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

Abstract

We study the Sobolev spaces H σ ⁢ ( K ) {H^{\sigma}(K)} and H 0 σ ⁢ ( K ) {H^{\sigma}_{0}(K)} on p.c.f. self-similar sets. First, for σ ∈ ℝ + {\sigma\in\mathbb{R}^{+}} , we make an exact description of the tangents of functions in H σ ⁢ ( K ) {H^{\sigma}(K)} at the boundary, and introduce a countable set of critical orders that arises naturally in the boundary behavior of functions. These critical orders are just 1 2 + ℤ + {\frac{1}{2}+\mathbb{Z}_{+}} in the Euclidean case, but become complicated on fractals. Second, we characterize H 0 σ ⁢ ( K ) {H_{0}^{\sigma}(K)} as the space of functions in H σ ⁢ ( K ) {H^{\sigma}(K)} with tangents of appropriate order, that depend on σ and critical orders, being 0. Last, we extend H σ ⁢ ( K ) {H^{\sigma}(K)} to σ ∈ ℝ {\sigma\in\mathbb{R}} , and obtain various interpolation theorems with σ ∈ ℝ + {\sigma\in\mathbb{R}^{+}} or σ ∈ ℝ {\sigma\in\mathbb{R}} . The interpolation space presents a critical phenomenon when the resulted order σ θ {\sigma_{\theta}} is critical. Moreover, for the interpolation couple ( H 0 σ ⁢ ( K ) , H 0 σ ′ ⁢ ( K ) ) {(H^{\sigma}_{0}(K),H^{\sigma^{\prime}}_{0}(K))} , more than the classical theorem, our interpolation theorem fully covers the teratological case that { σ , σ ′ } {\{\sigma,\sigma^{\prime}\}} contains at least one critical order.

Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

“…Finally, let us mention that there also exists a large amount of literature concerning the construction of Sobolev like and more generally Besov like functional spaces on fractals using the Laplace operator as a central object, see for instance [10], [11], [16], [49] and the references therein. When p = 2, some inclusions are known (following for instance from [5]), but making an exact identification between those spaces and the Korevaar-Schoen ones is a challenging interesting problem for the future.…”

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.…”