Harmonic Analysis, Function Theory and Partial Differential Equations

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The applicant proposes to continue research into the circle of ideas surrounding***1) characterizing two weight inequalities for singular integral operators, and***2) investigating interpolation and corona problems in higher dimensions,***3) developing the regularity theory of subelliptic partial differential*equations such as the Monge-Ampere and nonlinear wave equations,***4) investigating harmonic analysis questions for polynomial growth measures.***Major problems are overcome in these investigations through a multidisciplinary approach bringing together ideas from analysis, geometry, algebra, combinatorics and operator theory. Two examples include the solution to the corona problem for the Drury-Arveson space in higher dimensions and the solution to the two weight inequality for the Hilbert transform. The corona problem was solved using solutions to the d-bar problem in partial differential equations, the geometry of holomorphic functions, the algebra of forms, the combinatorics of rogue factors and estimates for positive operators on the ball. The two weight problem was solved with an interplay between Haar decompositions, the geometry of the Hilbert transform, and the algebra and combinatorics of the associated dyadic multiscale analysis. We propose to continue investigations toward settling the famous corona problem for bounded analytic functions in higher dimensions and the two weight inequalities for Riesz transforms and related operators in higher dimensions.**