(1)  Using a method of simultaneous uniformization, we parameterize the space of Weil-Petersson curves within the product of the integrable Teichmüller space, which is defined by the integrability of Beltrami coefficients. This approach establishes a biholomorphic homeomorphism from the space of such embeddings into the Banach space of Besov functions. Analogous results for chord-arc curves and the BMO Teichmüller space are also presented to support the above arguments.

Several consequences and applications of these results are derived:  

(i) The integrable Teichmüller space is shown to be real-analytically equivalent to the Besov space.  

(ii) The correspondence between the Riemann mapping parameterizations and the arc-length parameterizations of Weil-Petersson curves is explicitly formulated.  

(iii) The Cauchy transform of Besov functions on Weil-Petersson curves is expressed in terms of the derivative of this biholomorphic map.

In the case of chord-arc curves, this framework leads to the Calderon theorem. It also follows that the Cauchy transforms on Weil-Petersson and chord-arc curves depend holomorphically on their embeddings as these vary in the Bers coordinates.

(2)  On the two subsurfaces of a Riemann surface divided by a p-Weil-Petersson curve c, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of c. By requiring the coincidence of boundary values on c, we establish a correspondence between the harmonic functions in these Banach spaces.  We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of p-Besov functions on the unit circle.