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Equivalence of Integral Means

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Abstract

Let Aρf and Lρf be the integral means of f on a ball Bρ and its sphere, respectively. Frequently they are regarded as a bridge because of the smoothness. In this paper we will investigate their equivalence. More precisely, we will prove, among others, the following relationship: for any 1 ≤ p ≤ ∞ and d ≥ 2,

$$\parallel A_\rho f-f\parallel _p\sim \parallel L_\rho f-f\parallel _p,\quad \forall f\in L^p({\rm R}^d).$$

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Authors and Affiliations

  1. Department of Computer Sciencce, China Institute of Metrology, 310034, Hangzhou, China

    Tingfan Xie

  2. Department of Mathematics, University of Duisburg, 47057, Duisburg, Germany

    Xinlong Zhou

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  1. Tingfan Xie
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  2. Xinlong Zhou
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Correspondence to Xinlong Zhou.

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Xie, T., Zhou, X. Equivalence of Integral Means. Results. Math. 39, 357–373 (2001). https://doi.org/10.1007/BF03322696

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