Generic norm growth of powers of homogeneous unimodular Fourier multipliers

For an integer \(d\ge 2\), \(t\in \mathbb {R}\), and a 0-homogeneous function \(\Phi \in C^{\infty }(\mathbb {R}^{d}{\setminus }\{0\},\mathbb {R})\), we consider the family of Fourier multiplier operators \(T_{\Phi }^t\) associated with symbols \(\xi \mapsto \exp (it\Phi (\xi ))\) and prove that for a generic phase function \(\Phi \), one has the estimate \(\Vert T_{\Phi }^t\Vert _{L^p\rightarrow L^p} \gtrsim _{d,p, \Phi }\langle t\rangle ^{d|\frac{1}{p}-\frac{1}{2}|}\). That is the maximal possible order of growth in \(t\rightarrow \pm \infty \), according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions \(\Phi \) 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.