On an inequality of Tosio Kato for degenerate-elliptic operators

Abstract

Let Ω be a domain in R n and T = ∑ j, k = 1 n ( ∂ j − ib j ( x)) a jk ( x)( ∂ k − ib k ( x)), where the a jk and the b j are real valued functions in C 1(Ω) , and the matrix ( a jk ( x)) is symmetric and positive definite for every x ϵ Ω. If T 0 is the same as T but with b j = 0, j = 1,…, n, and if u and Tu are in L loc 1(Ω) , then T. Kato has established the distributional inequality T 0 ¦ u ¦ ⩾ Re[(sign ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T − q on R n . In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on R n .