Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Abstract

We study the Sobolev spaces H σ (K) and H σ 0 (K) on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for σ ∈ R + , we make an exact description of the tangents of functions in H σ (K) at the boundary. Second, we characterize H σ 0 (K) as the space of functions in H σ (K) with zero tangent of an appropriate order depending on σ. Last, we extend H σ (K) to σ ∈ R, and obtain various interpolation theorems with σ ∈ R + or σ ∈ R. We illustrate that there is a countable set of critical orders, that arises naturally in the boundary behavior of functions, such that H σ 0 (K) presents a critical phenomenon if σ is critical. These orders will play a crucial role in our study. They are just the values in 1 2 + Z+ in the classical case, but are much more complicated in the fractal case.

Equivalence of Besov spaces on p.c.f. self-similar sets

“…Then we will provide the definitions of the two classes of Besov spaces, B p,q σ (K) and Λ p,q σ (K). There is a large literature on function spaces on fractals or on more general metric measure spaces, see [1,2,3,10,11,12,13,15,20,30] and the references therein.…”

“…L 1 is naturally the critical line concerning the continuity of functions, and L 2 is the critical line concerning the Hölder continuity of functions and thus the existence of normal derivatives at boundaries. In the authors' related works [10,11,12], there is a discussion on the role of these critical lines concerning the relationship between Sobolev spaces and (heat) Besov spaces on p.c.f. self-similar sets with different boundary conditions.…”