Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group (Q2513736)

In this paper, the authors first introduce the test spaces \(\mathcal{M}_{\mathrm{flag}}^M(\mathbb{H}^n)\) associated with the flag structure on the Heisenberg group \(\mathbb{H}^n:=\mathbb{C}^n\times\mathbb{R}\) by projecting the corresponding product test spaces \(\mathcal{M}_{\mathrm{flag}}^M(\mathbb{H}^n\times\mathbb{R})\) onto \(\mathbb{H}^n\), where \(M\in\mathbb{N}\). Let \(\mathrm{Q}\) be the collection of all dyadic Heisenberg cubes and \(\mathrm{R}_{\mathrm{vert}}\) be the collection of all strictly vertical dyadic Heisenberg rectangles. For any \((z,u)\in\mathbb{H}^n\), the wavelet Littlewood-Paley \(g\)-function \(g_{\mathrm{flag}}\), which is adapted to the flag structure on \(\mathbb{H}^n\), is defined by \[ g_{\mathrm{flag}}(f)(z,u) :=\left\{\sum_{\mathcal{Q}\in\mathrm{Q}}|\psi^\prime_{\mathcal{Q}} \ast f(z_{\mathcal{Q}},u_{\mathcal{Q}})|^2\chi_{\mathcal{Q}}(z,u) +\sum_{\mathcal{R}\in\mathrm{R}_{\mathrm{vert}}}|\psi^\prime_{\mathcal{R}}\ast f(z_{\mathcal{R}}, u_{\mathcal{R}})|^2\chi_{\mathcal{R}}(z,u)\right\}^{\frac12}, \] where \(\psi^\prime_{\mathcal{Q}}\), \(\psi^\prime_{\mathcal{R}}\) are component functions, and \(z_{\mathcal{Q}},\,u_{\mathcal{Q}}\) and \(z_{\mathcal{R}}, u_{\mathcal{R}}\) are any fixed points, respectively, in \(\mathcal{Q}\) and \(\mathcal{R}\). Let \(p\in(0,\infty)\). The flag Hardy space \(H_{\mathrm{flag}}^p(\mathbb{H}^n)\) on the Heisenberg group \(\mathbb{H}^n\) is defined by \[ H_{\mathrm{flag}}^p(\mathbb{H}^n):=\left\{f\in{\mathcal{M}^{M+\delta}_{\mathrm{flag}}(\mathbb{H}^n)}^\prime:\;g_{\mathrm{flag}}(f)\in L^p(\mathbb{H}^n)\right\}, \] where \({\mathcal{M}^{M+\delta}_{\mathrm{flag}}(\mathbb{H}^n)}^\prime\) denotes the dual of \(\mathcal{M}^{M+\delta}_{\mathrm{flag}}(\mathbb{H}^n)\) and \(M\) is sufficiently large depending on \(n\) and \(p\). For \(f\in H_{\mathrm{flag}}^p(\mathbb{H}^n)\), let \[ \|f\|_{H_{\mathrm{flag}}^p(\mathbb{H}^n)}:=\|g_{\mathrm{flag}}(f)\|_{L^p(\mathbb{H}^n)}. \] A flag convolution kernel on \(\mathbb{H}^n\) is a distribution \(K\) on \(\mathbb{R}^{2n+1}\) which coincides with a \(C^\infty\) function away from the coordinate subspace \(\{(0,u):\;0\in\mathbb{C}^n,\;u\in\mathbb{R}\}\subset\mathbb{H}^n\), and satisfies the following two conditions: (i) For any multi-indices \(\alpha=(\alpha_1,\,\ldots,\,\alpha_n)\), \(\beta=(\beta_1,\,\ldots,\,\beta_m)\), \[ |\partial_z^\alpha\partial_u^\beta K(z,u)|\leq C_{\alpha,\beta}|z|^{-2n-|\alpha|} (|z|^2+|u|)^{-1-|\beta|} \] for all \((z,u)\in\mathbb{H}^n\) with \(z\neq0\). (ii) For every multi-index \(\alpha\) and every normalized bump function \(\phi_1\) on \(\mathbb{R}\) and every \(\delta>0\), \[ \left|\int_\mathbb{R}\partial_z^\alpha K(z,u)\phi_1(\delta u)\,du\right| \leq C_\alpha|z|^{-2n-|\alpha|}; \] for every multi-index \(\beta\), every normalized bump function \(\phi_2\) on \(\mathbb{C}^n\) and every \(\delta>0\), \[ \left|\int_{\mathbb{C}^n}\partial_u^\beta K(z,u)\phi_2(\delta z)\,dz\right| \leq C_\beta|u|^{-1-|\beta|}; \] and for every normalized bump function \(\phi_2\) on \(\mathbb{H}^n\) and every \(\delta_1>0\) and \(\delta_2>0\), \[ \left|\int_{\mathbb{H}^n} K(z,u)\phi_3(\delta_1z,\delta_2u)\,dz\,du\right|\leq C. \] In this paper, the authors mainly prove that, if \(T\) is a flag singular integral with a flag convolution kernel \(K\), then, for all \(p\in(0,1]\), there exists a constant \(C_{p,n}\), depending on \(p\) and \(n\), such that \[ \|Tf\|_{H_{\mathrm{flag}}^p}\leq C_{p,n}\|f\|_{H^p_{\mathrm{flag}}}. \] As a corollary, the Marcinkiewicz multipliers are bounded on \(H^p_{\mathrm{flag}}(\mathbb{H}^n)\). The authors also characterize the dual spaces of \(H^p_{\mathrm{flag}}(\mathbb{H}^n)\), for \(p\in(0,1]\), and establish a Calderón-Zygmund decomposition that yields standard interpolation theorems for \(H^p_{\mathrm{flag}}(\mathbb{H}^n)\).