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Littlewood-Paley operators on the generalized Lipschitz spaces. (English) Zbl 0893.42010

Let \(1<p<\infty\) and \(-n<\alpha<1\), and define \({\mathcal E}^{\alpha,p}\) to be the set of functions \(f\) such that \[ \sup_{Q\subset\mathbb{R}^n} \frac{1}{|Q|^{\alpha/n}} \Biggl(\frac{1}{|Q|}\int_Q|f(x)-f_Q|^p dx\Biggr)^{1/p}<\infty, \] with norm, \(\|\;\|_{\alpha,p}\), defined to be the supremum above. For \(0<\alpha< 1\), \({\mathcal E}^{\alpha, p}= \text{Lip}_\alpha\), with equivalent norms. When the supremum is taken only over \(Q\subset \mathbb{R}^n\) which are centered at the origin, one has the space \({\mathcal E}^{\alpha,p}_0\). Let \(T\) be one of the Littlewood-Paley operators, \(s\) or \(g^*_\lambda\). The main results of this paper are that, for \(f\in{\mathcal E}^{\alpha,p}_0\), either \(Tf= \infty\) a.e. or \(Tf\) is finite almost everywhere and \(\| Tf\|_{\alpha,p}\leq C\|f\|_{\alpha,p}\), with the value of \(\lambda\) depending on \(p\) and \(n\). This generalizes earlier results of K. Yabuta [Math. Jap. 43, No. 1, 143-150 (1996; Zbl 0843.42008)] and demonstrates that for the larger spaces \({\mathcal E}^{\alpha,p}_0\) a wider range of \(\alpha\) is applicable.

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0843.42008
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References:

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