On the existence and boundedness of square function operators on Campanato spaces. (English) Zbl 1056.42015

Let \(\psi\in C^1(\mathbb R^n)\cap L^1(\mathbb R^n)\) satisfy \(\int \psi(x)dx=0\), \(| \psi(x)| \leq C(1+| x| )^{-n-1}\) and \(| \nabla\psi(x)| \leq C(1+| x| )^{-n-2}\). For this \(\psi\), one defines Littlewood-Paley’s \(g(f)\), Lusin’s area function \(S(f)\) and Littlewood-Paley’s \(g_{\lambda}^*\). One of the author’s results is as follows: Suppose \(f\in L^{p,\alpha}\,(1<p<\infty, -\frac np\leq\alpha<1)\) (the Campanato spaces), \(\lambda>3+\frac2n\). Then, if \(g_\lambda^*(f)(x_0)<+\infty\) for some \(x_0\in \mathbb R^n\), it follows that \(g_\lambda^*(f)(x)\) exists almost everywhere in \(\mathbb R^n\) and there exists \(C>0\) such that \(\| g_\lambda^*(f)\| _{L^{p,\alpha}}\leq C\| f\| _{L^{p,\alpha}}\). He shows the same results for \(g(f)\) and \(S(f)\). The case \(\alpha=0\) (i.e., BMO) is known by S. Wang [Sci. China, Ser. A 10, 890–899 (1984), and S. Wang and J. Chen [Chin. Ann. Math., Ser. A 11, No. 5, 630–638 (1990; Zbl 0782.42018)]. The same results for existence of \(g(f)(x)\) etc for \(x\in E\) of positive measure are already known. He improves them. Further improvement is given by K. Yabuta [Sci. Math. Jpn. 59 No. 1, 93–112 (2004; Zbl 1043.42015)]. There \(\lambda>3+\frac2n\) is replaced by \(\lambda >1\) for \(0<\alpha<1/2\), and \(\lambda>1+2\alpha/n\) for \(1/2\leq \alpha<1\), and the same results for (generalized) Marcinkiewicz functions.


42B25 Maximal functions, Littlewood-Paley theory
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[1] DOI: 10.1090/S0002-9939-1987-0877035-1 · doi:10.1090/S0002-9939-1987-0877035-1
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