On rough Marcinkiewicz integrals along surfaces
Let \(\mathbb R^n\), \(n\geq2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) the unit sphere in \(\mathbb R^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree zero on \(\mathbb R^n\) with \(\Omega\in L^1(S^{n-1})\) and \[ \int_{S^{n-1}}\Omega(y')\,d\sigma(y')=0,\tag \(*\) \] where \(x'\equiv x/|x|\) for any \(x\neq0\). For a measurable real-valued function \(b\) on \(\mathbb R^+\), one says that \(b\in\Delta_\nu\), \(\nu>1\), if \[ \|b\|_{\Delta_\nu}\equiv\sup_{R>0}\left(R^{-1}\int_0^R |b(t)|^\nu\,dt\right)^{1/\nu}<\infty. \] For \(m>1\), let \(\Gamma(t)\equiv(\gamma_1(t),\cdots,\gamma_m(t))\) be a curve on \(\mathbb R^m\), where each \(\gamma_j\) is a real-valued continuous function on \(\mathbb R^+\). For \(x\in\mathbb R^n\) and \(x_* \in\mathbb R^m\), define the Marcinkiewicz integral operator \(\mu_{\Omega,b,\Gamma}\) in \(\mathbb R^{n+m}\) along \(\Gamma\) by \[ \mu_{\Omega,b,\Gamma}(f)(x,x_*)\equiv\left(\int_0^\infty\left| F_{t,\Gamma}^b(x,x_*)\right|^2t^{-3}\,dt\right)^{1/2}, \] where \[ F_{t,\Gamma}^b(x,x_*)\equiv\int_{\{y\in\mathbb R^m:\,|y|<t\}}b(|y|) \Omega(y')|y|^{1-n}f(x-y,x_*-\Gamma(|y|))\,dy. \] Also, define the lower dimensional maximal function \(M_\Gamma\) by setting, for all \(x_1\in{\mathbb R}\) and \(x_* \in\mathbb R^m\), \[ M_\Gamma(h)(x_1,x_*)\equiv\sup_{R>0}R^{-1}\int_0^R|h(x_1-t,x_*-\Gamma(t))|\,dt. \] Let \(H^1(S^{n-1})\) denote the Hardy space on the unit sphere. In this paper, the authors prove that if \(n\geq2\), \(m\geq1\), \(b\in\Delta_\nu\) for \(\nu>1\), and \(\Omega\in H^1(S^{n-1})\) satisfies (\(\ast\)), then there exists a positive constant \(C\), independent of \(f\), such that {\parindent=7mm \begin{itemize}\item[(i)] \(\|\mu_{\Omega,b,\Gamma}(f)\|_{L^2(\mathbb R^{n+m})}\leq C \|f\|_{L^2(\mathbb R^{n+m})}\), \item[(ii)] \(\|\mu_{\Omega,b,\Gamma}(f)\|_{L^p(\mathbb R^{n+m})}\leq C \|f\|_{L^p(\mathbb R^{n+m})}\) \end{itemize}} for \(|1/p-1/2|<\min\{1/2,\;1/\nu'\}\), provided that \(M_\Gamma\) is bounded on \(L^q(\mathbb R^{m+1})\) for all \(q>1\), where \(1/\nu+1/\nu'=1\). Furthermore, the authors also show that if the assumption \(\Omega\in H^1(S^{n-1})\) is replaced by the assumption \(\Omega\in L(\log^+L)^{1/2}(S^{n-1})\), the above conclusions still hold.
scientific article; zbMATH DE number 5700799
scientific article