The extension problem for fractional Sobolev spaces with a partial vanishing trace condition

We construct whole-space extensions of functions in a fractional Sobolev space of order s ∈ (0, 1) and integrability p ∈ (0,∞) on an open set 0 which vanish in a suitable sense on a portion D of the boundary ∂O of 0. The set 0 is supposed to satisfy the so-called interior thickness condition in ∂O\D, which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case D = ∅ using a geometric construction.