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Weighted inequalities for commutators of Schrödinger-Riesz transforms. (English) Zbl 1246.42018

Let V: d , d3, be a non-negative locally integrable function that belongs to a reverse Hölder class RH q for some q>d 2, that is, there exists C>0 such that the reverse Hölder inequality

1 |B| B V(x) q dx 1 q C |B| B V(x)dx(1)

holds for every ball B in d . The critical radius function ρ(x) is defined as follows:

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

Let =-Δ+V be a Schrödinger operator. And let = -1 2 be the associated Riesz transform vector and * be its adjoint operator.

The space BMO θ (ρ) is the set of locally integrable functions b satisfying

1 μ(B(x,r)) B(x,r) |b(y)-b B |dμ(y)C1 + r ρ(x) θ ,

for all x𝒳 and r>0, where θ>0 and b B =1 μ(B) B b(y)dμ(y). Denote BMO (ρ)= θ>0 BMO θ (ρ). Moreover, the authors introduce classes of weights that are given in terms of the critical radius function ρ(x) and they are larger than the classical A p weights.

For bBMO (ρ), the authors consider the commutator

T b (f)(x)=T(bf)(x)-b(x)Tf(x),x𝒳,

where T= or * .

Their main results are to obtain weighted L p , 1<p<, and weak L log L estimates for the commutator T b .

Reviewer: Yu Liu (Beijing)
MSC:
42B20Singular and oscillatory integrals, several variables
47F05Partial differential operators
47B47Commutators, derivations, elementary operators, etc.
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