On functions of integrable mean oscillation (Q2574717)

Given \(f\in L^1({\mathbb T})\), the authors consider the modulus of mean oscillation \(w_{\text{mo}}(f)\) defined by \[ \big(w_{\text{mo}}(f)\big)(t)= \sup_{| I| \leq t}\frac1{| I| } \int_I| f(\zeta)-f_I| \,dm(\zeta), \] where \(I\) is an arc, \(m\) is the normalized Lebesgue measure on the unit circle \({\mathbb T}\), and \(f_I\) is the mean value of \(f\) over \(I\). Similarly, \(w_{\text{ho}}(f)\) denotes the modulus of harmonic oscillation of \(f\). The authors show that for \(0<p<\infty\), there exists \(C_p>0\) such that \[ \int_0^1\big((w_{\text{mo}}(f))(t)\big)^p \,\frac{dt}{t}\leq \int_0^1\big((w_{\text{ho}}(f))(t)\big)^p \,\frac{dt}{t}\leq C_p \int_0^1\big((w_{\text{mo}}(f))(t)\big)^p\, \frac{dt}{t}. \]