Kato square root problem with unbounded leading coefficients

2017/12/28 by Escauriaza, Luis, Hofmann, Steve
#35B45 #35J15 #35J25 #35J70 #42B20 #42B37 #47A07 #47B44 #47D06 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.09808

Abstract

We prove the Kato conjecture for elliptic operators, L=-∇⋅((\mathbf A+\mathbf D)∇ ), with \mathbf A a complex measurable bounded coercive matrix and \mathbf D a measurable real-valued skew-symmetric matrix in ℝⁿ with entries in BMO(ℝⁿ); i.e., the domain of √(L) is the Sobolev space H¹(ℝⁿ) in any dimension, with the estimate ‖√(L) f‖₂\lesssim ‖ ∇ f‖₂.

Related

• The Kato square root problem on an arbitrary domain of ℝ^d
• The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds
• The Kato square root problem for parabolic operators with an anti-symmetric part in BMO
• The Kato problem for weighted elliptic and parabolic operators of higher order
• Kato's square root problem in Banach spaces