Abstract

This dissertation presents a rigorous and comprehensive wavelet-based framework for the characterization of function spaces, focusing in depth on the local Hardy space h1(Rn) as well as the non-homogeneous bounded mean oscillation space bmo(Rn). Building on the foundational work of Meyer, we develop a localized version of the theory of dyadic tent spaces, introducing a new space t1 reflecting, for arbitrary L1 functions, the behavior of their wavelet coefficients at small scales and marked by an abstraction of their large scale features. We prove that t1 is a Banach space, establish its atomic decomposition theory, and construct a continuous embedding from Meyer’s dyadic tent space T1, viewed as a quotient space, into t1. Using this framework, we provide a complete wavelet-based characterization of h1(Rn), including equivalent norms and necessary conditions for membership. At the same time, we revisit and expand upon Meyer’s characterizations of H1(Rn), offering detailed proofs and new insights. Furthermore, we extend Goldberg’s results on h1(Rn) to include the case of test functions which are bounded with compact support and normalized integral, beyond the classical normalized Schwartz class. In the final chapter, we establish a complete wavelet-based characterization of bmo(Rn), the dual of h1(Rn), including equivalent norms, and extend Goldberg’s non-homogeneous bmo results to include explicit constructions involving bounded functions with compact support and normalized integral. To our knowledge, the use of the theory of tent spaces t1 to achieve the wavelet characterizations of h1(Rn) and bmo(Rn), and the extensions of Meyer’s and Goldberg’s theorems have not appeared in the literature. Throughout the different sections, we put emphasis on the regularity, localization, and cancellation properties inherent in wavelet systems, demonstrating their analytical power in both homogeneous and non-homogeneous settings.