Holomorphic Singular Integrals in Several Complex Variables and Applications

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This research project seeks to develop mathematical tools that allow to draw information on large sets of data located in places that are hard to reach, and for which actual reach would require disrupting the site of study, by collecting smaller data sets that are within easy reach and do not require disrupting the site. For instance, the information we seek may be the temperature at the core of a tree (the wood underneath the bark). Or, say, the temperature of soil located deep down underground. In both examples, performing direct measurements would require disrupting the object of study (drilling the tree; drilling the ground) which is expensive and disruptive. Instead, with the mathematics-based methods developed in this project it is enough to measure temperature on the tree's bark, or on the earth's surface (and neither requires drilling). Then one plots the easily-collected data into an integral (the "big sister" of the integrals studied in calculus) and the output of this integral will be the temperature at the core of the tree (or temperature of the soil deep underground). The part of mathematics that deals with these problems is called "harmonic analysis" and "singular integrals"; the methods employed are called "integral representation formulas." These methods work even for e.g., trees that have very rough bark ("fractal-like") as opposed to smooth bark ("integral formulas for non-smooth domains").This project brings together techniques and problems from different parts of the general field of analysis, with primary emphasis on complex function theory in one and several complex variables, and on real harmonic analysis on Euclidean space. One of the main goals is to develop a theory of Cauchy-like singular integrals with holomorphic kernel and for non-smooth domains in n-dimensional complex Euclidean space that successfully blends the complex structure of the ambient domain with the Calderon-Zygmund theory for singular integrals on non-smooth domains in 2n-dimensional real space. Stripping away the smoothness assumptions brings to the fore the geometric interplay between the operators and the domains on which they act: recent advances by Lanzani and her collaborators have set the ground for pushing this theory well below the C2-category and for bringing it on par with the real analysis techniques that were developed over the past thirty years in Geometric Measure Theory, Partial Differential Equations and Quasiconformal mapping. A novel component builds upon recent work by the principal investigator and her collaborator M. Pramanik and seeks to investigate the aforementioned holomorphic singular integral operators by exploring suitable symmetrized forms of their Schwartz kernels. Further problems include the investigation of div-curl inequalities in the context of the d-bar complex in complex Euclidean space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.