manuscripta math. 127, 219–234 (2008) © Springer-Verlag 2008
Xuan Thinh Duong · Lixin Yan
Commutators of Riesz transforms of magnetic
Schrödinger operators
Received: 27 March 2007 / Revised: 25 February 2008
Published online: 18 July 2008
Abstract. Let A =-(∇- i a) · (∇- i a) + V be a magnetic Schrödinger operator acting
on L
2
(R
n
), n ≥ 1, where a = (a
1
,..., a
n
) ∈ L
2
loc
(R
n
, R
n
) and 0 ≤ V ∈ L
1
loc
(R
n
).
In this paper, we show that when a function b ∈ BMO(R
n
), the commutators [b, T
k
] f =
T
k
(bf ) - bT
k
f , k = 1,..., n, are bounded on L
p
(R
n
) for all 1 < p < 2, where the
operators T
k
are Riesz transforms (∂/∂ x
k
- ia
k
) A
-1/2
associated with A.
1. Introduction
Consider a real vector potential a = (a
1
,..., a
n
) and an electric potential V . In
this paper, we assume that
a
k
∈ L
2
loc
(R
n
), ∀k = 1,..., n, (1.1)
0 ≤ V ∈ L
1
loc
(R
n
). (1.2)
Let L
k
= ∂/∂ x
k
- ia
k
. We define the form Q by
Q( f , g) =
n
k =1
R
n
L
k
f L
k
gdx +
R
n
Vf gdx
with domain
D( Q) ={ f ∈ L
2
(R
n
), L
k
f ∈ L
2
(R
n
) for k = 1,..., n and
√
Vf ∈ L
2
(R
n
)}.
It is well known that this symmetric form is closed. Note also that it was shown
by Simon [13] that this form coincides with the minimal closure of the form given
X. T. Duong is supported by a grant from Australia Research Council. L. X. Yan is
supported by NCET of Ministry of Education of China and NNSF of China (Grant No.
10571182/10771221).
X. T. Duong (B ): Department of Mathematics, Macquarie University, North Ryde, NSW
2109, Australia. e-mail: duong@ics.mq.edu.au
L. Yan: Department of Mathematics, Zhongshan University, 510275 Guangzhou,
People’s Republic of China. e-mail: mcsylx@mail.sysu.edu.cn
Mathematics Subject Classification (2000): Primary 42B20; Secondary 42B25, 47B38
DOI: 10.1007/s00229-008-0202-y