A note on the maximal Gurov-Reshetnyak condition

In a recent paper [17] we established an equivalence between the Gurov-Reshetnyak and A_∞ conditions for arbitrary absolutely continuous measures. In the present paper we study a weaker condition called the maximal Gurov-Reshetnyak condition. Although this condition is not equivalent to A_∞ even for Lebesgue measure, we show that for a large class of measures satisfying Busemann-Feller type condition it will be self-improving as is the usual Gurov-Reshetnyak condition. This answers a question raised independently by Iwaniec and Kolyada.