Hardy Averaging Operator on Generalized Banach Function Spaces and Duality
Yoshihiro Mizuta
Hiroshima Institute of Technology, JapanAleš Nekvinda
Czech Technical University, Praha, Czech RepublicTetsu Shimomura
Hiroshima University, Graduate School of Education, Higashi-Hiroshima, Japan
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Abstract
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.
Cite this article
Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura, Hardy Averaging Operator on Generalized Banach Function Spaces and Duality. Z. Anal. Anwend. 32 (2013), no. 2, pp. 233–255
DOI 10.4171/ZAA/1483