JournalszaaVol. 32, No. 2pp. 233–255

Hardy Averaging Operator on Generalized Banach Function Spaces and Duality

  • Yoshihiro Mizuta

    Hiroshima Institute of Technology, Japan
  • Aleš Nekvinda

    Czech Technical University, Praha, Czech Republic
  • Tetsu Shimomura

    Hiroshima University, Graduate School of Education, Higashi-Hiroshima, Japan
Hardy Averaging Operator on Generalized Banach Function Spaces and Duality cover
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Abstract

Let Af(x):=1B(0,x)B(0,x)f(t) dtAf(x):=\frac{1}{|B(0,|x|)|} \int_{B(0,|x|)} f(t) \ dt be the nn-dimensional Hardy averaging operator. It is well known that AA is bounded on L\spp(Ω)L\sp p(\Omega) with an open set ΩRn\Omega \subset \R^n whenever 1<p1<p\leq\infty. We improve this result within the framework of generalized Banach function spaces. We in fact find the "source'' space SXS_X, which is strictly larger than XX, and the "target'' space TXT_X, which is strictly smaller than XX, under the assumption that the Hardy-Littlewood maximal operator MM is bounded from XX into XX, and prove that AA is bounded from SXS_X into TXT_X. We prove optimality results for the action of AA and its associate operator AA' on such spaces and present applications of our results to variable Lebesgue spaces Lp()(Ω)L^{p(\cdot)}(\Omega), 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 n=1n=1 and Ω\Omega 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