Approximate extension in Sobolev space

Let Lm,p(Rn) be the homogeneous Sobolev space for p is an element of (n, infinity), mu be a Borel regular measure on Rn, and Lm,p(Rn) + Lp(d mu) be the space of Borel measurable functions with finite seminorm ;Lm,p(Rn)+Lp(d mu) := inff1+f2=f 1pLm,p(Rn) + fRn |f2|pd mu}1/p. We construct a linear operator T : Lm,p(Rn) + Lp(d mu) -> Lm,p(Rn), that nearly optimally decomposes every function in the sum space: pLm,p(Rn) + ilen|Tf - f|pd mu <= C & pLm,p(Rn)+Lp(d mu) with C dependent on m, n, and p only. For E subset of Rn, let Lm,p(E) denote the space of all restrictions to E of functions F is an element of Lm,p(Rn), equipped with the standard trace seminorm. For p is an element of (n, infinity), we construct a linear extension operator T : Lm,p(E) -> Lm,p(Rn) satisfying Tf|E = f|E and Lm,p(Rn) <= C & Lm,p(E), where C depends only on n, m, and p. We show these operators can be expressed through a collection of linear functionals whose supports have bounded overlap. (c) 2023 Elsevier Inc. All rights reserved.