A note on generalized Fujii-Wilson conditions and BMO spaces
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A NOTE ON GENERALIZED FUJII–WILSON CONDITIONS AND BMO SPACES BY
Sheldy Ombrosi∗ Instituto de Matem´ atica de Bah´ıa Blanca (INMABB), Departamento de Matem´ atica Universidad Nacional del Sur (UNS) - CONICET Av. Alem 1253, Bah´ıa Blanca, Argentina e-mail: [email protected] AND
Carlos P´ erez∗∗ Department of Mathematics, University of the Basque Country IKERBASQUE (Basque Foundation for Science) and BCAM –Basque Center for Applied Mathematics, Bilbao, Spain e-mail: [email protected] AND
Ezequiel Rela† Departamento de Matem´ atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria Pabell´ on I, Buenos Aires 1428 Capital Federal, Argentina e-mail: [email protected] AND
Israel P. Rivera-R´ıos∗ Instituto de Matem´ atica de Bah´ıa Blanca (INMABB), Departamento de Matem´ atica Universidad Nacional del Sur (UNS) - CONICET Av. Alem 1253, Bah´ıa Blanca, Argentina e-mail: [email protected] ∗ S. O. and I. P. R.-R. are supported by grants PIP (CONICET) 11220130100329CO
and PICT 2018-02501.
∗∗ This work was supported by the Spanish Ministry of Economy and Competitive-
ness, MTM2017-82160-C2-2-P and SEV-2017-0718.
† E. R. is partially supported by grants UBACyT 20020170200057BA and PICT-
2015-3675. Received April 2, 2019
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S. OMBROSI ET AL.
Isr. J. Math.
ABSTRACT
In this note we generalize the definition of the Fujii–Wilson condition providing quantitative characterizations of some interesting classes of weights, such as A∞ , Aweak and Cp , in terms of BMO type spaces suited to them. ∞ We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom’s BMO type spaces.
1. Introduction and main results Given a weight v, namely, a non-negative locally integrable function in Rn , and a functional Y : Q → (0, ∞) defined over the family of all cubes in Rn with sides parallel to the axes, we define the class of functions BMOv,Y by BMOv,Y = {f ∈ L1loc (Rn ) : f BMOv,Y < ∞} where f BMOv,Y := sup Q
1 Y (Q)
and, as usual, 1 fQ = |Q|
|f − fQ |v < ∞, Q
f Q
denotes the average of f over Q. In the case that Y (Q) = v(Q) for every cube Q, a classical result due to Muckenhoupt and Wheeden in [MW76] asserts that BMO = BMOv,v holds whenever v ∈ A∞ . Also, the case of Y (Q) = w(Q) for some weight w and v = 1 was considered in [MW76] and independently by J. Garc´ıa-Cuerva [GC79] in the context of Hardy spaces.1 Later on, S. Bloom [Blo85] also considered this special case in the context of commutators and used the notation BMOw to denote the space BMO1,w . We refer the reader to [GCHST91, HLW17, LORR17, LORR18, Hyt, AMPRR] for the latest advances and related results in that direction. The unweighted case v = 1 and Y (Q) = |Q| corresponds, obviously, to the classical BMO space of John–Nirenberg [JN61] and in that case we shall drop the subscripts. 1 Results involving that space appear in an abstract of that author in Notices of the AMS, February 1974, p. A-309.
Vol. TBD, 2020
GENERALIZED FUJII–WILSON
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