Two geometric obstruction results in harmonic analysis

Abstract

This dissertation consists of two independent parts. In the first, we prove an uncertainty principle in the setting of Gabor frames that generalizes the classical Balian-Low Theorem. Namely, with H^s denoting the L²-Sobolev space with s derivatives, we show that if f ∈ H^p/2([special characters omitted]) and fˆ ∈ H^{p' /2}([special characters omitted]) with 1 < p < ∞ and [special characters omitted] = 1, then the Gabor system [special characters omitted](f, 1, 1) generated by time-frequency translates of f from the lattice [special characters omitted] is not a frame for L²([special characters omitted]). In combination with previously-known results, this completes the classification of the L²-Sobolev time-frequency regularity conditions f ∈ [special characters omitted], fˆ ∈ [special characters omitted] under which [special characters omitted](f, 1, 1) can constitute a frame for L ²([special characters omitted]). In the "endpoint" case p = 1, we also obtain a generalization of previously-known decay conditions on f ∈ H^1/2([special characters omitted]) to guarantee a Balian-Low-type obstruction result. These results are established by first proving a variant of the endpoint Sobolev embedding into VMO, which is then combined with a topological VMO-degree argument on the Zak transform of the function f.

In the second part of the dissertation, we provide sufficient normal curvature conditions on the boundary of a domain D ⊂ [special characters omitted] to guarantee unboundedness of the bilinear Fourier multiplier operator with symbol χ_D, outside the local L² setting (i.e. from [special characters omitted] × [special characters omitted] to [special characters omitted] with [special characters omitted] = 1 and p_j < 2 for exactly one value of j or [special characters omitted] < 1). In particular, these curvature conditions are satisfied by any domain D that is locally strictly convex at a single boundary point.