On Instability of the Nikodym Maximal Function Bounds over Riemannian Manifolds

Abstract

We show that, for odd d, the L d+2 2 bounds of Sogge [10] and Xi [13] for the Nikodym maximal function over manifolds of constant sectional curvature, are unstable with respect to metric perturbation, in the spirit of the work of Sogge and Minicozzi [7]. A direct consequence is the instability of the bounds for the corresponding oscillatory integral operator. Furthermore, we extend our construction to show that the same phenomenon appears for any d-dimensional Riemannian manifold with a local totally geodesic submanifold of dimension ⌈ d+1 2 ⌉ if d ≥ 3. In contrast, Sogge's L 7 3 bound for the Nikodym maximal function on 3-dimensional variably curved manifolds is stable with respect to metric perturbation.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

“…3 It is likely, however, that the range of p is not optimal. For instance, Minicozzi and the third author [32] (see also [40]) found specific manifolds for which (1.3) can hold for all σ < 1/p only if p ≥ 2(3n+1) 3n−3 for n odd or p ≥ 2(3n+2) 3n−2 for n even; it is not unreasonable to speculate that these necessary conditions should, for general M , be sufficient. 4 The examples of [32] rely on Kakeya compression phenomena for families of geodesics; the (euclidean) Kakeya conjecture, if valid, would preclude such behaviour over R n .…”

“…In this case Φ(x, y) is given by the associated Riemannian distance function d g (x, y) minus a constant. By Proposition 4.1 and the counterexamples of Minicozzi and the third author [32] (see also [40]), there exist metrics for which optimal local smoothing is not possible when p < pn,+ where pn,+ := if n is even.…”

Sharp Local Smoothing Estimates for Fourier Integral Operators

“…The white region remains open. p σ S [54] 4 ε 0 Mockenhoupt-Seeger-S [44] 4 1/8 Bourgain [6] 4 1/8 + ε 0 Tao-Vargas [60] + Wolff [68] 4 1/8 + 1/88− Wolff [66] 74+ 1/p− Garrigós-Seeger [21] 190/3+ 1/p− Garrigós-Seeger-Schlag [21] 20+ 1/p− S. Lee-Vargas [39] 3 1/6− Bourgain-Demeter [8] 6 1/6− J. Lee [36] 4 3/16− Table 1. Chronological progress on Conjecture 2.4 for n = 2.…”

Square function estimates and local smoothing for Fourier integral operators

“…The situation for a general compact Riemannian manifold (𝑀, g) is quite different. It was shown by Minicozzi and Sogge [19] (see also [23]) that there exist compact manifolds (𝑀, g) that exhibit a certain Kakeya compression phenomenon that forbids favorable Kakeya-type estimates. Therefore, for such manifolds, local smoothing fails for all 𝜎 < 1∕𝑝 if…”