Some remarks on Marcinkiewicz integrals along submanifolds (Q1758301)

Let \(n\geq2\) and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the induced Lebesgue measure \(d\sigma=d\sigma(\cdot)\). For \(x\in\mathbb{R}^n\setminus\{0\}\), let \(x'=x/|x|\). Let \(\Omega\) be a function in \(L^1(S^{n-1})\) satisfying the cancellation condition \(\int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0\). For \(\gamma\in[1,\infty]\), let \(\Delta_\gamma(\mathbb{R}_+)\) denote the collection of all measurable functions \(h:\;[0,\infty)\rightarrow\mathbb{C}\) satisfying \(\|h\|_{\Delta_\gamma}:=\sup_{R>0}[R^{-1}\int_0^R|h(t)| ^\gamma\,dt]^{1/\gamma}<\infty\). For \(\alpha\in(0,\infty)\), let \(L(\log L)^\alpha(S^{n-1})\) denote the class of all measurable functions \(\Omega\) such that \[ \|\Omega\|_{L(\log L)^\alpha(S^{n-1})}:=\int_{S^{n-1}} |\Omega(y')|\log^\alpha(2+|\Omega(y')|)\,d\sigma(y')<\infty. \] For \(q\in[1,\infty)\), let \(B_q^{(0,\gamma)}(S^{n-1})\) denote the block space generated by \(q-\)blocks. In this paper, the authors study parametric Marcinkiewicz integral operators of the form \[ \begin{multlined}\mu_{\Omega,\phi,\psi,h}^\rho f(x,x_{n+1})\\ =\left[\int_0^\infty\left|\frac{1}{t^p}\int_{|y|\leq t} f(x-\phi(|y|)y',x_{n+1}-\psi(|y|))\frac{\Omega(y')} {|y|^{n-\rho}}h(|y|)\,dy\right|^2\,\frac{dt}{t}\right]^{1/2},\end{multlined} \] where \(\rho\in(0,\infty)\), \((x,x_{n+1})\in\mathbb{R}^n\times\mathbb{R}=:\mathbb{R}^{n+1}\), \(\phi\) and \(\psi\) are suitable real-valued functions defined on \(\mathbb{R}_+\), and \(f\in\mathcal{S}(\mathbb{R}^{n+1})\), the space of Schwartz functions. Let \(h\in\Delta_\gamma(\mathbb{R}_+)\) for some \(\gamma\in(1,\infty]\) and \(\phi\), \(\psi\) satisfy certain assumptions. The authors establish the \(L^p(\mathbb{R}^{n+1})\) boundedness of \(\mu_{\Omega,\phi,\psi,h}^\rho\) with \(\Omega\in L(\log L)^\alpha(S^{n-1})\) for certain \(\alpha\), where \(p\in (1,\infty)\) may depend on \(\gamma\). If \(h\) further satisfies a more restrictive condition which is \(h\in L^\gamma(\mathbb{R}_+,dt/t)\) for some \(\gamma\in(1,\infty]\), the authors have sharper results with respect to the conditions on \(\Omega\) and \(p\). The authors also obtain same results for the following maximal operator related to the Marcinkiewicz integral, \[ \mathcal{M}^{(\gamma)}_{\Omega,\phi,\psi}f(x,x_{n+1}):=\sup_h |\mu_{\Omega,\phi,\psi,h}^\rho f(x,x_{n+1})|, \] where the supremum is taken over all measurable radial functions \(h\) satisfying \(\|h\|_{L^{\gamma}(\mathbb{R}_+,dt/t)}\leq1\) for some \(\gamma\in(1,\infty]\). Moreover, the authors also consider the counterpart for the \(L^p(\mathbb{R}^{n+1})\) boundedness of \(\mu_{\Omega,\phi,\psi,h}^\rho\) with \(\Omega\in B_q^{(0,\gamma)}(S^{n-1})\) for certain \(\gamma\).