Uncertainty principles of Heisenberg type for the Bargmann transform

In this work, we introduce a family of weighted Bergman spaces {A(alpha,n)}(n is an element of N). This family satisfies the continuous inclusions A(alpha,n) subset of ... subset of A(alpha,2) subset of A(alpha,1) subset of A(alpha,0) = A(alpha), where A(alpha) is the classical weighted Bergman space. Next, we define and study the derivative operator del = d/dz and its adjoint operator L-alpha = z(2) d/dz + (alpha + 2)z on the weighted Bergman space A(alpha), and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space A(alpha,n) is also given when we considered the operators del(n) = del(n) and L-alpha,L-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 A(alpha) and the Lebesgue space L-2(R+,d mu(alpha)), where d mu(alpha) is an appropriate measure.