Holomorphic Singular Integral techniques for Non-Smooth domains and applications

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This mathematics research project deals with the study of so-called integral formulas in complex and harmonic analysis. Integral formulas are important tools for recovering information on large data sets that are located in hard-to-reach places by collecting very small samples that are within easy reach. For instance, integral formulas can be used to recover the temperature in the interior of a solid body (say a tree, or even a planet) without having to probe holes in the body (that is, in the tree example, without having to drill holes in the trunk). Instead, one measures the temperature at surface level (say on the tree's bark) and plots these values in the integral formula: the output will be the value of the temperature inside. One of the novelties of this project is that it allows to deal with objects whose outer surface is very rough (as opposed to very smooth).The PI and her collaborators will bring together techniques and problems from different parts of the general field of analysis, specifically real harmonic analysis and complex function theory in one and several complex variables. One of the main goals is to develop a theory of Cauchy-like singular integrals with holomorphic kernel and for non-smooth domains in Euclidean complex space that successfully blends the complex structure of the ambient domain with the Calderon-Zygmund theory for singular integrals on non-smooth domains in real space. Doing so requires coming to terms with the additional rigidity imposed by the underlying complex structure, in particular the geometric properties that are collectively known as (or linked to) pseudo-convexity. Applications to several complex variables include the regularity of certain orthogonal projection operators, such as the Bergman and Szego projections, in the novel context of non-smooth domains.