On the existence and boundedness of square function operators on Campanato spaces

Abstract

Let $g(f)$ be a Littlewood-Paley square function of $f$, which belongs to Campanato spaces $L^{p,\alpha}(1 < p <\infty,-{n\over p} \leq\alpha <1)$. We prove that if $g(f)(x_0)$ exists (i.e. $g(f)(x_0)<\infty$) for a single point $x_0 \in R^n$, then $g(f)(x)$ exists almost everywhere in $R^n$ and $\|g(f)\|_{L^{p,\alpha}}\leq C\|f\|_{L^{p,\alpha}}$.Thus we give an improvement of some earlier results such as in [8], where it is always needed to assume $g(f)(x)$ exists in a set of positive measure in order to get the a.e. existence and boundedness of $g(f)(x)$.