The CMO-Dirichlet Problem for the Schrodinger Equation in the Upper Half-Space and Characterizations of CMO

Let L be a Schrodinger operator of the form L = -Delta + V acting on L-2 (R-n) where the non-negative potential V belongs to the reverse Holder class RHq for some q >= (n + 1)/2. Let CMOL(R-n) denote the function space of vanishing mean oscillation associated to L. In this article, we will show that a function f of CMOL(R-n) is the trace of the solution to Lu = -u(tt) + Lu = 0, u(x, 0) = f (x), if and only if, u satisfies a Carleson condition sup(B:balls) C-u,C-B := sup(B(xB, rB): balls) r(B)(-n) integral(rb)(0) integral(B(xB, rB)) vertical bar t del u(x, t)vertical bar(2)dx dt/t < infinity, and lim(a -> 0) sup(B:rb <= a) C-u,C-B = lim(a ->infinity) sup(B:rb >= a) C-u,C-B = lim(a ->infinity) sup(B:B subset of(B(0,a))c) C-u,C-B=0. This continues the lines of the previous characterizations by Duong et al. (J Funct Anal 266(4):2053-2085, 2014) and Jiang and Li (ArXiv:2006.05248v1) for the BMOL spaces, which were founded by Fabes et al. (Indiana Univ Math J 25:159-170, 1976) for the classical BMO space. For this purpose, we will prove two new characterizations of the CMOL(R-n) space, in terms of mean oscillation and the theory of tent spaces, respectively.