Mapping properties of the Marcinkiewicz integrals on homogeneous groups (Q2785256)

Let \(\mathbb{H}\) be a homogeneous group with homogeneous dimension \(Q\) and homogeneous norm \(|.|\). The unit sphere \(S\) in \(\mathbb{H}\) is defined as \(S= \{x\in\mathbb{H}:|x|= 1\}\). Let \(\Omega\) be a measurable function on \(\mathbb{H}\) that satisfies \(\Omega(\delta_\Gamma x)= \Omega(x)\) for a.e. \(x\in \mathbb{H}\setminus\{0\}\) and all \(r> 0\), where \(\{\delta_r\}_{r>0}\) is the dilation group on \(\mathbb{H}\). The Marcinkiewicz integral \(\mu_\Omega f\) on \(\mathbb{H}\) is defined as NEWLINE\[NEWLINE\mu_\Omega f(x)= \Biggl(\int^\infty_0|F_t(x)|^2 t^{-3} dt\Biggr)^{1/2},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEF_t(x)= \int_{B_t(x)} \Omega(x\circ y^{-1})|x\circ y^{-1}|^{-Q+1} f(y) dyNEWLINE\]NEWLINE and \(B_t(x)\) is the ball centered at \(x\) with radius \(t\). The following is the main result in the paper.NEWLINENEWLINENEWLINETheorem: If \(\Omega\) satisfies (1) \(\Omega\in\Lambda^\alpha(S)\), \(\alpha> 0\), where \(\Lambda^\alpha\) is the space of Lipschitz functions on \(S\); (2) \(\int_S \Omega= 0\); (3) \(\mu_\Omega\) is bounded on \(L^q(\mathbb{H})\) for some \(q> 1\), then the following inequalities hold: NEWLINE\[NEWLINE\begin{aligned} & \|\mu_\Omega f\|_{L^1}\leq C_1\|f\|_{H^1}\quad\text{for all }f\in H^1(\mathbb{H}),\\ & \|\mu_\Omega f\|_{\text{BMO}}\leq C_\infty\|f\|_{L^\infty_c}\quad\text{for all }f\in L^\infty_c(\mathbb{H}),\\ & \|\mu_\Omega f\|_{L^p}\leq C_p\|f\|_{L^p}\quad\text{for }1< p< \infty\quad\text{and }f\in L^p(\mathbb{H}).\end{aligned}NEWLINE\]NEWLINE Without condition (3), the theorem is an extension of a well-known classical result when \(\mathbb{H}= \mathbb{R}^n\).