Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We 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.

The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝ d ) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

Bilinear identities involving the k-plane transform and Fourier extension operators

We prove certain L2(ℝn) bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the k-plane transform. As the estimates are L2-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical L2(ℝ2)-bilinear identity for Fourier extension operators associated to curves in ℝ2.

A comparison principle for convolution measures with applications

We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].