A counterexample to a multilinear endpoint question of Christ and Kiselev

Abstract

Christ and Kiselev 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 ${\sum }_n{T}_n(F,...,F)$ and carefully bounding each term $T_n(F,...,F)$. It is conjectured that the results of Christ and Kiselev 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.

arXiv comment9 pages, no figures, to appear, Math. Res. Letters. More detailed remarks, many minor changes

References in corpus (2)

• $L^p$ estimates for the biest II. The Fourier case
• $L^p$ estimates for the biest I. The Walsh case

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