Trace and extension theorems for Sobolev-type functions in metric spaces

Abstract

Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-$\theta$ bound are considered. Based on a Poincaré inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds $\theta$. The trace $T$ is shown to be a compact operator mapping a Sobolev-type space on a domain into a Besov space on the boundary. Sufficient conditions for $T$ to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to $\theta$, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in $L^{\theta }$ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.

References in corpus (2)

• Besov spaces via hyperbolic fillings
• Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity

Cited by in corpus (10)

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• Traces of Newton-Sobolev, Hajlasz-Sobolev and BV functions on metric spaces
• Extension and trace theorems for noncompact doubling spaces
• Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity
• Neumann $p$-Laplacian problems with a reaction term on metric spaces
• $p$-harmonic mappings between metric spaces
• Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces
• Quantitative characterization of traces of Sobolev maps
• Characterization of trace spaces on regular trees via dyadic norms