A counterexample to a multilinear endpoint question of Christ and Kiselev

By Camil Muscalu, Terence Tao, C. Thiele. 2001

60 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f5ccf05103609c804fc8aa6a4cfe3dfadeaf1746

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In Mathematical Physics

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Citations in the corpus

• Anisotropic Hardy-Lorentz spaces and their applications (2015, background): “It is well known that, due to their fine structures, Lorentz spaces play an irreplaceable role in the study on various critical or endpoint analysis problems from many different research fields and there exists a lot of literatures on this subject, here we only mention several recent papers from har”