Problems in complex and harmonic analysis related to weighted norm inequalities

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The applicant proposes to continue researching weighted norm inequalities for Calderon-Zygmund operators (CZOs), and their applications in quasiconformal maps, and Whitney extension problems. CZOs appear as natural tools when understanding the most critical cases of virtually all partial differential equations that describe our physical world, from Schrodinger operators in quantum mechanics to Navier-Stokes equations in fluid flow. Researching into two weight norm inequalities for these operators extends the applications and furthers our understanding of the individual operators under consideration, in ways that would remain completely obscure without this research. More specifically, the applicant and collaborators recently attained an important success by proving in a two-part paper (the applicant is a coauthor in part one) the well-known Nazarov-Treil-Volberg T1 conjecture for the 30+ year-old two weight problem for the Hilbert transform (the simplest non-trivial CZO). The resolution of this 1D problem opens the door for the (more flexible in applications) local Tb version, and higher dimensional versions. Crucial tools in the solution of the 1D case are missing in these versions, leaving enough room for further research, some of it to be performed with students. Another recent success in which weighted norm inequalities proved to be crucial was the solution, by the applicant and coauthors, of the 16-year-old conjecture of Astala regarding distortion of (Hausdorff measure of) sets under planar quasiconformal maps. These maps are good models for elasticity. Astala had understood how these maps distort area and planar sets of smaller dimension (e.g. 1D). But the fine properties of the distortion of the corresponding smaller measures (e.g. length) remained a mystery, which was solved with the type of weighted norm inequalities described above. Unfortunately, some of these tools are missing for the physically relevant 3D case, which the applicant proposes to study. However, enough intuition is given by the weighted inequality proof in 2D to adapt into a plan of attack in 3D. The applicant and collaborators' research on weighted norm inequalities gave rise to an unexpected application in Whitney extension problems. These classical "fitting a function to data" problems ask when a function, a priori only defined on a small set, can be viewed (extended) as defined (with good behaviour) in the whole space. This extension is a key step in many relevant problems to be able to apply the modern tools for the analysis of the partial differential equations governing physical phenomena. While understanding the Whitney extension problems for Sobolev spaces (these are the natural setting for the modern approach for partial differential equations), it turned out that certain tools from the applicant and collaborators' research on weighted norm inequalities appear as very promising and relevant. The applicant proposes to continue this line of research.