Uncertainty inequality on weighted Hardy spaces

Abstract

Abstract In this paper we introduce a weighted Hardy space ℋ β {\mathscr{H}_{\beta}} . This space generalizes some complex Hilbert spaces like the Dirichlet space 𝒟 {\mathscr{D}} , the Bergman space 𝒜 {\mathscr{A}} and the Segal–Bargmann space ℱ {\mathscr{F}} . It plays the role of background for our contribution. In particular, we study the derivative operator D and its adjoint operator L β {L_{\beta}} on ℋ β {\mathscr{H}_{\beta}} . Furthermore, we establish a general uncertainty inequality of Heisenberg type for the space ℋ β {\mathscr{H}_{\beta}} .