Interpolation of a regular subspace complementing the span of a radially singular function

We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^\alpha$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the subspace of regular functions is still a closed subspace. The main tool is here a result by Ivanov and Kalton on interpolation of subspaces. To apply it, we study the $K$-functional of the $r^\alpha$-singular function. It turns out that the $K$-functional possesses upper and lower bounds that have a common decay rate at zero.