2023/11/06 by Cong Chen, Hua Wang, Chen, Cong +1
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Advanced Harmonic Analysis Research
paper · pdf · doi:10.48550/arxiv.2311.03407
We consider the Schrödinger operator L=-Δ+V on \mathbb Rd, d≥3, where the nonnegative potential V belongs to the reverse Hölder class RHs for some s≥ d/2. A real-valued function f∈ L1loc(\mathbb Rd) belongs to the (BMO) space BMOρ,θ(\mathbb Rd) with 0<θ<∞ if ‖f‖_BMOρ,θ :=supB(x0,r)(1+(r)/(ρ(x0)))-θ((1)/(|B(x0,r)|)∫B(x0,r)|f(x)-fB| dx), where the supremum is taken over all balls B(x0,r)⊂\mathbb Rd, ρ(⋅) is the critical radius function in the Schrödinger context and fB:=(1)/(|B(x0,r)|)∫B(x0,r)f(y) dy. A real-valued function f∈ L1loc(\mathbb Rd) belongs to the (Lipschitz) space Lipβρ,θ(\mathbb Rd) with 0<β<1 and 0<θ<∞ if ‖f‖Lipβρ,θ :=supB(x0,r)(1+(r)/(ρ(x0)))-θ (\frac1|B(x0,r)|1+β/d∫B(x0,r)|f(x)-fB| dx). It can be easily seen that BMOρ,θ(\mathbb Rd) (or Lipβρ,θ(\mathbb Rd)) is a function space which is larger than the classical BMO (or Lipschitz) space. In this paper, we give some new characterizations of BMO and Lipschitz spaces associated with the Schrödinger operator L. We extend some previous works of Bongioanni--Harboure--Salinas and Liu--Sheng to the weighted case. The classes of weights considered here are larger than the classical Muckenhoupt classes.