A theorem attributed to Beurling for the Fourier transform pairs asserts that for any nontrivial function $f$ on $\mathbb{R}$ the bivariate function $ f(x) \hat{f}(y) e^{|xy|} $ is never integrable over $ \mathbb{R}^2.$ Well known uncertainty principles such as theorems of Hardy, Cowling–Price etc. follow from this interesting result. In this talk we explore the possibility of formulating (and proving!) an analogue of Beurling’s theorem for the operator valued Fourier transform on the Heisenberg group.

The video of this talk is available on the IISc Math Department channel.