Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces

Abstract

Let  be a Schrödinger operator of the form  = −Δ + 𝑉 acting on 𝐿 2 (ℝ 𝑛 ) where the nonnegative potential 𝑉 belongs to the reverse Hölder class 𝐵 𝑞 for some 𝑞 ≥ 𝑛. Let BMO  (ℝ 𝑛 ) denote the BMO space on ℝ 𝑛 associated to the Schrödinger operator . In this article we will show that a function 𝑓 ∈ BMO  (ℝ 𝑛 ) is the trace of the solution of 𝕃𝑢 ∶= 𝑢 𝑡 + 𝑢 = 0, 𝑢(𝑥, 0) = 𝑓(𝑥), where 𝑢 satisfies a Carleson-type conditionConversely, this Carleson-type condition characterizes all the 𝕃-carolic functions whose traces belong to the space BMO  (ℝ 𝑛 ).

Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators

“…Now, we prove this proposition for the case |h| < t 1/2α . For |h| < |x − y|/2 < t 1/2α or |h| < t 1/2α < |x − y|/2, the desired estimate can be deduced from (19) and (20). The case |x − y|/2 < |h| < t 1/2α remains to be considered.…”

“…Ma-Stinga-Torrea-Zhang [14] characterized the Campanato-type spaces associated with L via the fractional derivatives of the Poisson semigroup. For further information on this topic, we refer to [15][16][17][18][19][20][21] and the references therein. Assume that L = −∆ + V, with V ∈ B q , q > n. By the regularity estimates obtained in Section 3, we establish the following equivalent characterizations: for 0 < γ < min{2α, 2αβ},…”

Regularity of fractional heat semigroup associated with Schrödinger operators

“…Ma-Stinga-Torrea-Zhang [23] characterized the Campanato type spaces associated with L via the fractional derivatives of the Poisson semigroup. For further information on this topic, we refer to [4,5,17,19,28,31,32] and the references therein. Assume that L = −∆+V with V ∈ B q , q > n. By the regularity estimates obtained in Section 3, we establish the following equivalent characterizations: for 0 < γ < min{2α, 2αβ},…”