×

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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1090/S0002-9939-1987-0877035-1 · doi:10.1090/S0002-9939-1987-0877035-1
[2] Spaces of harmonic functions representable by Poisson integrals of functions in BMO and Lp,{\(\lambda\)} 25 pp 159– (1976)
[3] Ann. Scuola Norm. Sup. Pisa 17 pp 175– (1963)
[4] Realvariable methods in harmonic analysis (1986)
[5] Chinese Annals of Mathematics, Series A 11 pp 630– (1990)
[6] Science in China, Series A 10 pp 890– (1984)
[7] Ill. J. Math. 33 pp 531– (1989)
[8] Math. Japonica 43 pp 143– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.