  In this paper, we show almost global existence of small solutions to the
Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or
cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the
size of initial data, we employ a weighted Sobolev norm whose regularity index
is the smallest among all the admissible Sobolev norms of integer order. We
must overcome the difficulty caused by the absence of the $H^1$-$L^p$
Klainerman-Sobolev type inequality, in order to obtain a required a priori
bound in the low-order Sobolev norm. The introduction of good substitutes for
this inequality is therefore at the core of this paper. Using the idea of
showing the well-known Lady\v{z}enskaja inequality, we prove some weighted
inequalities, which, together with the generalized Strauss inequality, play a
role as the good substitute.
