On reverse weak (1,1) type inequalities (Q5932448)

For a locally finite non-negative Borel measure \(\nu \), the maximal operator is defined at a~locally integrable function \(f\) on \(\mathbb R\) by NEWLINE\[NEWLINE M_\nu (f)(x)=\sup _{x<b,\nu [x,b)>0}\frac 1{\nu [x,b)}\int _{[x,b)}|f|d\nu . NEWLINE\]NEWLINE When \(\nu \) is the Lebesgue measure, then it easily follows from the Whitney covering theorem that \(M_\nu \) is locally of reverse weak (1,1) type, that is, NEWLINE\[NEWLINE \nu (\{M_\nu (\chi _If)>\lambda \}\cap I) \geq \frac c\lambda \int _{\{M_\nu (\chi _If)>\lambda \}\cap I}|f|d\nu NEWLINE\]NEWLINE for some \(c>0\), all locally integrable \(f\) and all \(\lambda >0\) big enough. This result is not true for all Borel measures \(\nu \). The author characterizes those \(\nu \) for which the reverse weak type inequality holds and gives also corresponding results for the one-sided version of the operator.