Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces
The authors study the boundedness of singular integral operators \(T\) with kernel \(\mathcal K\) in the Journé's class on \(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3}\), with regularity exponent \(\varepsilon>0\) [\textit{J. L. Journé}, Rev. Mat. Iberoam. 1, No. 3, 55--91 (1985; Zbl 0634.42015)], acting on the multiparameter Hardy space \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\). Their main result reads as follows: For \(1\leq i\leq 3\), if \(T^*_i(1)=0\) denotes the condition that, for every \(\varphi^j\in C_{0,0}^\infty(\mathbb R^{n_j})\), \[ \int_{\mathbb R^{n_i}}\int_{\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3}} \mathcal{K}(x_1,x_2,x_3,y_1,y_2,y_3) \varphi^1(y_1)\varphi^2(y_2)\varphi^3(y_3)\,dy_1dy_2dy_3dx_i=0, \] then an operator \(T\) as above is bounded on \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\) for the range of indexes \(\max\{\frac{n_i}{n_i+\varepsilon}:1\leq i\leq 3\}<p\leq 1\), if and only if \(T^*_i(1)=0\), \(1\leq i\leq 3\). The necessary conditions are obtained from the cancellation properties of functions in the classical Hardy space, together with its maximal characterization. To prove the sufficiency, they reduce the problem to showing the boundedness of \(\{T_{t_1,t_2,t_3}\}_{t_1,t_2,t_3>0}\) from \(H^p(\mathbb R^{n_1}\times \mathbb R^{n_2}\times \mathbb R^{n_3})\) to \(L^p_{\mathcal H}(\mathbb R^{n_1+n_2+n_3})\) by using \(H^p\)-atoms, where \(T_{t_1,t_2,t_3}(f)=\psi_{t_1,t_2,t_3}*(T(f))\), \(\psi_{t_1,t_2,t_3}(x_1,x_2,x_3)=\psi_{t_1}^1(x_1)\psi_{t_2}^2(x_2)\psi_{t_3}^3(x_3)\), \(\psi_{t_i}^i(x_i)=t_i^{-n_i}\psi^i(x_i/t_i)\), \(\psi^i\in C_{0,0}^\infty(\mathbb R^{n_i})\) with \[ \int_0^\infty|\widehat{\psi^i}(t\xi)|^2\,\frac{dt}t=1,\quad\text{for all }\xi\in\mathbb R^{n_i}\setminus \{0\}, \] \(L^p_{\mathcal H}\) is the \(\mathcal H\)-valued \(L^p\) space and \(\mathcal H\) is the Hilbert space defined by \[ \mathcal{H}=\bigg\{\{h_{t_1,t_2,t_3}\}_{t_1,t_2,t_3>0}:\| \{h_{t_1,t_2,t_3}\}\|_{\mathcal H}= \bigg(\int_0^\infty\int_0^\infty\int_0^\infty| h_{t_1,t_2,t_3}|^2\,\frac{dt_1}{t_1}\frac{dt_2}{t_2}\frac{dt_3}{t_3}\bigg)^{1/2}<\infty\bigg\}. \] This result extends the two-parameter case [\textit{Y. Han} et al., J. Funct. Anal. 258, No. 8, 2834--2861 (2010; Zbl 1197.42006)]. In the last part of the paper, they also study regularity properties of \(\{T_{t_1,t_2,t_3}\}_{t_1,t_2,t_3>0}\).
scientific article; zbMATH DE number 6143076
scientific article