  Suppose q is a complex number of modulus one and different from 1,-1. Let
O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the
relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on
Hilbert space and the Weyl calculus of pseudodifferential operators we
construct *-algebras of ``functions'' on the quantum quarter plane R^{++}_q and
on the real quantum plane R^2_q which are left module *-algebras for the Hopf
*-algebra U_q(gl_2(R)). We define a family h_k, k\in Z^2, of covariant positive
linear functionals on these *-algebras and study the actions of the *-algebras
O(R^2_q) and U_q(gl_2(R)) on the associated Hilbert spaces. Quantum analogs of
the partial Fourier transforms and the Fourier transform are found. A
differential calculus on the ``function'' *-algebras is also developed and
investigated.
  A shorter version appeared in International Journal of Mathematics 13 (2002),
279-321.
