A counterexample to a multilinear endpoint question of Christ and Kiselev

Christ and Kiselev \cite{ck0},\cite{ck1} have established that the generalized eigenfunctions of one-dimensional Dirac operators with $L^p$ potential $F$ are bounded for almost all energies for $p <2$. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series $\sum_n T_n(F, \ldots, F)$ and carefully bounding each term $T_n(F, \ldots, F)$. It is conjectured that the results in \cite{ck1} also hold for $L^2$ potentials $F$. However in this note we show that the bilinear term $T_2(F,F)$ and the trilinear term $T_3(F,F,F)$ are badly behaved on $L^2$, which seems to indicate that multilinear expansions are not the right tool for tackling this endpoint case.

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Published 1 January 2003