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

Let n >= 2, w be a Muckenhoupt A(2)(R-n) weight, Omega a bounded Lipschitz domain of R-n, and L := -w(-1) div(A Delta center dot) the degenerate elliptic operator on Omega 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)||(Lp(Omega, vw)) similar to ||del f||(Lp(Omega, vw)) for any f is an element of W-0(1, p) (Omega, vw) when L satisfies the Dirichlet boundary condition, or, for any f is an element of W-1,W- p (Omega,W- vw) with integral(Omega) f(x)dx = 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 Holder class with respect to w, W-0(1, p) (Omega, vw) and W-1,W- p (Omega, vw) denote the weighted Sobolev spaces on Omega, 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)||(L2(Omega)) similar to ||del f||(L2(Omega)) for any f is an element of W-0(1, p) (Omega) when L satisfies the Dirichlet boundary condition, or, for any f is an element of W-0(1, 2) with (Omega) f(Omega) f(x)dx = 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 Omega with the Dirichlet or the Neumann boundary condition are also established. (c) 2023 Elsevier Inc. All rights reserved.