On the Commutator of Marcinkiewicz Integrals with Rough Kernels in Variable Morrey-Type Spaces
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ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS IN VARIABLE MORREY-TYPE SPACES M. Qu1 and L. Wang2,3
UDC 517.5
Within the framework of variable-exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The proposed approach is based on the theory of variable exponents and on the generalization of the BMO-norms.
1. Introduction Let Rn be an n-dimensional Euclidean space of points x = (x1 , . . . , xn ) with the norm n ⇣X
|x| =
i=1
x2i
⌘1/2
.
Suppose that Sn−1 is a unit sphere in Rn , n ≥ 2, equipped with the normalized Lebesgue measure dσ(x0 ). Let ⌦ 2 L1 (Sn−1 ) be homogeneous of degree zero and such that Z
⌦(x0 )dσ(x0 ) = 0,
(1)
Sn−1
where x0 = x/|x| for any x 6= 0. Then the Marcinkiewicz integral operator µ⌦ of higher dimension is defined by 0
where
µ⌦ (f )(x) = @ F⌦,t (f )(x) =
Z1 0
11 2
dt |F⌦,t (f )(x)|2 3 A t
Z
|x−y|t
,
⌦(x − y) f (y) dy. |x − y|n−1
A locally integrable function b is said to be a BMO(Rn ) function if it satisfies the equality 1 kbk⇤ := sup n |B| x2R ,r>0
Z
|b(y) − bB | dy < 1,
B
1
School of Mathematics and Statistics, Anhui Normal University, Wuhu, China; e-mail: [email protected]. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, China; e-mail: [email protected]. 3 Corresponding author. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 928–944, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.6023. Original article submitted June 6, 2017; revision submitted October 29, 2018. 1080
0041-5995/20/7207–1080
© 2020
Springer Science+Business Media, LLC
O N THE C OMMUTATOR OF M ARCINKIEWICZ I NTEGRALS WITH ROUGH K ERNELS
1081
Z 1 where B is a ball of radius r centered at x, bB = b(t) dt and kbk⇤ is the norm in BMO(Rn ). Thus, for |B| B b 2 BMO(Rn ), the commutator of the Marcinkiewicz integral operator µ⌦,b is defined by � �2 112 � Z1 �� Z � dt ⌦(x − y) B � � C (b(x) − b(y))f (y) dy µ⌦,b (f ) = @ � � 3A . n−1 � � |x − y| t � 0 � |x−y|t 0
It is well known that Stein [23] first proved that if ⌦ 2 Lipγ (Sn−1 ), 0 < γ 1, then µ⌦ is of the type (p, p) for 1 < p 2 and of the weak type (1, 1). Later, Ding, Fan, and Pan [7] removed the assumption of smoothness for ⌦ and showed that µ⌦ is bounded on Lp (Rn ) for 1 < p < 1 if ⌦ 2 H 1 (Sn−1 ). Here, H 1 (Sn−1 ) denotes the classical Hardy space on Sn−1 . On the other hand, by using a good-λ inequality, Torchinsky and Wang [25] established the weighted Lp -boundedness of µ⌦ and µ⌦,b for ⌦ 2 Lipγ (Sn−1 ), 0 < γ 1. For some recent results, we refer the reader to [2, 8, 14–18] and the references therein. In recent years, following the fundamental work by Kov´acˇ ik and R´akosn´ık [13], function spaces with variable exponents, such as the variable-exponent Lebesgue, Herz and Morrey spaces etc., have attracted much attention mainly due to their useful applications in the fluid dynamics, image restoration, and differential equations with p(x)-growth (see [1, 3, 11, 30–32] and th
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