Uncertainty principles of Heisenberg type on Dirichlet space

Abstract

In this paper, we introduce a family of Dirichlet spaces $$\{{\mathcal {D}}_n\}_{n\in {\mathbb {N}}}$$ . This family satisfies the continuous inclusions $${\mathcal {D}}_{n}\subset \cdots \subset {\mathcal {D}}_{2}\subset {\mathcal {D}}_{1}\subset {\mathcal {D}}_0={\mathcal {D}}$$ , where $${\mathcal {D}}$$ is the classical Dirichlet space. Next, we define and study the operator $$Xf(z):=f'(z)-f'(0)$$ and its adjoint operator $$Yf(z)=z^2f'(z)$$ on the Dirichlet space $${\mathcal {D}}$$ , and we establish an uncertainty inequality of Heisenberg type for this space. A more general uncertainty inequality for the space $${\mathcal {D}}_n$$ is also given when we considered the operators $$X_n=X^n$$ and $$Y_n=Y^n$$ .