On the Hardy-Littlewood majorant problem

Details On the Hardy-Littlewood majorant problem On the Hardy-Littlewood majorant problem

2003-03-19

arxiv.org/abs/math/0303244v1

Mathematics

Classical Analysis and ODEs

8 pages

Scientific paper

10.1017/S0305004104007911

Consider a trigonometric polynomial f of degree N, and associate to it the polynomial F in which each coefficient of f is replaced by its absolute value. F is called the majorant of f. We show that the L^3 norm of f can be larger than that of F by a power of N. Similar examples could be constructed in which 3 is replaced by any p > 2 not equal to an even integer. This provides a negative answer to a question, the "Hardy-Littlewood majorant problem", which has been the object of some study. Had the answer been affirmative, there would have followed exciting results connected with the restriction and Kakeya conjectures.

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