Boundedness And Elliptic Operators Linked Littlewood-Paley Function

In this thesis, some basic knowledges of Riemannian manifolds are introduced. Using the doubling volume and the weighted estimates of derivatives of the heat kernel, we obtain the boundedness of Littlewood-Paley-Stein function associated with the divergence form operator on LP(M) for1<p<2. Then we consider the boundedness of hL(f) function associated with the second order elliptic operator on LP(Rn). At the last, by using the Gaffney type estimates, we prove that hLf is bounded from H1(Rn) to L1(Rn).