On Kolmogorov's theorem, the Hardy-Littlewood maximal function and the radial maximal function
The main object of study in this paper is the weak maximal function. This is the function \(M_ w\phi\) defined for any weak integrable function \(\phi\) as \(M_ w\phi (x)=\sup \{\| \phi \|^ I_{l,\infty}/| I|:x\in I\}\), where \(\| \phi \|^ I_{l,\infty}\) denotes the \(``weak\)-L\({}^ 1\) norm'' of \(\phi\) on the interval I, i.e., \(\| \phi \|^ I_{l,\infty}=\sup \{\delta | \{t\in I:| \phi (t)| >\delta \}|;\delta >0\}\). In case \(\phi\) is the modulus of the boundary values of an holomorphic function f with positive real part, it is shown that \(M_ w\phi\) is pointwise equivalent to the radial maximal function of f. The main result of the paper is that the weak (1,1) estimate for both the Hardy-Littlewood maximal function and the conjugate function of a measure \(\mu\) can be improved in the following sense: if \(\phi\) is one of those functions, then \(\| M^ m_ w\phi \|_{l,\infty}\leq C_ 1C_ 2^ m\| \mu \|\), \(m=1,2,...\). for some absolute constants \(C_ 1,C_ 2\). This property does not hold for an arbitrary weak-integrable function. The generalization of this result to a Calderón-Zygmund singular convolution operator appeared in Bull Am. Math. Soc., New Ser. 16, 271-273 (1987; Zbl 0617.42010).
Calderón-Zygmund singular convolution operator
scientific article; zbMATH DE number 4075749