Multi-Linear Singular Integrals

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Over the past two years I have been working on multi-linear singular integral operators. Recently, in a joint work with T.Tao andC.Thiele we generalized the previous results on the bilinear Hilberttransform to the case of multilinear operators with much more singularsymbols. The striking fact about these developments is that it seems that thesetechniques could have a very nice and important application to thespectral theory of Schroedinger operators. The final goal of our project, is to show that for the classicalSchroedinger operators with square integrable potential and for almostevery positive eigenvalue, their corresponding eigenfunctions are bounded. The answer to the above conjecture would imply a better understandingof the behaviour of the Schroedinger operators, which lie in the heart ofQuantum Mechanics.