A counterexample to a multilinear endpoint question of Christ and Kiselev

Abstract

Christ and Kiselev [2], [3] 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 n T n (F, . . ., F ) and carefully bounding each term T n (F, . . ., F ).It is conjectured that the results in [3] 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.

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View
• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View
• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View