Let \( \Omega \) be a bounded Lipschitz domain , and \( 1 < p < \infty \). The trace operator \( T \) is a bounded map

\[ T : W^{1,p}(\Omega) \to W^{1 - \frac{1}{p}, p}(\partial \Omega) \]

[1, Satz 6.41].

Remarks

  • The trace operator

    \[ T : W^{1,1}(\Omega) \to L^{1}(\partial \Omega) \]

    is surjective [2].

  • The trace operator \(T\colon W^{1,p}(\Omega)\to W^{1-1/p,p}(\partial \Omega)\) is surjective. In particular, there is a bounded trace extension operator

    \[ E : W^{1 - \frac{1}{p}, p}(\partial \Omega) \to W^{1,p}(\Omega) \]

    with

    \[ T \circ E = \mathrm{id} \]

    [1, Satz 6.41].

References Link to heading

  1. M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. doi:10.1007/978-3-642-15269-6
  2. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rendiconti del Seminario Matematico della Università di Padova, vol. 27, pp. 284–305, 1957. [Online]. Available: https://www.numdam.org/item/?id=RSMUP_1957__27__284_0 [Accessed: Oct. 8, 2025].