Transference principles for divergence sets of dispersive equations and their applications

Abstract

In this article, we study the fractal refinement of the Carleson's problem: (1) We introduce an equivalence relation on linear dispersive PDEs on $\mathbb{R}^{n}$ based on the high-frequency oscillation of their Fourier multipliers, proving that equations in the same class share the same Hausdorff dimension of divergence sets. As applications, we estimate these dimensions for abstract dispersive families. Consequently, we recover known $1$-dimensional results for linearized Boussinesq and Beam propagations, while establishing new results for both in higher dimensions and for Klein-Gordon propagations in all dimensions. (2) Next, we study the effect of radial symmetry by obtaining dimension estimates for fractional Schrödinger equations with concave phase and radial initial data. Applying a transference principle, dimension estimates are extended to families of abstract dispersive equations. As special cases, we recover the known result for the Beam propagation and also obtain new results for the linearized Boussinesq and the Klein-Gordon propagations in the radial setting. (3) We study analogues of (2) for the $3$-dimensional real hyperbolic space by obtaining dimension estimates for a wide class of dispersive propagations: the Schrödinger, fractional Schrödinger (with convex and concave phases), Beam, linearized Boussinesq, Wave and Klein-Gordon equations. This is the first result of its kind, outside the setting of flat Euclidean geometry. Our arguments are based on an inter-spatial transference principle of geometric-analytic flavour. In the process, we also obtain a quantitative result on the singular sets of Sobolev functions, in the spirit of Žubrinić, which may be of independent interest.