New studies on a family of q-weighted Bergman spaces on the unit disk and applications

In this paper, we give a family of q-weightedBergman spaces\A{, ₍, ₐ\}₍ whichsatisfies the continuous inclusionA, ₍, ₐ. . . , ₁, ₐ, ₀, ₐ=A, ₐ, whereA, ₐ the q-weighted Bergman space. Moreover, a more general uncertainty inequality of theHeisenberg-type for the space A, ₍, ₐ isgiven by considering the operators, ₍, ₐ: =ⁿ, ₐ andL, ₍, ₐ: =Lⁿ, ₐ. Also, we study onA, ₐ the q-Toeplitz operators, theq-Hankel operators and the q-Berezin operators. Finally, anapplication of the theory of extremal function and reproducingkernel of Hilbert space is given and we use it to establish theextremal function associated to an bounded linear operatorT: A, ₐ H, for any Hilbert spaceH. As application, we come up with some results regarding theextremal functions associated to the difference operatorTf (z): =1z (f (z) -f (0) ) andTf (z): =11+q (f (z) -f (-z) ).