Weighted L^p Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators

2015/09/18 by Dachun Yang, Junqiang Zhang, Yang, Dachun +1
Mathematics · #35J70 #42B30 #42B35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Primary 47B06 #Secondary 46E30 #math.CA #math.FA #msc:35J70 #msc:42B30 #msc:42B35 #msc:46E30 #msc:47B06

paper · pdf · doi:10.48550/arxiv.1509.05478

40 pages, Submitted

arxiv created 2015/09/18 · openalex publication_date 2015/09/18 · arxiv updated 2015/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let w be a Muckenhoupt A₂(ℝⁿ) weight and L_w:=-w^-1\mathopdiv(A∇) the degenerate elliptic operator on the Euclidean space ℝⁿ, n≥ 2. In this article, the authors establish some weighted L^p estimates of Kato square roots associated to the degenerate elliptic operators L_w. More precisely, the authors prove that, for w∈ A_p(ℝⁿ), p∈((2n)/(n+1), 2] and any f∈ C^∞_c(ℝⁿ), ‖L_w^1/2(f)‖_{L^p(w, ℝⁿ)}∼ ‖∇ f‖_{L^p(w, ℝⁿ)}, where C_c^∞(ℝⁿ) denotes the set of all infinitely differential functions with compact supports.

Citations

• Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Related

• The Kato problem and extensions for degenerate elliptic operators of higher order in weighted spaces
• The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds
• Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators
• The Kato square root problem for weighted parabolic operators
• The Kato problem for weighted elliptic and parabolic operators of higher order