The CMO-Dirichlet problem for the Schrödinger equation in the upper half-space and characterizations of CMO

2021/07/01 by Song, Liang, Wu, Liangchuan
#35J10 #42B35 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.00496

Abstract

Let L be a Schrödinger operator of the form L=-Δ+V acting on L²(\mathbb Rⁿ) where the nonnegative potential V belongs to the reverse Hölder class RH_q for some q≥ (n+1)/2. Let CMO_L(ℝⁿ) denote the function space of vanishing mean oscillation associated to L. In this article we will show that a function f of CMO_L(ℝⁿ) is the trace of the solution to \mathbbLu=-u_tt+L u=0, u(x,0)=f(x), if and only if, u satisfies a Carleson condition sup_B: ballsC_u,B :=sup_B(x_B,r_B): balls r_B^-n∫₀^r_B∫_{B(xB, rB)} |t ∇ u(x,t)|² \frac dx dt t

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