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Commutators of Riesz transforms related to Schrödinger operators. (English) Zbl 1213.42075

The authors consider the Schrödinger operator

𝔏=-Δ+V

on d , d3, where the nonnegative potential V(x) belongs to RH q , which is a class satisfying the following reverse Hölder inequality

1 |B| B V(x) q dx 1 q C1 |B| B V(x)dx(e1)

for every ball B in n . The auxiliary function ρ(x) related to V(x) is defined as

ρ(x)=1 m(x,V)= ˙sup r>0 r : 1 r n-2 B(x,r) V (y) d y 1,x n ·

For θ>0, a locally integrable function b is said to be in the class BMO θ (ρ) if

1 |B(x,r)| B(x,r) |b(y)-b B |dyC(1+r ρ(x)) θ ,

for all x d and r>0, where b B =1 |B| B b(y)dy· Define BMO (ρ)= θ>0 BMO θ (ρ). Clearly BMOBMO (ρ). For θ>0, a locally integrable function b is said to be in the class BMO θ log (ρ) if

1 |B(x,r)| B(x,r) |b(y)-b B |dyC(1+r ρ(x)) θ 1+log + (ρ(x)/r),

for all x d and r>0.

Denote =(-Δ+V) -1 2 . For a locally integrable function b, the commutators in this paper are defined as

b f(x)=(bf)(x)-b(x)f(x)

and

b * f(x)= * (bf)(x)-b(x) * f(x),

under the assumption that VRH q 0 for q 0 >d/2 and bBMO (ρ). The authors study the L p boundedness of the commutators b and b * . Especially, they prove that b * :L BMO 𝔏 is bounded if and only if bBMO θ log (ρ)· Moreover, this conclusion holds true for b when VRH d .

Reviewer: Liu Yu (Beijing)

MSC:
42B35Function spaces arising in harmonic analysis
35J10Schrödinger operator
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