L2 Schrödinger Maximal Estimates Associated with Finite Type Phases in ℝ2

Abstract

In this paper, we establish Schrödinger maximal estimates associated with the finite type phase

$$\phi(\xi_{1},\xi_{2}):=\xi_{1}^{m}+\xi_{2}^{m},$$

where m ≥ 4 is an even number. Following [12], we prove an L² fractal restriction estimate associated with the surface

$$\{(\xi_{1},\xi_{2},\phi(\xi_{1},\xi_{2}))\ :\ (\xi_{1},\xi_{2})\in[0,1]^{2}\}$$

as the main result, which also gives results on the average Fourier decay of fractal measures associated with these surfaces. The key ingredients of the proof include the rescaling technique from [16], Bourgain–Demeter’s ℓ² decoupling inequality, the reduction of dimension arguments from [17] and induction on scales. We notice that our Theorem 1.1 has some similarities with the results in [8]. However, their results do not cover ours. Their arguments depend on the positive definiteness of the Hessian matrix of the phase function, while our phase functions are degenerate.