Linear extension operators for restrictions of function spaces to irregular open sets
Let an open set R n satisfy for some 0 d n and " > 0 the condition: the d-Hausdorr content H d (\ B) "jBj d=n for any ball B centered at of volume jBj 1. Let H s p denote the Bessel potential space on R n (1 < p < 1, s > 0), and let H s p ] be the linear space of restrictions of elements of H s p to endowed with the quotient space norm. In this paper we nd suucient conditions for the existence of a linear extension operator for H s p ], i.e., a linear bounded operator ext : H s p ] ! H s p such that ext fj = f for all f. The main result is that such an operator exists if (i) d > n ? 1 and s > (n ? d)= min(p; 2), or (ii) d n ? 1 and s ? s] > (n ? d)= min(p; 2). It is an open problem whether these assumptions are sharp.