On the Muckenhoupt condition for Hutchinson orbits of some iterated function systems

AIMAR HUGO, CARENA MARILINA, IAFFEI BIBIANA

Santiago de Chile

Congreso; III Congreso Latinoamericano de Matemáticos; 2009

The purpose of this note is twofold. First, we show that some very classical contractive similitudes systems  $Phi$ on [0,1] do not propagate the Muckenhoupt condition along the Hutchinson orbits.Actually they do not even propagate the doubling condition for a measure. Second, we show how an adequate change $Psi$ in the system $Phi$, preserving all the geometric properties of $Phi$ itself, allows us to prove an upper estimate for a Hardy-Littlewood maximal operator in terms of a iteration of two operators of the same sort. We apply this inequality to prove that Muckenhoupt condition uniformly propagates along the Hutchinson orbits for the new system $Psi$.