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

Abstract

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.