Weighted norm inequalities for singular integrals

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The applicant proposes to continue investigations into weighted norm inequalities for singular, fractional and maximal integrals, and into existence and regularity for degenerate elliptic partial differential equations. Calderon-Zygmund singular integrals and fractional integrals arise in the most critical cases of the study of virtually all partial differential equations, from Schrodinger operators to quantum mechanics to Navier-Stokes equations in fluid flow. The study of two weight norm inequalities for these operators not only extends the scope of application, but reveals important properties of the kernels of the individual operators under consideration, that often remain hidden without such investigation. A number of new tools have arisen from the candidate's recent research, such as weighted Alpert wavelets with higher vanishing moments and testing restricted to special functions for doubling cubes, and a long term goal is to build these ideas and others into the fabric of existing theory to provide a comprehensive treatment of two weight norm inequalities for singular integrals, and their application to geometric measure theory and partial differential equations. In the short term, we will work more specifically to (1) extend our solution to the NTV conjecture characterizing boundedness of the Hilbert transform from one L2 space to another one, to higher dimensional Calderon-Zygmund operators in more general settings, and (2) to extend our recent local Tb theorem for the Hilbert transform to higher dimensions in collaboration with students, and pursue potential applications. In regard to degenerate partial differential equations, we propose a broad program of extending our recent work on local boundedness, continuity and maximum principles for infinitely degenerate operators, which could help point the way to investigating the classical smooth case as well. The impact of these investigations should help motivate the extension of the classical regularity results on elliptic equations to the infinitely degenerate regime.