Approximate Extension in Sobolev Space

2020/11/21 by Drake, Marjorie K.
#41A05 (Secondary) #42B99 #46E35 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2011.10855

Abstract

Let L^m,p(ℝⁿ) be the homogeneous Sobolev space for p ∈ (n,∞), μ be a Borel regular measure on ℝⁿ, and L^m,p(ℝⁿ) + L^p(dμ) be the space of Borel measurable functions with finite seminorm ‖f‖_{L^m,p(ℝⁿ) + L^p(dμ)} := inf_{f1 +f2 = f} \ ‖f₁‖_{L^m,p(ℝⁿ)}^p + ∫_ℝⁿ |f₂|^p dμ\^1/p. We construct a linear operator T:L^m,p(ℝⁿ) + L^p(dμ) → L^m,p(ℝⁿ), that nearly optimally decomposes every function in the sum space: ‖Tf‖_{L^m,p(ℝⁿ)}^p + ∫_ℝⁿ |Tf-f|^p dμ≤ C ‖f‖_{L^m,p(ℝⁿ) + L^p(dμ)}^p with C dependent on m, n, and p only. For E ⊂ ℝⁿ, let L^m,p(E) denote the space of all restrictions to E of functions F ∈ L^m,p(ℝⁿ), equipped with the standard trace seminorm. For p ∈ (n, ∞), we construct a linear extension operator T:L^m,p(E) → L^m,p(ℝⁿ) satisfying Tf|_E = f|_E and ‖Tf‖_{L^m,p(ℝⁿ)} ≤ C ‖f‖_L^m,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.

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