), let \(\omega\) be a weight on \(\Omega\), let \(\varphi: (0,p-1)\rightarrow\mathbb{R}_+\) be a finite non-decreasing function such that \(\lim_{t\rightarrow 0}\varphi (t)=0\). The weighted grand Lebesgue space \(L_{p),\varphi,\omega}\) is the set of measurable functions \(f\) such that \(\|f\|_{L_{p),\varphi,\omega}}:=\sup_{0<\varepsilon<p-1}\varphi (\varepsilon)\frac{1}{|\Omega|}\int_\Omega|f|^{p-\varepsilon}\omega\,dx<\infty\). On the weighted grand Lebesgue space, the authors introduce the minimal operator NEWLINE\[NEWLINEmf(x):= \inf_{x\in Q}\frac{1}{|Q|}\int_Q|f(y)|\,dy,NEWLINE\]NEWLINE where the infimum is taken over all cubes \(Q\) which contain \(x\) with sides parallel to the coordinate axes. Many properties of the minimal operator are investigated and an analogue of the John-Nirenberg theorem related to functions of bounded mean oscillation (BMO) is proved.