  We prove necessary conditions for Fredholmness of singular integral operators
with piecewise continuous coefficients on weighted Banach function spaces.
These conditions are formulated in terms of indices of submultiplicative
functions associated with local properties of the space, of the curve, and of
the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$
(weighted Lebesgue spaces with variable exponent). Moreover, our necessary
conditions become also sufficient for weighted Nakano spaces over nice curves
whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$
satisfies the estimate $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.
