Uncertainty principles of Heisenberg type for the Bargmann transform

Abstract

In this work, we introduce a family of weighted Bergman spaces $\{{\mathcal {A}}_{\alpha ,n}\}_{n\in {\mathbb {N}}}$. This family satisfies the continuous inclusions ${\mathcal {A}}_{\alpha ,n}\subset \cdots \subset {\mathcal {A}}_{\alpha ,2}\subset {\mathcal {A}}_{\alpha ,1}\subset {\mathcal {A}}_{\alpha ,0}={\mathcal {A}}_{\alpha }$, where ${\mathcal {A}}_{\alpha }$ is the classical weighted Bergman space. Next, we define and study the derivative operator $\nabla =\frac{\text{ d }}{\text{ d }z}$ and its adjoint operator $L_{\alpha }=z^2\frac{\text{ d }}{\text{ d }z}+(\alpha +2) z$ on the weighted Bergman space ${\mathcal {A}}_{\alpha }$, and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space ${\mathcal {A}}_{\alpha ,n}$ is also given when we considered the operators $\nabla _n=\nabla ^n$ and $L_{\alpha ,n}:=(L_{\alpha })^n$. Afterward, we give Heisenberg-type and Laeng-Morpurgo-type uncertainty inequalities for the Bargmann transform $B_{\alpha }$, which is an isometric isomorphism between the space ${\mathcal {A}}_{\alpha }$ and the Lebesgue space $L^2({\mathbb {R}}_+,\text{ d }\mu _{\alpha })$, where $\text{ d }\mu _{\alpha }$ is an appropriate measure.