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Let E(A ) denote the shift-invariant space associated with a countable family A of functions in L 2 (H n ) with mutually orthogonal generators, where H n denotes the Heisenberg group. The characterizations for the collection E(A ) to be orthonormal, Bessel sequence, Parseval frame and so on are obtained in terms of the group Fourier transform of the Heisenberg group. These results are derived using such type of results which were proved for twisted shift-invariant spaces and characterized in terms of Weyl transform. In the last section of the paper, some results on oblique dual of the left translates of a single function ϕ is discussed in the context of principal shift-invariant space V (ϕ).

In this paper, we prove the existence of an extremal for the Dunkltype Sobolev inequality in case of p = 2. Also we prove the existence of an extremal of the Stein-Weiss inequality for the D-Riesz potential in case of r = 2.

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