Generic norm growth of powers of homogeneous unimodular Fourier multipliers

Abstract

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 $\Vert T_{\Phi }^t{\Vert }_{L^p\to L^p}{\gtrsim }_{d,p,\Phi }⟨t{⟩}^{d|\frac{1}{p}-\frac{1}{2}|}$. That is the maximal possible order of growth in $t\to \pm \infty$, according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions $\Phi$ 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.

9 pages