A counterexample to a multilinear endpoint question of Christ and Kiselev

Abstract

Christ and Kiselev [2],[3] have established that the generalized eigen-functions of one-dimensional Dirac operators with Lp potential F are bounded for almost all energies for p < 2. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series ∑nTn(F, . . . , F) and carefully bounding each term Tn(F, . . . , F). It is conjectured that the results in [3] also hold for L2 potentials F. However in this note we show that the bilinear term T2(F, F) and the trilinear term T3(F, F, F) are badly behaved on L2, which seems to indicate that multilinear expansions are not the right tool for tackling this endpoint case.