Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols

Abstract

We study Fourier multiplier operators associated with symbols $\xi \mapsto \exp (i\lambda \varphi (\xi /|\xi |))$, where $\lambda$ is a real number and $\varphi$ is a real-valued $C^{\infty }$ function on the standard unit sphere ${\mathbb{S}}^{n-1}\subset {\mathbb{R}}^n$. For $1<p<\infty$ we investigate asymptotic behavior of norms of these operators on $L^p({\mathbb{R}}^n)$ as $|\lambda |\to \infty$. We show that these norms are always $O((p^{\ast }-1)|\lambda {|}^{n|1/p-1/2|})$, where $p^{\ast }$ is the larger number between $p$ and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces ${\mathbb{R}}^n$. In particular, this gives a negative answer to a question posed by Maz'ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols $r\exp (i\phi )\mapsto \exp (i\lambda \cos \phi )$. We show that their $L^p$ norms are comparable to $(p^{\ast }-1)|\lambda {|}^{2|1/p-1/2|}$ for large $|\lambda |$, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.

25 pages; v2: references added, submitted for publication; v3: another reference added, accepted for publication