  Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume
that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein
vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and
$d\in(0,\infty)$. In this article, the authors prove that, for any $f\in
\mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$ (the ball Campanato-type function space
associated with $X$), the Littlewood--Paley $g$-function $g(f)$ is either
infinite everywhere or finite almost everywhere and, in the latter case, $g(f)$
is bounded on $\mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$. Similar results for both
the Lusin-area function and the Littlewood--Paley $g_\lambda^*$-function are
also obtained. All these results have a wide range of applications.
Particularly, even when $X$ is the weighted Lebesgue space, or the mixed-norm
Lebesgue space, or the variable Lebesgue space, or the Orlicz space, or the
Orlicz-slice space, all these results are new. The proofs of all these results
strongly depend on several delicate estimates of Littlewood--Paley operators on
the mean oscillation of the locally integrable function $f$ on $\mathbb{R}^n$.
Moreover, the same ideas are also used to obtain the corresponding results for
the special John--Nirenberg--Campanato space via congruent cubes.
