In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L2(Hd) by using self-similar tilings for the acceptable dilations on the Heisenberg group.
Let L=-Δ + V be a Schrdinger operator in R^d d > 3, where the nonnegative potentialV belongs to the reverse Hlder class B_(d). We establish the BMO_L-boundedness of Riesz transforms(■)L^(-1/2), and give the Fefferman-Stein type decomposition of BMO_L functions.
We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L p to L p if 1 ≤ p ≤ 4 3 . This is different from the Heisenberg group, on which the restriction operator is not bounded from L p to L p unless p = 1.
