Examples of metric measure spaces related to modified Hardy-Littlewood maximal operators (Q2790824)

In [Ann. Acad. Sci. Fenn., Math. 40, No. 1, 443--448 (2015; Zbl 1403.42023)], the author considered the modified Hardy-Littlewood maximal operator, which is defined for a parameter \(k\geq 1\) as NEWLINE\[NEWLINE M_k f (x)=\sup_{x\in B} \frac{1}{\mu(kB)} \int_B |f| \;d\mu, \;x\in X, NEWLINE\]NEWLINE where the supremum is taken over all balls \(B\) containing \(x\). It was proved there that \(M_3\) and the centered version \(M_2^c\) of \(M_2\) are of weak type \((1,1)\). The results were mentioned to be sharp in the sense that, in general, any \(k<3\) or any \(k<2\) is not enough in the uncentered or the centered case, respectively, for \(M_k\) or \(M_k^c\) to be of weak type \((1,1)\).NEWLINENEWLINEThe paper shows that, for \(X\) being a countable, connected and acyclic planar graph equipped with the geodesic distance, the corresponding modified Hardy-Littlewood maximal operators \(M_k\) and \(M_k^c\) fail to be of weak type \((1,1)\) for \(1\leq k <3\) and \(1\leq k <2\), respectively.