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

Abstract:

We study Fourier multiplier operators associated with symbols $\xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |))$, where $\lambda$ is a real number and $\phi$ is a real-valued $\mathrm {C}^\infty$ 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 $\mathrm {L}^p(\mathbb {R}^n)$ 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 $r\exp (\mathbb {i}\varphi ) \mapsto \exp (\mathbb {i}\lambda \cos \varphi )$. We show that their $\mathrm {L}^p$ 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.