On Littlewood-Paley functions and singular integrals (Q1592615)

For an \(L^1(\mathbb{R}^n)\) function \(\Phi\) one defines \(\Phi_t(x)=2^{-nt}\Phi(x/2^t)\). The authors consider the Littlewood-Paley \(g\)-function \(g_\Phi f(x)=\bigl(\int_{0}^\infty |\Phi_t\ast f(x)|^2 dt\bigr)^{1/2}\), and the singular integral \(T_\Phi\) defined by \(T_\Phi f(x)=\int_{0}^{\infty}\Phi_t\ast f(x) dx\). The main result is: Suppose \(\Phi\in L^1(\mathbb{R}^n)\) satisfies (i) for some \(p_0>1\) \(\|\sup_{t\in \mathbb{R}}|\Phi_t|\ast f\|_{L^q(\mathbb{R}^n)}\leq C_q\|f\|_{L^q(\mathbb{R}^n)}\) for all \(q>p_0\), (ii) there are \(\alpha, \beta>0\), \(d\in \{1,2,\ldots, n\}\) such that \(|\hat\Phi(\xi)|\leq C\min\{|\Pi_d\xi|^\alpha, |\Pi_d\xi|^{-\beta}\}\) for all \(\xi\in \mathbb{R}^n\), where \(\Pi_d\xi=(\xi_1,\ldots,\xi_d)\). Then, for every \(p\), \(2p_0(p_0+1)^{-1}<p<2p_0(p_0-1)^{-1}\), there exists \(C_p>0\) such that \(\|g_\Phi(f)\|_{L^p(\mathbb{R}^n)}\leq C_p\|f\|_{L^p(\mathbb{R}^n)}\) and \(\|T_\Phi(f)\|_{L^p(\mathbb{R}^n)}\leq C_p\|f\|_{L^p(\mathbb{R}^n)}\). The proofs are modifications of the proof in \textit{J. Duoandikoetxea} and \textit{J. L. Rubio de Francia} [Invent. Math. 84, 541-561 (1986; Zbl 0568.42012)]. Some applications to the Marcinkiewicz integrals and singular integrals with rough kernel are given. Torus cases are also considered.