Uncertainty principles of Heisenberg type on Dirichlet space

In this paper, we introduce a family of Dirichlet spaces {Dn}n∈N. This family satisfies the continuous inclusions Dn⊂ ⋯ ⊂ D2⊂ D1⊂ D= D, where 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) = z2f′(z) on the Dirichlet space D, and we establish an uncertainty inequality of Heisenberg type for this space. A more general uncertainty inequality for the space Dn is also given when we considered the operators Xn= Xn and Yn= Yn. © 2021, Università degli Studi di Ferrara.