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Title: Double Hilbert transforms along surfaces in the Heisenberg group
Author: Vitturi, Marco
ISNI:       0000 0004 6421 3267
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2017
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We provide an L² theory for the local double Hilbert transform along an analytic surface (s, t ,φ(s, t )) in the Heisenberg group H¹, that is operator f ↦ Hφ f (x) := p.v.∫∣s∣,∣t∣≤1 f (x ∙ (s, t ,φ(s, t ))-¹) ds/s dt/t, where ∙ denotes the group operation in H1. This operator combines several features: it is amulti-parameter singular integral, its kernel is supported along a submanifold, and convolution is with respect to a homogeneous group structure. We reprove Hφ is always L²(H¹)→L²(H¹) bounded (a result first obtained in [Str12]) to illustrate the method and then refine it to characterize the largest class of polynomials P of degree less than d such that the operator HP is uniformly bounded when P ranges in the class. Finally, we provide examples of surfaces that can be treated by our method but not by the theory of [Str12].
Supervisor: Wright, Jim ; Blue, Pieter Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: harmonic analysis ; multi-parameter singular integrals ; Heisenberg group