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

We study Fourier multiplier operators associated with symbols $\xi\mapsto \exp(i\lambda\phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\phi$ 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 $r\exp(i\varphi) \mapsto \exp(i\lambda\cos\varphi)$. 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 Dragi\v{c}evi\'{c}, Petermichl, and Volberg.Comment: 25 pages; v2: references added, submitted for publicatio