Littelwood-Paley function and Lipschitz spaces (Q5942843)

This paper deals with the Littlewood-Paley-Stein functional \(g\) associated with the Poisson semigroup of the Laplace operator on \(\mathbb R^n\). It is well known that \(g\) is \(L^p(\mathbb R^n)\)-bounded for all \(1<p<+\infty\) [see \textit{E. M. Stein}, ``Singular integrals and differentiability properties of functions'' (1970; Zbl 0207.13501); ``Topics in harmonic analysis related to the Littlewood-Paley theory'' (1970; Zbl 0193.10502)]. Here, the author is interested in the action of \(g\) on Lipschitz spaces \(\text{Lip}_{\alpha}(\mathbb R^n)\) for \(0<\alpha<1\). More precisely, it is proved that if \(f\in \text{Lip}_{\alpha}(\mathbb R^n)\) and if \(g^{2}(f)(x)<+\infty\) for some \(x\in \mathbb R^n\), then \(g^{2}(f)\in \text{Lip}_{2\alpha}(\mathbb R^n)\) if \(0<\alpha<\frac 12\), and \(\nabla g^{2}(f)\in \text{Lip}_{2\alpha-1}(\mathbb R^n)\) if \(\frac 12<\alpha<1\). For \(\alpha=\frac 12\), \(g^{2}(f)\) belongs to the Zygmund class \(\Lambda_{\ast}\). The proof is elementary, and the results make more precise and complete a theorem previously obtained by \textit{S. Wang} [Ill. J. Math. 33, No. 4, 531-541 (1989; Zbl 0671.42018)]. Finally, the author gives an example of a function \(f\in \text{Lip}_{\frac 12}(\mathbb R)\) such that \(g^{2}(f)\) is neither Lipschitz nor differentiable.