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Singular spherical maximal operators on a class of two step nilpotent lie groups

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Abstract

LetH n≅ℝ2n⋉ℝ be the Heisenberg group and letμ t be the normalized surface measure for the sphere of radiust in ℝ2n. Consider the maximal function defined byM f=supt>0|f*μ t |. We prove forn≥2 thatM defines an operator bounded onL p(H n) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.

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Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Detlef Müller

  2. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, 53706, Madison, WI, USA

    Andreas Seeger

Authors
  1. Detlef Müller
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  2. Andreas Seeger
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Correspondence to Detlef Müller.

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The second author was supported in part by the National Science Foundation.

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Müller, D., Seeger, A. Singular spherical maximal operators on a class of two step nilpotent lie groups. Isr. J. Math. 141, 315–340 (2004). https://doi.org/10.1007/BF02772226

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  • DOI: https://doi.org/10.1007/BF02772226

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