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.