Distribution function estimates for Marcinkiewicz integrals and differentiability
The authors use distribution function estimates to study the Marcinkiewicz integral on \({\mathbb{R}}^ n:\) \[ M_{\alpha}(F)(x)=(\int_{{\mathbb{R}}^ n}\frac{| F(x+u)+F(x-u)- 2F(x)|^ 2}{| u|^{n+2\alpha}}du)^{1/2}, \] 0\(<\alpha <2\), and its relation to differentiability. In particular, they consider results of \textit{E. M. Stein} and \textit{A. Zygmund} [see, e.g., Studia Math. 23, 247-283 (1963; Zbl 0122.302)] on the finiteness almost everywhere of the Marcinkiewicz integral in a given measurable set. Additionally, they use their estimates to derive \(L^ p\) and weak \(L^ p\) equivalences relating \(M_{\alpha}\) and operators which are known to measure differentiability, such as the Peano maximal function: \[ P_{\alpha}(F)(x)=\sup_{h>0}h^{-\alpha}(h^{-n}\int_{| u| <h}| F(x+u)-F(x)|^ 2du)^{1/2},\quad 0<\alpha \leq 1, \] with a similar definition for \(1<\alpha <2\). Several other functions are used in this study, including a nonsymmetric version of the Marcinkiewicz integral, extensions of \(M_{\alpha}\) and \(P_{\alpha}\) to harmonic functions on \({\mathbb{R}}_+^{n+1}\), and nontangential maximal functions. One of the interesting features of the analysis is that control by the Marcinkiewicz integral is obtained in terms of the Hardy-Littlewood maximal function of the set where \(M_{\alpha}\) (or \(M_{\alpha}\) plus a nontangential maximal function) exceeds a constant.
scientific article; zbMATH DE number 3967022