On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces |
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基金项目: | The first author is supported in part by Guangdong Natural Science Funds for Distinguished Young Scholar (Grant No. 2016A030306040), NSF of Guangdong (Grant No. 2014A030313417) and NNSF of China (Grant Nos. 11471338 and 11622113), the third author is supported by the NNSF of China (Grant Nos. 11371378 and 11521101) and Guangdong Special Support Program |
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摘 要: | Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.
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