  We study upper estimates of the martingale dimension $d_m$ of diffusion
processes associated with strong local Dirichlet forms. By applying a general
strategy to self-similar Dirichlet forms on self-similar fractals, we prove
that $d_m=1$ for natural diffusions on post-critically finite self-similar sets
and that $d_m$ is dominated by the spectral dimension for the Brownian motion
on Sierpinski carpets.
