| In this thesis we derive sufficient conditions for L('2)-boundedness of convolution by a tempered distribution on a 3-step nilpotent Lie group. This is achieved by producing estimates on the symbol of the distribution, which is the Fourier transform of the pull-back of the distribution to the Lie algebra of the group via the exponential map. This symbolic calculus is related to the Weyl calculus and the Kohn-Nirenberg calculus for pseudo-differential operators.; A significant feature in the study of these symbols is that they live on the dual of the Lie algebra. In the theory of Kirillov, the irreducible unitary representations of nilpotent Lie groups are classified by the orbits in the dual of the Lie algebra under the co-adjoint representation. Roger Howe has shown that convolution by a distribution on the Heisenberg group is a bounded operator if the co-adjoint derivatives of its symbol satisfy certain estimates.; Let H(,n) be the (2n + 1)-dimensional Heisenberg group. Then there is an action on H(,n) of the subgroup S(,n) of Sp(2n) which is isomorphic to the abelian group of n x n symmetric matrices. The semidirect product SH(,n) of S(,n) and H(,n) is a 3-step nilpotent Lie group with one-dimensional centre. We show that any such group is a subgroup of SH(,n) for some n. By relating distributions on SH(,n) to distributions on H(,n), and discovering a connection between their symbols, we have derived estimates involving the co-adjoint derivatives of symbols on SH(,n)('*). These estimates are then carried over to certain subgroups of SH(,n). |
