Generic norm growth of powers of homogeneous unimodular Fourier multipliers

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