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A characterization of Sobolev spaces via local derivatives

David Swanson — 2010

Colloquium Mathematicae

Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

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