Approximate extension in Sobolev space

Let Lm,p(Rn) be the homogeneous Sobolev space for p?(n,?), ? a Borel regular measure on Rn, and Lm,p(Rn)+Lp(d?) of measurable functions with finite seminorm ?f?Lm,p(Rn)+Lp(d?):=inff1+f2=f?{?f1?Lm,p(Rn)p+?Rn|f2|pd?}1/p. We construct linear operator T:Lm,p(Rn)+Lp(d?)?Lm,p(Rn), that nearly optimally decomposes every function in sum space: ?Tf?Lm,p(Rn)p+?Rn|Tf?f|pd??C?f?Lm,p(Rn)+Lp(d?)p C dependent m, n, p only. For E?Rn, let Lm,p(E) denote all restrictions to E F?Lm,p(Rn), equipped standard trace seminorm. we extension T:Lm,p(E)?Lm,p(Rn) satisfying Tf|E=f|E ?Tf?Lm,p(Rn)?C?f?Lm,p(E), where depends only p. show these operators can expressed through collection functionals whose supports have bounded overlap.