Some Problems Of Besov And Triebel-Lizorkin Spaces

In this dissertation, we study the theory of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, and investigate the properties of the multi-parameter Besov and Triebel-Lizorkin spaces associated with Zygmund dilation. More precisely, we concentrate on the following topics.First, using inhomogeneous Calderon’s reproducing formulas and spaces of test functions associated with a para-accretive function, we introduce the inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function on spaces of homogeneous type. We prove the Tb theorem of these spaces and apply it to show that the operator of Riesz potential type can be used as the lifting operator of these spaces. Furthermore, we also study the pointwise multiplier theorem of these spaces.We provide new Littlewood-Paley characterizations of the homogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type with the "reverse" doubling property. To achieve the goal, the key step is to prove a T1 theorem, where the kernel of the operator with only "half" the usual smoothness condition. Moreover, using Calderon’s reproducing formulas and almost orthogonal estimates, the T1 theorem for the inhomogeneous Besov and Triebel-Lizorkin spaces on RD spaces are also obtained. As applications, new Littlewood-Paley char-acterizations of these spaces with only "half" the usual conditions on the approximates to the identity are presented.Furthermore, let (X. d. μ) be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ, satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of L2(X) constructed recently by Auscher and Hytonen, we in-troduce the homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces on such a general setting. Moreover, we establish the wavelet characterizations and provide the duali-ties for these spaces. We also consider the pointwise multiplier theory for the inhomogeneous Besov and Triebel-Lizorkin spaces.Finally, we use the discrete Littlewood-Paley-Stein theory to introduce the multi-parameter Besov and Triebel-Lizorkin spaces associated with Zygmund dilation and consider the bound-edness of Ricci-Stein singular integral operators on these spaces. Furthermore, we only apply Calderon’s reproducing formulas and almost orthogonal estimates associated with Zygmund dilation to prove the lifting property of these spaces.