Comparison between maximal and square functions of 1-forms and its applications (Q1200111)

Let \(M\) be a complete noncompact Riemannian manifold with \(\text{Ric}(M)\geq 0\). Let \(\{\mathbb{P}_ t\}_{t>0}\) be the Poisson semi- groups on \(M\) and \(\omega\) be a \(C_ c^ \infty\) 1-form on \(M\). Define \[ \begin{cases} A_{P,\alpha}(\omega)(x)=\left(\int^ \infty_ 0\int_{B(x,\alpha t)}|\nabla\mathbb{P}_ t(\omega)(y)|^ 2V_ x^{-1}(\alpha t) t dy dt\right)^{1/2}, \\ N_{P,\alpha}(\omega)(x)=\sup_{d(x,y)<\alpha t}|\mathbb{P}_ t(\omega)(y)|, \\ g_ P(\omega)(x)=\left(\int^ \infty_ 0\left|{\partial\over\partial t}\mathbb{P}_ t(\omega)(x)\right|^ 2t dt\right)^{1/2}. \end{cases} \] The main results of the paper under review are: Theorem 1. \(\| A_{P,\alpha}(\omega)\|_ p\leq C\| N_{P,\beta}(\omega)\|_ p\) for \(0<\alpha,\beta,p<\infty\). Theorem 2. Riesz transform \(\nabla(+\Delta)^{-1/2}\) is \(L^ p\)-bounded for \(1<p<+\infty\). Theorem 3. For \(C_ c^ \infty\) 1-form, there holds \[ |\{x:g_ P(\omega)(x)>\lambda\}|\leq C(|\{x:N_{P,\alpha}(\omega)(x)> \lambda\}|+ |\{x:A_{P,\alpha}(\omega)(x)> \lambda\})|; \] especially, \(g_ P\) is \(L^ p\)-bounded for \(1<p<\infty\) and weak type (1,1) bounded and \(\| g_ P(\omega)\|_ p\leq C\| N_{P,\alpha}(\omega)\|_ p\) for \(0<p\),\(\alpha<\infty\).