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

Let L be a Schrödinger operator of the form L = - Δ + V acting on L 2 ( R n ) where the non-negative potential V belongs to the reverse Hölder class RH q for some q ≥ ( n + 1 ) / 2 . Let CMO L ( R n ) denote the function space of vanishing mean oscillation associated to L . In this article, we will show that a function f of CMO L ( R n ) is the trace of the solution to L u = - u tt + L u = 0 , u ( x , 0 ) = f ( x ) , if and only if, u satisfies a Carleson condition sup B : balls C u , B : = sup B ( x B , r B ) : balls r B - n ∫ 0 r B ∫ B ( x B , r B ) | t ∇ u ( x , t ) | 2 dx dt t < ∞ , and lim a → 0 sup B : r B ≤ a C u , B = lim a → ∞ sup B : r B ≥ a C u , B = lim a → ∞ sup B : B ⊆ B ( 0 , a ) c C u , 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 BMO L 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 CMO L ( R n ) space, in terms of mean oscillation and the theory of tent spaces, respectively.