Abstract
Let a function b belong to the space operatorname{BMO}_{theta }(rho ), which is larger than the space operatorname{BMO}(mathbb{R}^{n}), and let a nonnegative potential V belong to the reverse Hölder class mathit{RH}_{s} with n/2< s< n, ngeq 3. Define the commutator [b,T_{beta }]f=bT_{ beta }f-T_{beta }(bf), where the operator T_{beta }=V^{alpha } nabla mathcal{L}^{-beta }, beta -alpha =frac{1}{2}, frac{1}{2}< beta leq 1, and mathcal{L}=-Delta +V is the Schrödinger operator. We have obtained the L^{p}-boundedness of the commutator [b,T_{beta }]f and we have proved that the commutator is bounded from the Hardy space H^{1}_{mathcal{L}}(mathbb{R}^{n}) into weak L^{1}(mathbb{R}^{n}).
Highlights
Introduction and results LetL = – + V be the Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class RHs with s > n/2, n ≥ 3
We have obtained the Lp-boundedness of the commutator [b, Tβ ]f and we have proved that the commutator is bounded from the Hardy space HL1 (Rn) into weak L1(Rn)
1 Introduction and results Let L = – + V be the Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class RHs with s > n/2, n ≥ 3
Summary
The maximal function with respect to the semigroup {e–tL}t>0 is defined by MLf (x) = supt>0 |e–tLf (x)|. The Hardy space associated with L is defined as follows (see [3, 4]). Definition 1 We say that f is an element of HL1 (Rn) if the maximal function MLf belongs to L1(Rn). The quasi-norm of f is defined by f HL1 (Rn) = MLf L1(Rn). The space HL1 (Rn) admits the following atomic decomposition (see [3, 4]). Following [10], the space BMOθ (ρ) with θ ≥ 0 is defined as the set of all locally integrable functions b such that 1 |B(x, r)|
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