Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains.

Let n ≥ 2 , w be a Muckenhoupt A 2 (R n) weight, Ω a bounded Lipschitz domain of R n , and L : = − w − 1 div (A ∇ ⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted L p estimate for the Kato square root of L : ‖ L 1 / 2 (f) ‖ L p (Ω , v w) ∼ ‖ ∇ f ‖ L p (Ω , v w) for any f ∈ W 0 1 , p (Ω , v w) when L satisfies the Dirichlet boundary condition, or, for any f ∈ W 1 , p (Ω , v w) with ∫ Ω f (x) d x = 0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w , W 0 1 , p (Ω , v w) and W 1 , p (Ω , v w) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w , via letting v : = w − 1 , the unweighted L 2 estimate for the Kato square root of L that ‖ L 1 / 2 (f) ‖ L 2 (Ω) ∼ ‖ ∇ f ‖ L 2 (Ω) for any f ∈ W 0 1 , 2 (Ω) when L satisfies the Dirichlet boundary condition, or, for any f ∈ W 1 , 2 (Ω) with ∫ Ω f (x) d x = 0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L 2 estimates, the unweighted L 2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established. [ABSTRACT FROM AUTHOR]