  On fractals, different measures (mutually singular in general) are involved
to measure volumes of sets and energies of functions. Singularity of measures
brings difficulties in (especially non-linear) analysis on fractals. In this
paper, we prove a type of Sobolev inequalities, which involve different and
possibly mutually singular measures, on product Sierpinski spaces. Sufficient
and necessary conditions for the validity of these Sobolev inequalities are
given. Furthermore, we compute the sharp exponents which appears in the
sufficient and necessary conditions for the product Kusuoka measure, i.e. the
reference energy measure on Sierpinski spaces.
