Comparison between maximal function and square function on positively curved manifolds
Suppose \(M_ i\) is a compact Riemannian manifold, \(\nabla_ i\) its gradient operator and \(\{P_{t_ i}\}_{t_ i>0}\) its Poisson semigroup, where \(i=1,2\). Also let \(\bar M=M_ 1\times M_ 2\), \(\bar M^{\perp}=M^{\perp}_ 1\times M^{\perp}_ 2\), \[ \nabla =\nabla_ 1\nabla_ 2,\quad P_ t=P_{t_ 1}P_{t_ 2}\text{ and } \Gamma_{{\bar \alpha}}(\bar x)=\{(\bar y,\bar t)\in \bar M^{\perp}:\;\bar x\in B_{\bar y}({\bar \alpha}\bar t)\}. \] For a suitable function f on M set \(u(\bar x,\bar t)=P_{\bar t}(f)(\bar x)\). The nontangential maximal function of f is defined by \[ N_{p,{\bar \alpha}}(f)(\bar x)=\sup_{\bar t\in (R^ 1_+)^ 2}\sup_{\bar y\in B_{\bar x}(\bar t)}| u(\bar y,\bar t)|, \] and the area integral function \[ A_{p,{\bar \beta}}(f)(\bar x)=[\int_{\Gamma_{{\bar \beta}}(\bar x)}| \nabla u(\bar y,\bar t)|^ 2 V_ x^{-1}({\bar \beta}\bar t)d\bar y d\bar t]^{1/2}. \] In this paper, the authors use the method of Carleson measure on the manifold to derive mainly that there is a constant \(C={\mathcal C}(\bar M,p,{\bar \alpha},{\bar \beta})>0\) such that \[ \| A_{p,{\bar \beta}}(f)\|_ p\leq {\mathcal C}\| N_{p,{\bar \alpha}}(f)\|_ p\text{ for } f\in L^{\infty}_ c(\bar M)\text{ and } 0<p<\infty \] if \(Ric(\bar M)\geq 0\), \({\bar \alpha}\) and \({\bar \beta}\in (R^ 1_+)^ 2\). This result is a generalization of the similar fact in the usual \(H^ p\)-space.
scientific article; zbMATH DE number 4159041