Uncertainty principles of Heisenberg type for the Bargmann transform.

In this work, we introduce a family of weighted Bergman spaces { A α , n } n ∈ N . This family satisfies the continuous inclusions A α , n ⊂ ⋯ ⊂ A α , 2 ⊂ A α , 1 ⊂ A α , 0 = A α , where A α is the classical weighted Bergman space. Next, we define and study the derivative operator ∇ = d d z and its adjoint operator L α = z 2 d d z + (α + 2) z on the weighted Bergman space A α , and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space A α , n is also given when we considered the operators ∇ n = ∇ n and L α , n : = (L α) n . Afterward, we give Heisenberg-type and Laeng-Morpurgo-type uncertainty inequalities for the Bargmann transform B α , which is an isometric isomorphism between the space A α and the Lebesgue space L 2 (R + , d μ α) , where d μ α is an appropriate measure. [ABSTRACT FROM AUTHOR]