Let be the -dimensional Hardy averaging operator. It is well known that is bounded on with an open set whenever . We improve this result within the framework of generalized Banach function spaces. We in fact find the "source'' space , which is strictly larger than , and the "target'' space , which is strictly smaller than , under the assumption that the Hardy-Littlewood maximal operator is bounded from into , and prove that is bounded from into . We prove optimality results for the action of and its associate operator on such spaces and present applications of our results to variable Lebesgue spaces , as an extension of A.~Nekvinda and L.~Pick [Math.~Nachr. 283 (2010), 262–271; Z.~Anal.~Anwend. 30 (2011), 435–456] in the case when and is a bounded interval.
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Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura, Hardy Averaging Operator on Generalized Banach Function Spaces and Duality. Z. Anal. Anwend. 32 (2013), no. 2, pp. 233–255DOI 10.4171/ZAA/1483