  The main purpose of this paper is to introduce moduli of smoothness with
Jacobi weights $(1-x)^\alpha(1+x)^\beta$ for functions in the Jacobi weighted
$L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the
smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If
$1\le p\le\infty$, then these moduli are equivalent to certain weighted
$K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik
moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to
certain "Realization functionals".
