Hard Problems in Hard Analysis

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We investigate four major unsolved problems in or bordering on harmonicanalysis. These are the Kakeya problem, the Lipschitz differentiationproblem, the restriction problem, and global solvability for theNavier Stokes equation. For the Kakeya problem, we continue ourwork with Tao on improving exponents in the sums-differences approach.For the problem of differentiation by Lipschitz vector fields, we tryto apply our work on maximal functions in arbitrary directions tounderstand what are the limitations on a counterexample and hopefullythat none can exist. For the restriction problem, we try to applythe new results on Kakeya and to better understand Bourgain'smachine for converting Kakeya results to ones about restriction.For Navier Stokes, we first discretize everything in theform of a kind of generalized wavelet coefficients. In work withPavlovic, this has already produced a generalization ofthe Caffarelli-Kohn-Nirenberg theorem to the case of hyperdissipation.We hope from this point of view to discover a sort of local dispersionproperty for the cascading effect from the nonlinear term. Then wehope to tie in the Clay problem with a dyadic model in which thisdispersion is a given. As might be imagined, we are unlikely to solveall these problems.Analysis concerns the proof of estimates on interesting systems byexaming the contributions of all their parts. One such system isthe Navier Stokes equation which governs the behaviour of incompressibleviscous fluids. An important open problem is whether this equationstarting with smooth initial data can develop singularities withouta forcing term. This would akin to a cyclone beginning spontaneouslyin one's bathtub. It seems rather unlikely but the tools of analysisare not yet strong enough to rule it out. Our approach is todiscretize the problem, that is to try to approximate the problemby one about a finite number of objects and investigate possibleinteractions of those objects by means of combinatorics. Mostphysically arising mathematics can be looked at this way because matteris not continuous but rather composed of particles. We will workon the above problem and some other important problems which may beapproached with the same point of view.