  We prove an analogue of Wolff's inequality for the so-called intrinsic
nonlinear potentials associated with the quasilinear elliptic equation \[
-\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the
sub-natural growth case $0<q< p-1$, where $\Delta_{p}u = \text{div}( |\nabla
u|^{p-2} \nabla u )$ is the $p$-Laplacian, and $\sigma$ is a nonnegative
measurable function (or measure) on $\mathbb{R}^n$.
  As an application, we give a necessary and sufficient condition for the
existence of a positive solution $u \in L^{r}(\mathbb{R}^{n})$ ($0<r<\infty$)
to this problem, which was open even in the case $p=2$.
  Our version of Wolff's inequality for intrinsic nonlinear potentials relies
on a new characterization of discrete Littlewood-Paley spaces $f^{p,
q}(\sigma)$ defined in terms of characteristic functions of dyadic cubes in
$\mathbb{R}^n$.
