| 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 Hs denoting the L2-Sobolev space with s derivatives, we show that if f ∈ Hp/2( R ) and fˆ ∈ Hp'/2R with 1 < p < infinity and 1p+1p' = 1, then the Gabor system G (f, 1 ,1) generated by time-frequency translates of f from the lattice ZxZ is not a frame for L2( R ). In combination with previously-known results, this completes the classification of the L2-Sobolev time-frequency regularity conditions f ∈ Hs1R , fˆ ∈ Hs2R under which G (f, 1, 1) can constitute a frame for L 2( R ). In the "endpoint" case p = 1, we also obtain a generalization of previously-known decay conditions on f ∈ H1/2( R ) 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 ⊂ R4 to guarantee unboundedness of the bilinear Fourier multiplier operator with symbol chiD, outside the local L2 setting (i.e. from Lp1R2 x Lp2R2 to Lp'3 R2 with 1pj = 1 and pj < 2 for exactly one value of j or p'3 < 1). In particular, these curvature conditions are satisfied by any domain D that is locally strictly convex at a single boundary point. |
