We formulate $\lambda$-deformed $\sigma$-models as QFTs in the upper-half plane. For different boundary conditions we compute correlation functions of currents and primary operators, exactly deformation parameter $\lambda$ for large values level $k$ underlying WZW model. To perform our computations use either conformal perturbation theory association with Cardy's doubling trick, well meromorphi...

In this paper, we obtain a sucient condition for boundedness of composition operators betweenweighted spaces of holomorphic functions on the upper half-plane whenever our weights are standardanalytic weights, but they don't necessarily satisfy any growth condition.