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Functions from Sobolev and Besov Spaces with Maximal Hausdorff Dimension of the Exceptional Lebesgue Set

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Abstract

We prove that for p > 1 and 0 < α < n/p there exists a function from the Bessel potentials class J α (L p(ℝn)) such that the Hausdorff dimension of its exceptional Lebesgue set is n − αp. We also show that such a function may be taken from the Besov class B α p,q (L p(ℝn)) with any q > 0.

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Correspondence to V. G. Krotov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 5, pp. 145–153, 2013.

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Krotov, V.G., Prokhorovich, M.A. Functions from Sobolev and Besov Spaces with Maximal Hausdorff Dimension of the Exceptional Lebesgue Set. J Math Sci 209, 108–114 (2015). https://doi.org/10.1007/s10958-015-2488-0

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Keywords

  • BESOV Space
  • Lebesgue Space
  • Lizorkin Space
  • Disjoint Support
  • Bessel Potential