  The most general form of a marginal extended perturbation in a
two-dimensional system is deduced from scaling considerations. It includes as
particular cases extended perturbations decaying either from a surface, a line
or a point for which exact results have been previously obtained. The
first-order corrections to the local exponents, which are functions of the
amplitude of the defect, are deduced from a perturbation expansion of the
two-point correlation functions. Assuming covariance under conformal
transformation, the perturbed system is mapped onto a cylinder. Working in the
Hamiltonian limit, the first-order corrections to the lowest gaps are
calculated for the Ising model. The results confirm the validity of the
gap-exponent relations for the perturbed system.
