David Rottensteiner (Imperial) - "Modulation spaces on the Heisenberg group"

Abstract

For the nilpotent Heisenberg group Hⁿ, applications from hyperbolic PDE theory suggest the construction of modulation spaces on Hⁿ. In the ℝⁿ-case modulation spaces are most commonly known to be those function spaces that can be characterized by L^p-L^q-integrability of the short time Fourier transform of their members f and their invariance under the group translation on ℝⁿ. Well-known representation theoretic approaches to modulation spaces on locally compact Abelian groups as well as coorbit space theory on generic locally compact groups give a feeling for what the spaces might look like, which features they should share with the ℝⁿ-case and what is lost.

Further valuable insight is gained through the study of the different equivalent descriptions of Besov spaces, both on general nilpotent Lie groups and ℝⁿ, and their relation (including embedding properties) to Modulation spaces in the particular case of ℝⁿ.