The Wigner transformation associated with the Hankel multidimensional operator

Abstract

Abstract We consider the Hankel multidimensional operator Δ α \Delta_{\alpha} , α = ( α 1 , … , α n ) ∈ ] − 1 2 , + ∞ [ n \alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathopen{]}-\frac{1}{2},+\infty\mathclose{[}^{n} , defined on ] 0 , + ∞ [ n \mathopen{]}0,+\infty\mathclose{[}^{n} by Δ α = ∑ j = 1 n ( ∂ 2 ∂ x j 2 + 2 ⁢ α j + 1 x j ⁢ ∂ ∂ x j ) . \Delta_{\alpha}=\sum_{j=1}^{n}\biggl{(}\frac{\partial^{2}}{\partial x_{j}^{2}}+\frac{2\alpha_{j}+1}{x_{j}}\frac{\partial}{\partial x_{j}}\biggr{)}. We define and study the Wigner transformation V g \mathscr{V}_{g} (called also Gabor transform), where g ∈ L 2 ⁢ ( d ⁢ μ α ) g\in L^{2}(d\mu_{\alpha}) and d ⁢ μ α d\mu_{\alpha} is the measure defined on [ 0 , + ∞ [ n [0,+\infty\mathclose{[}^{n} by d ⁢ μ α ⁢ ( x ) = ⨂ j = 1 n x j 2 ⁢ α j + 1 2 α j ⁢ Γ ⁢ ( α j + 1 ) ⁢ d ⁢ x j . d\mu_{\alpha}(x)=\bigotimes_{j=1}^{n}\frac{x_{j}^{2\alpha_{j}+1}}{2^{\alpha_{j}}\Gamma(\alpha_{j}+1)}\,dx_{j}. Using harmonic analysis related to the Hankel operator Δ α \Delta_{\alpha} , we prove a Plancherel theorem and an orthogonality property for the transformation V g \mathscr{V}_{g} . Next, we establish a reconstruction formula for V g \mathscr{V}_{g} and give some applications. In the second part of this work, as applications of the Wigner transformation V g \mathscr{V}_{g} , we define and study the anti-Wick operators A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) , where g 1 , g 2 ∈ L 2 ⁢ ( d ⁢ μ α ) g_{1},g_{2}\in L^{2}(d\mu_{\alpha}) are called window functions and σ ∈ L p ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) \sigma\in L^{p}(d\mu_{\alpha}\otimes d\mu_{\alpha}) is a signal. Building on the properties of the Wigner transformation V g \mathscr{V}_{g} , we prove that the operators A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) are bounded linear operators and compact on the Hilbert space L 2 ⁢ ( d ⁢ μ α ) L^{2}(d\mu_{\alpha}) . Finally, we establish a formula of the trace for the anti-Wick operator A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) when the signal 𝜎 belongs to L 1 ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) L^{1}(d\mu_{\alpha}\otimes d\mu_{\alpha}) .