Maximal functions measuring smoothness: counterexamples (Q1947238)

Let \(n\in\mathbb{N}\), \(s>0\), and \(p\geq 1\). The author considers two maximal operators measuring smoothness. The first one is defined by \[ M^\flat_{s,p}f(x)= \sup_{Q\ni x}\;\underset{\deg(\pi)\leq s-1}{}{\text{inf}}|Q|^{-s/n}\Biggl(|Q|^{-1} \int_Q |f-\pi|^p\Biggr)^{1/p}, \] where \(Q\) is any cube with sides parallel to the axes and \(\pi\) any polynomial. By \(M^\sharp_{s,p}f\) is denoted the maximal operator with polynomials \(\pi\) having degree at most \(s\). Clearly \(M^\sharp_{s,p}f\) can be pointwise smaller than \(M^\flat_{s,p}f\). The author constructs examples of compactly supported \(f\in L^\infty(\mathbb{R}^n)\) for which \(M^\sharp_{s,p}f\in L^q(\mathbb{R}^n)\) and \(\| M^\flat_{s,p}f\|_{L^q(\mathbb{R}^n)}= \infty\), for \(q>{p\over sp+n}\).