We observe that the classical theorem of Hardy on Fourier transform pairs can be reformulated in terms of the heat kernel associated with the Laplacian on the Euclidean space. This leads to an interesting version of Hardy's theorem for the sublaplacian on the Heisenberg group. We also consider certain Rockland operators on the Heisenberg group and Schrödinger operators on ℝⁿ related to them.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm87-1-9,
author = {S. Thangavelu},
title = {An analogue of Hardy's theorem for the Heisenberg group},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {137-145},
zbl = {0965.43002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm87-1-9}
}
S. Thangavelu. An analogue of Hardy's theorem for the Heisenberg group. Colloquium Mathematicae, Tome 89 (2001) pp. 137-145. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm87-1-9/