Generic norm growth of powers of homogeneous unimodular Fourier multipliers

TL;DR

This paper demonstrates that for a broad class of Fourier multipliers with homogeneous phase functions, the operator norms grow at a maximal rate as the parameter tends to infinity, confirming a general phenomenon previously observed in specific cases.

Contribution

It establishes that the maximal growth rate of Fourier multiplier operator norms is generic for a wide class of phase functions, extending prior specific examples to a general setting.

Findings

Operator norms grow at the maximal rate for generic phase functions.

Special examples of phase functions exhibit the same maximal growth.

Results disprove a conjecture of Maz'ya by showing the general phenomenon.

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 $\lVert T_{\Phi}^t\rVert_{L^p\to L^p} \gtrsim_{d,p, \Phi}\langle t\rangle ^{d\lvert\frac{1}{p}-\frac{1}{2}\rvert}$. That is the maximal possible order of growth in $t\to \pm \infty$, according to the previous work by V. Kova\v{c} and the author and the result shows that the two special examples of functions $\Phi$ that induce the maximal growth, given by V. Kova\v{c} and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon.