AbstractWe establish weighted $$L^p$$
L
p
-Fourier extension estimates for $$O(N-k) \times O(k)$$
O
(
N
-
k
)
×
O
(
k
)
-invariant functions defined on the unit sphere $${\mathbb {S}}^{N-1}$$
S
N
-
1
, allowing for exponents p below the Stein–Tomas critical exponent $$\frac{2(N+1)}{N-1}$$
2
(
N
+
1
)
N
-
1
. Moreover, in the more general setting of an arbitrary closed subgroup $$G \subset O(N)$$
G
⊂
O
(
N
)
and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation $$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$
-
Δ
u
-
u
=
Q
(
x
)
|
u
|
p
-
2
u
,
u
∈
W
2
,
p
(
R
N
)
,
where Q is a nonnegative bounded and G-invariant weight function.
Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations
