Generic norm growth of powers of homogeneous unimodular Fourier multipliers.

For an integer d ≥ 2 , t ∈ R , and a 0-homogeneous function Φ ∈ C ∞ (R d \ { 0 } , R) , we consider the family of Fourier multiplier operators T Φ t associated with symbols ξ ↦ exp (i t Φ (ξ)) and prove that for a generic phase function Φ , one has the estimate ‖ T Φ t ‖ L p → L p ≳ d , p , Φ ⟨ t ⟩ d | 1 p - 1 2 | . That is the maximal possible order of growth in t → ± ∞ , according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions Φ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon. [ABSTRACT FROM AUTHOR]