Weak type \((1,1)\) estimates for a class of discrete rough maximal functions (Q2459311)

The authors consider the maximal function \({\mathcal M}^*\) defined by \[ ({\mathcal M}^*f)(x)=\sup_{M>0}\frac{1}{M}\sum_{0<m<M}f(x-[m^\alpha]), \quad z\in{\mathbb Z}. \] The main result shows that if \(1<\alpha<1+1/1000\), then \({\mathcal M}^*\) has weak type \((1,1)\).