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. In particular, 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 any 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.