Two geometric obstruction results in harmonic analysis. Sushrut Zubin Sulaksh Gautam

ISBN: 9781109356014

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Two geometric obstruction results in harmonic analysis.  by  Sushrut Zubin Sulaksh Gautam

Two geometric obstruction results in harmonic analysis. by Sushrut Zubin Sulaksh Gautam
| NOOKstudy eTextbook | PDF, EPUB, FB2, DjVu, audiobook, mp3, RTF | 62 pages | ISBN: 9781109356014 | 7.35 Mb

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,MoreThis 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 /2( R ) 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 Lp3 R2 with 1pj = 1 and pj < 2 for exactly one value of j or p3 < 1). In particular, these curvature conditions are satisfied by any domain D that is locally strictly convex at a single boundary point.



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