On The Sum Of A Sobolev Space And A Weighted $L_P$-Space

Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $\mu$ on $R^n$ we give a constructive characterization of the space $L^1_p(R^n)+L_p(R^n;\mu)$. We express the norm in this space in terms of certain oscillations with respect to the measure $\mu$. This enables us to describe the $K$-functional for the couple $(L_p(R^n;\mu),L^1_p(R^n))$ in terms of these oscillations, and to prove that this couple is quasi-linearizable.