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

Abstract

Let $p>n$ and let $L_p^1(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $\mu$ on $R^n$ we give a constructive characterization of the space $L_p^1(R^n)+L_p(R^n;\mu )$. 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 $(L_p(R^n;\mu ),L_p^1(R^n))$ in terms of these oscillations, and to prove that this couple is quasi-linearizable.

63 pages

References in corpus (1)

• Sobolev Extension By Linear Operators

Cited by in corpus (2)

• Sobolev Extension By Linear Operators
• Sobolev $L_p^2$-functions on closed subsets of $R^2$