Estimates of maximal functions measuring local smoothness (Q5928593)

The maximal function mentioned in the title is defined by \[ N _\eta f(x) = {\sup}(|Q|\eta(|Q|^{1/n}))^{-1}\int_Q |f(t)-f(x)|dt, \] where \(\eta\) is monotone nondecreasing on \((0, 1]\), \(\eta (+0)=0\), \(\eta (t)/t\) is monotone decreasing; the supremum is taken over all cubes \(Q\) containing \(x\). By demanding that \(N _\eta f\) belong to some standard space of measurable functions (such as a Lorentz space or an Orlicz space), we obtain various conditions expressing the smoothness of \(f\). The author studies interrelations of these conditions, and also their relationship with other definitions of smoothness (for instance, with definitions in terms of integral moduli of continuity).