Fourier restriction in low fractal dimensions

Let $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$ , and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$ . If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$ : there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$ . On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$ . Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$ , and studying the resulting estimate in three different regimes: small values of $\alpha$ , intermediate values of $\alpha$ , and large values of $\alpha$ . In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$ .

Mesh Quality Preserving Shape Optimization Using Nonlinear Extension Operators

In this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings, we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving linear-elliptic operators for the extension and the effect of additional nonlinear advection on the set of reachable shapes. It is moreover discussed how the computational complexity of the proposed algorithm can be reduced. The benefit of the nonlinearity in the extension operator is substantiated by several numerical test cases of stationary, incompressible Navier–Stokes flows in 2d and 3d.

Tomography bounds for the Fourier extension operator and applications

We explore the extent to which the Fourier transform of an $$L^p$$ L p density supported on the sphere in $$\mathbb {R}^n$$ R n can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $$X(|\widehat{gd\sigma }|^2)$$ X ( | g d σ ^ | 2 ) and $$\mathcal {R}(|\widehat{gd\sigma }|^2)$$ R ( | g d σ ^ | 2 ) , where X and $$\mathcal {R}$$ R denote the X-ray and Radon transforms respectively; here $$d\sigma $$ d σ denotes Lebesgue measure on the unit sphere $$\mathbb {S}^{n-1}$$ S n - 1 , and $$g\in L^p(\mathbb {S}^{n-1})$$ g ∈ L p ( S n - 1 ) . We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $$\widehat{gd\sigma }$$ g d σ ^ , establishing some natural variants of conjectures of Stein and Mizohata–Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel’s theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.

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.