Boundedness and unboundedness results for some maximal operators on functions of bounded variation

  1. Aldaz, J.M. 1
  2. Pérez Lázaro, J. 2
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Universidad Pública de Navarra
    info

    Universidad Pública de Navarra

    Pamplona, España

    ROR https://ror.org/02z0cah89

Journal:
Journal of Mathematical Analysis and Applications

ISSN: 0022-247X

Year of publication: 2008

Volume: 337

Issue: 1

Pages: 130-143

Type: Article

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DOI: 10.1016/J.JMAA.2007.03.097 SCOPUS: 2-s2.0-34548218765 WoS: WOS:000255425400013 GOOGLE SCHOLAR

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Abstract

We characterize the space BV (I) of functions of bounded variation on an arbitrary interval I ⊂ R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV (I) into the Sobolev space W1, 1 (I). By restriction, the corresponding characterization holds for W1, 1 (I). We also show that if U is open in Rd, d > 1, then boundedness from BV (U) into W1, 1 (U) fails for the local directional maximal operator MTv, the local strong maximal operator MTS, and the iterated local directional maximal operator MTd ○ ⋯ ○ MT1. Nevertheless, if U satisfies a cone condition, then MTS : BV (U) → L1 (U) boundedly, and the same happens with MTv, MTd ○ ⋯ ○ MT1, and MR. © 2007 Elsevier Inc. All rights reserved.