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Abstract
Let L be a linear operator in L
2(ℝ
n
) and generate an analytic semigroup {e
−tL
}
t≥0 with kernels satisfying an upper bound of Poisson type. In this article, the authors introduce the localized Hardy space via molecules and show that
, where
and
are Hardy spaces associated with L and L + I, respectively. Characterizations of
via the localized Lusin area function and
are established. Then, the authors introduce the localized BMO space bmo
L
(ℝ
n
) and prove that the dual of
is bmo
L*(ℝ
n
), where L* denotes the adjoint operator of L in L
2(ℝ
n
). The John–Nirenberg inequality for elements in bmo
L
(ℝ
n
) and a characterization of bmo
L
(ℝ
n
) via BMO
L
(ℝ
n
) are also established, where BMO
L
(ℝ
n
) is the BMO space associated with L. As applications, the authors obtain the characterizations of the localized Hardy space
associated to the Schrödinger operator L = −Δ + V, where
is a nonnegative potential, in terms of the localized Lusin-area functions and the localized radial maximal functions.
AMS Subject Classifications:
Acknowledgement
Dachun Yang is supported by the National Natural Science Foundation (Grant No. 10871025) of China.