Estimates for Littlewood--Paley Operators on Ball Campanato-Type Function Spaces

Abstract

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.