Sharp Estimates for Eigenvalues, Integrals, and Sobolev Imbeddings

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The author proposes to work on various extremal problemsfor functionals which are defined in terms of integralsor eigenvalues. These problems include: Lieb-Thirringinequalities for the number bound states of Schrodinger operators; sharp Moser-Trudinger and Sobolev inequalitiesfor the CR sphere, with applications to determinants of CR invariant operators; sharp inequalities for integralsinvolving cross-ratios, with applications to zeta functionsof Laplacians on the sphere; sharp inequalities of log-Sobolevtype on the disk, with applications to zeta functions ofDirichlet and Neumann Laplacians on the disk; dimension-free Carleson measure inequalities. The research willtouch on important questions arising in differentialgeometry, harmonic analysis, and mathematical physics.It is difficult to understate the importance of eigenvalueestimates in both applied and theoretical sciences. It isvery often the case that to given physical structures, suchas systems of particles, vibrating membranes, or strings,one can attach certain characteristic numerical values,called "eigenvalues". These numbers are important, as somerelevant properties of a given structure can be often codedinto numerical functions of the eigenvalues, called"spectral functionals". In an extremal problem one triesto find out which structures of the same type wouldyield the highest (or lowest) possible spectral functionals; very often, they are the ones with exhibit the mostsymmetrical geometry.