On the \(L^ p\)-bounds for maximal functions associated to convex bodies in \({\mathbb{R}}^ n\) (Q1821288)

The author continues his very interesting work on maximal operators associated to convex bodies in \({\mathbb{R}}^ N\). If B is a convex, symmetric body in \({\mathbb{R}}^ N\), then define: \[ Mf(x)=\sup_{t>0}\frac{1}{Vol B}\int_{B}| f(x+ty)| dy,\quad f\in L^ 1_{loc}({\mathbb{R}}^ N); \] \[ M_ 1f(x)=\sup_{j\in Z}\frac{1}{Vol B}\int_{B}| f(x+2^ j_ y)| dy. \] The main results are: (i) For \(1<p<\infty\), \(\| M_ 1f\|_ p\leq C_ p\| f\|_ p,\) with \(C_ p\) independent of the dimension N and the body B. (ii) For \(3/2<p<\infty\) there is a constant \(C_ p'\) such that \(\| Mf\|_ p\leq C_ p'\| f\|_ p,\) with \(C_ p'\) independent of N and B.