It was proved independently by Foschi and by Hundertmark and Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for
L
2
L^2
functions on the paraboloid in the two lowest-dimensional cases. Here we prove that Gaussians are critical points for the
L
p
L^p
to
L
q
L^q
adjoint Fourier restriction inequalities if and only if
p
=
2
p=2
. Also, Gaussians are critial points for the
L
2
L^2
to
L
t
r
L
x
q
L^r_t L^q_x
Strichartz inequalities for all admissible pairs
(
r
,
q
)
∈
(
1
,
∞
)
2
(r,q) \in (1,\infty )^2
.
Abstract. We study the problem of existence of extremizers for the L 2 to L p adjoint Fourier restriction inequalities for the hyperboloid in dimensions 3 and 4 in the case p is an even integer. We use the method developed by Foschi in [5] to show that extremizers do not exist.
Abstract. It is known that extremizers for the L 2 to L 6 adjoint Fourier restriction inequality on the cone in Ê 3 exist. Here, we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the cone. If we use the knowledge of the exact form of the extremizers, as found by Carneiro, then we can show that nonnegative extremizing sequences converge, after the application of symmetries.
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