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Bulletin of the London Mathematical Society Advance Access published online on September 3, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp073
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© 2009 London Mathematical Society

Heat-flow monotonicity related to the Hausdorff–Young inequality

Jonathan Bennett

School of Mathematics
The University of Birmingham
The Watson Building
Edgbaston
Birmingham
B15 2TT
United Kingdom
J.Bennett@bham.ac.uk
N.Bez@bham.ac.uk

Neal Bez

Department of Mathematics
University Gardens
University of Glasgow
Glasgow
G12 8QW
United Kingdom

Anthony Carbery

School of Mathematics and Maxwell Institute for Mathematical Sciences
The University of Edinburgh
James Clerk Maxwell Building
King's Buildings
Edinburgh
EH3 9JZ
United Kingdom
A.Carbery@ed.ac.uk

Received 14 April 2008.

It is known that if q is an even integer, then the Lq(Rd) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres ‘simultaneously slide’ to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff–Young inequality.

Dedicated to the memory of Laura Wisewell, 1975–2007

2000 Mathematics Subject Classification 42A38 (primary).

The first and second authors were supported by EPSRC grant EP/E022340/1.


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