  We consider a trace theorem for self-similar Dirichlet forms on self-similar
sets to self-similar subsets. In particular, we characterize the trace of the
domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets
to their boundaries, where boundaries mean the triangles and rectangles which
confine gaskets and carpets. As an application, we construct diffusion
processes on a collection of fractals called fractal fields, which behave as
the appropriate fractal diffusion within each fractal component of the field.
