Maximal operators related to the Radon transform and the Calderon-Zygmund method of rotations
This interesting paper is concerned with maximal operators involving several parameters. For a function f on \({\mathbb{R}}^ n\), and x in \({\mathbb{R}}^ n\) and \(\theta\) in \(\Sigma_{n-1}\), the unit sphere, set \[ Mf(x,\theta)=\sup \{(2r)^{-1}\int^{r}_{-r}| f(x-t\theta)| dt:r>0\}, \] \[ Hf(x,\theta)=p.v.\quad \pi^{-1}\int^{\infty}_{- \infty}f(x-t\theta)dt/t,\quad and \] \[ H^*f(x,\theta)=\sup \{\pi^{- 1}| \int_{| t| >\epsilon}f(x-t\theta)dt/t|:\epsilon >0\}. \] Then the authors obtain mixed norm estimates for Mf, Hf, and \(H^*f\), in spaces \(L^ p({\mathbb{R}}^ n;L^ q(\Sigma_{n-1}))\). The main result is that if \(1<p\leq \max \{2,(n+1)/2\}\) and \(q<p(n-1)/(n-p)\), then M, H and \(H^*\) are bounded from \(L^ p\) to \(L^ p(L^ q)\). Similar results are obtained when \(\theta\) is restricted to a subset of \(\Sigma_{n-1}\) of small dimension, and for the Radon transform. The important new element in this paper is that, for n large enough, results for \(p>2\) are obtained.
scientific article; zbMATH DE number 4071613
scientific article