A Trace Theorem For Sobolev Spaces On The Sierpinski Gasket

Abstract

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L 2 domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

“…and f H σ (SG) h 2 L 2 (SG) + ∞ m=0 3 −m 5 σm x∈V m+1 \Vm |c x | 2 1/2 ; while the expansions can be extended to the case σ ≥ 2 by repeatedly applying the Green's operator. As an application of this result, we invite readers to refer to [6] for the trace spaces of H σ (SG) onto the bottom line segment of SG, which extends Jonsson's work on domE [18]. Readers may also compare our decomposition theorem with the theorem of multi-harmonic splines on p.c.f.…”