  The conformal mapping w=(L/2\pi)\ln z transforms the critical plane with a
radial perturbation \alpha\rho^{-y} into a cylinder with width L and a constant
deviation \alpha(2\pi/L)^y from the bulk critical point when the decay exponent
y is such that the perturbation is marginal. From the known behavior of the
homogeneous off-critical system on the cylinder, one may deduce the correlation
functions and defect exponents on the perturbed plane. The results are
supported by an exact solution for the Gaussian model.
