We study Fourier multiplier operators associated with symbols
ξ
↦
exp
(
i
λ
ϕ
(
ξ
/
|
ξ
|
)
)
\xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |))
, where
λ
\lambda
is a real number and
ϕ
\phi
is a real-valued
C
∞
\mathrm {C}^\infty
function on the standard unit sphere
S
n
−
1
⊂
R
n
\mathbb {S}^{n-1}\subset \mathbb {R}^n
. For
1
>
p
>
∞
1>p>\infty
we investigate asymptotic behavior of norms of these operators on
L
p
(
R
n
)
\mathrm {L}^p(\mathbb {R}^n)
as
|
λ
|
→
∞
|\lambda |\to \infty
. We show that these norms are always
O
(
(
p
∗
−
1
)
|
λ
|
n
|
1
/
p
−
1
/
2
|
)
O((p^\ast -1) |\lambda |^{n|1/p-1/2|})
, where
p
∗
p^\ast
is the larger number between
p
p
and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces
R
n
\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
φ
)
↦
exp
(
i
λ
cos
φ
)
r\exp (\mathbb {i}\varphi ) \mapsto \exp (\mathbb {i}\lambda \cos \varphi )
. We show that their
L
p
\mathrm {L}^p
norms are comparable to
(
p
∗
−
1
)
|
λ
|
2
|
1
/
p
−
1
/
2
|
(p^\ast -1) |\lambda |^{2|1/p-1/2|}
for large
|
λ
|
|\lambda |
, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.