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There are known trace and extension theorems relating functions in a weightedSobolev space in a domain U to functions in a Besov space on the boundary bU.We extend these theorems to the case where the Sobolev exponent p is less thanone by modifying our Sobolev spaces to consider averages of functions inWhitney balls. Averaged Sobolev spaces are also of interest in the applicationsin the case where p>1, and so we also provide trace and extension results inthat case. Finally, we provide some comparable results for Neumann traces andextensions.

Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces