2024/11/07 by Chen, Cong, Wang, Hua
#35J10 #42B25 #42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.04377
Let us consider the Schrödinger operator L=-Δ+V on \mathbb Rd with d≥3, where Δ is the Laplacian operator on \mathbb Rd and the nonnegative potential V belongs to certain reverse Hölder class RHs with s≥ d/2. In this paper, the authors first introduce two kinds of function spaces related to the Schrödinger operator L. A real-valued function f∈ L1loc(\mathbb Rd) belongs to the (BLO) space BLOρ,θ(\mathbb Rd) with 0≤θ<∞ if ‖f‖_BLOρ,θ :=supQ(1+(r)/(ρ(x0)))-θ((1)/(|Q(x0,r)|) ∫Q(x0,r)[f(x)-\undersety\inQess inf f(y)] dx), where the supremum is taken over all cubes Q=Q(x0,r) in \mathbb Rd, ρ(⋅) is the critical radius function in the Schrödinger context. For 0<β<1, a real-valued function f∈ L1loc(\mathbb Rd) belongs to the (Campanato) space Cβ,∗ρ,θ(\mathbb Rd) with 0≤θ<∞ if ‖f‖_Cβ,∗ρ,θ :=supB(1+(r)/(ρ(x0)))-θ (\frac1|B(x0,r)|1+β/d∫B(x0,r)[f(x)-\undersety\inBess inf f(y)] dx), where the supremum is taken over all balls B=B(x0,r) in \mathbb Rd. Then we establish the corresponding John--Nirenberg inequality suitable for the space BLOρ,θ(\mathbb Rd) with 0≤θ<∞ and d≥3. Moreover, we give some new characterizations of the BLO and Campanato spaces related to L on weighted Lebesgue spaces, which is the extension of some earlier results.