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The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover , the Hardy-Littlewood-Sobolev inequality is established.
Let a a1, a2, . . . , am ∈ C be an m-dimensional vector. Then, it can be identified with an m ×m circulant matrix. By using the theory of matrix-valued wavelet analysis Walden and Serroukh, 2002 , we discuss the vector-valued multiresolution analysis. Also, we derive several different designs of finite length of vector-valued filters. The corresponding… (More)
In this article, we introduce a kind of unitary operator U associated with the involution on the Heisenberg group, invariant closed subspaces are identified with the characterization spaces of sub-Laplacian operators. In the sense of vector-valued functions, we study the theory of continuous wavelet transform. Also, we obtain a new inversion formula of… (More)
In this article, two types of Hardy's inequalities for the twisted convolution with Laguerre functions are studied. The proofs are mainly based on an estimate for the Heisenberg left-invariant vectors of the special Hermite functions deduced by the Heisenberg group approach.
In the past decade research on the multiresolution analysis has made considerable progress due to its wide applications. For the basic theory of multiresolution we refer readers to the work in 1, 2 . Recently, we find that a lot of authors try to extend the theory of wavelets on the Euclidean space to nilpotent Lie groups see 3–6 . In this paper we will… (More)
where φt x t−1φ x/t , ψt x t−1ψ x/t , and ∗ denotes the convolution on R. The Calderón reproducing formula is a useful tool in pure and applied mathematics see 1– 4 , particularly in wavelet theory see 5, 6 . We always call 1.1 an inverse formula of wavelet transform. In 7 , the authors generalized 1.1 to R when φ and ψ are sufficiently nice normalized… (More)