Pointwise characterization of Sobolev classes

  • B. Bojarski


We prove that a function f is in the Sobolev class W loc m,p (ℝ n ) or W m,p (Q) for some cube Q ⊂ ℝ n if and only if the formal (m − 1)-Taylor remainder R m−1 f(x,y) of f satisfies the pointwise inequality |R m−1 f(x,y)| ≤ |xy| m [a(x) + a(y)] for some a ε L p (Q) outside a set NQ of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ n .


Sobolev Space STEKLOV Institute Pointwise Estimate Main Lemma Sobolev Class 
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© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • B. Bojarski
    • 1
  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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