Sharp inequalities for Weyl operators and Heisenberg groups

Abstract

We study inequalities in harmonic analysis in the context of non-commutative non-compact locally compact groups. Our main result is the determination of the best constant in the Hausdorff-Young inequality for Heisenberg groups. We also obtain the somewhat surprising fact that the resulting sharp inequality does not admit any extremal functions. These results are obtained after a detailed study of the operators which occur in the Fourier decomposition of the regular representation of the Heisenberg groups. These are called Weyl operators and are of independent interest. We also obtain bounds for the best constants in the Hausdorff-Young inequality and in Young's inequality on semi-direct product groups, including non-unimodular groups. In particular, for real nilpotent groups of dimension n those best constants are shown to be dominated by the corresponding best constants for IR n. Although some of our preliminary lemmas are valid for all values ofp~(1, 2) the methods we use for our main results require that p belong to the sequence 4/3, 6/5, 8/7 ..... i.e. that p', the conjugate index, be an even integer. The contents of this paper are as follows. In Section 1 we discuss Weyl operators and determine, for p' even, the best constant in a Hausdorff-Young type inequality (Theorem 1). We also show the non-existence of extremal functions for this inequality. In Section 2 we prove some general results for locally compact groups which includes a form of Young's inequality for convolution appropriate for non-unimodular groups. This is applied to arbitrary semi-direct products. Then using a duality argument which relates the inequalities of Young and of Hausdorff-Young we obtain bounds for the Hausdorff-Young inequality (Theorem 2) on unimodular semi-direct product groups (for p' even). An interesting consequence of these results is that for a connected simply connected real nilpotent Lie group of dimension n, the best constants in the inequalities of Young and Hausdorff-Young are dominated by the corresponding best constants for IR n. In Section 3 we show (Theorem 3), using the theory of Weyl operators developed