Boundedness of \(g\)-functions on Triebel-Lizorkin spaces (Q842020)

Let \(\Omega\in H^1(S^{n-1})\) with \(\int\Omega(x')\,d\sigma(x')> 0\). The problem is to find sufficient conditions in order that the \(g\)-function \[ g_\Phi(f)(x)= \Biggl(\int^\infty_0 |\Phi_t* f(x)|^2\,dt/dt\Biggr)^{1/2} \] with \(\Phi_t(x)= t^{-h}\Phi(x/t)\) is bounded on the Triebel-Lizorkin space \(F^{\alpha,q}_p\), \(0<\alpha<1\), \(1< p\), \(q<\infty\). It is assumed that \(\Phi(x)= h(|x|)\Omega(x)\), where the function \(h\) is continuous on \(\mathbb{R}^+\). It is shown that this sufficient condition is the following: there exists an \(\varepsilon> 0\) such that \(|h(s)|\leq Cs^{-n+\varepsilon}(1+ s)^{-2\varepsilon}\) and a \(\gamma> 0\) satisfying the inequality \[ \int_{\mathbb{R}} |(s+ m)^{n-1} h(s+ m)- s^{n-1} h(s)|\,ds\leq C_{|m|}\gamma. \]