This thesis is composed of three interrelated parts: ω-Calderón-Zygmond operators, a wavelet characterization for weighted Hardy spaces, and the analytic dependence of minimal surfaces on their boundaries.
In the first part of the thesis, we prove a T1 theorem and develop a version of the Calderón-Zygmund theory for ω-CZO when ω belongs to A∞. As an application, we use our results to indicate some estimates for fractional integrals.
In the second part of this thesis, we give a wavelet area integral characterization for weighted Hardy spaces Hp(ω), 0 < p < ∞, with ω in A∞. At the same time, our wavelet characterization establishes the identification between Hp(ω) and T2p(ω), the weighted discrete tent space, for 0 < p < ∞ and ω in A∞. This allows us to use all the results of tent spaces for weighted Hardy spaces. In particular, we obtain the isomorphism between Hp(ω) and the dual space Hp'(ω), where 1 < p < ∞ and 1/p + 1/p' = 1, and the wavelet and the Carleson measure characterizations of BMOω. Moreover, we obtain interpolation between A∞-weighted Hardy spaces Hp1(ω) and Hp2(ω), 1 < p1 < p2 < ∞, which completes the proof of Calderón-Zygmund theory of the first part.
In the third part of this thesis, we prove that the functional which takes a closed Lavrentiev curve to the corresponding Riemann mapping is locally Lip1 on the set Ω of all closed Lavrentiev curves. This set is a subset of BMO(T). However, it is not open in BMO(T). We also prove that the previous functional is analytic for certain classes of closed Lavrentiev curves, including the class of curves which have some symmetry w.r.t. the unit circle. These classes of curves are submanifolds of BMO(T). Finally, we consider the functional which takes a Lavrentiev curve (closed or not) in n-dimensional Euclidean space to the corresponding minimal surface, and we study the differentiability and analyticity of this functional on certain function spaces.