Two-wavelet Localization Operators on L^p for the Weyl-Heisenberg Group

The goal of this paper is to investigate the boundedness and compactness on Lp(Rn) of some linear operators that are closely related to Daubechies operators with two admissible wavelets for the Weyl-Heisenberg group . We give two different proofs for the fact that Daubechies operators associated to symbols in L1(Rn × Rn) and two admissible wavelets in L1(Rn) ∩ L∞(Rn) are bounded linear operators on Lp(Rn), 1 ≤ p ≤ ∞. If the symbols are in Lr(Rn × Rn), 1 ≤ r ≤ 2, and the two admissible wavelets are in L1(Rn) ∩ L2(Rn) ∩ L∞(Rn), then the associated Daubechies operators are proved in Section 3 to be bounded linear operators on Lp(Rn), r ≤ p ≤ r', where r' is the conjugate index of r. The boundedness results in Sections 2 and 3 are then sharpened in Section 4 to results on compactness. Notwithstanding the gain from boundedness to compactness, the boundedness results in Sections 2 and 3 are of interest in their own right because the technique that we use to obtain compactness does not give us explicit estimates for the operator norms of the Daubechies operators in terms of the norms of the symbols and the two admissible wavelets.