Examples of singular maximal functions unbounded on \(L^ p\) (Q756066)

Let \(\gamma =\{(t,\gamma (t))\), \(t>0\}\) be a continuous curve in \({\mathbb{R}}^ 2\). Consider the maximal operator, associated to this curve \[ M_{\gamma}: f\to M_{\gamma}f(x)=\sup_{r>0}r^{- 1}\int^{r}_{0}| f(x_ 1-t,x_ 2-\gamma (t))| dt. \] The author gives examples of curves \(\gamma\) such that the corresponding operator \(M_{\gamma}\) is unbounded in natural functional spaces of function in \({\mathbb{R}}^ 2\). It appeared possible to construct such curves as piecewise linear ones and to obtain the desirable properties by choosing the slope of their parts. In particular there exists a curve \(\gamma\) such that \(M_{\gamma}\) is not of weak type \((1,1)\) while it is bounded in \(L^ p({\mathbb{R}}^ 2)\) for all \(p>1.\) Other examples demonstrate that \(L^ p\) boundedness of the operator \(M_{\gamma}\) depends on p in the whole range \(1<p<\infty\).