Complex Analysis and related topics

We present a Research Project in an area where Complex Analysis interacts with Harmonic Analysis, Operator Theory and Stochastic Processes. The framework of this proposal is the symbiosis in results and techniques between these topics which has already led to several major contributions in our field. This proposal is a natural continuation of a series of eight consecutive projects developed by the senior researchers of our group since 1998 but we have also added several new research topics of significant international relevance such as Wasserstein distances and Cristalline measures, which arise from the diversification and widening of our scientific interests in the last years, as well as the recent additions to our research team. Our team has established scientific relations with some of the best researchers in our area with whom we share interest in some of the goals of this project. This gives an obvious added value to the proposal. The objectives of this proposal can be divided into five interrelated research lines. First, we will try to find substantial results that relate the compression of an analytic self-mapping of the unit disc with the behavior of its Aleksandrov-Clark measures. Second, we will study the size of its hyperbolic Littlewood-Paley square function. Third, we will study L^p estimates of the Bergman projection and the corresponding Bekollé-Bonami weights in rough domains. Fourth, we will explore the Wasserstein distances between Aleksandrov-Clark measures and the Cristalline measures obtained using the curved model set technique of I. Meyer. Finally we plan to understand the role of harmonicity in the Law of the Iterated Logarithm of N. Makarov which has been extremely influential in our area. In the first research line, we plan to relate properties of an analytic self-mapping of the unit disc with the doubling behavior of its Aleksandrov-Clark measures. We will explore the relation between the hyperbolic compression of the mapping and the Bekollé-Bonami condition of the harmonic extension of its Aleksandrov-Clark measures. In the second, we plan to study L^p estimates of the hyperbolic Littlewood-Paley square function of an analytic self-mapping of the unit disc. In the third research line, we will consider the boundedness of the harmonic Bergman projection in the Lebesgue spaces L^p in rough domains as well as the corresponding weighted estimates for Bekollé-Bonami weights. We will also consider extensions to the euclidean space. In the fourth, given an analytic self-mapping of the unit disc we will study the Wasserstein distance between its Aleksandrov-Clark measures as well as the relation between the optimal transport plans and the behavior of the self-mapping. We plan to relate uniform approximation between self-mappings and the size of these distances. We will also study the cristalline measures arising from the Aleksandrov-Clark measures using the curved model set technique of I. Meyer. Finally, we plan to understand the role of harmonicity in the version of the Law of the Iterated Logarithm of Makarov and to find the right assumptions on a smooth (non-harmonic) function in an upper-half space so that it fulfills versions of LIL. We expect to find the right conditions on the laplacian of the function to translate the problem to a discrete situation where one can use Martingale Theory.