Regularity of fractional heat semigroup associated with Schrödinger operators

2020/12/14 by Li, P., Wang, Z., Qian, T. +1 · 1 citation
#35J10 #42B20 #42B30 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.07234

Abstract

Let L=-Δ+V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup \e^-tL^α\_t>0, α>0, associated with L. By the aid of the fundamental solution of the heat equation: ∂_tu+L u=∂_tu -Δu+Vu=0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K^L_α,t(⋅, ⋅), respectively. This method is independent of the Fourier transform, and can be applied to the second order differential operators whose heat kernels satisfying Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato type space BMO^γ_L(ℝⁿ) via \e^-tL^α\_t>0.

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