Product BMO, Little BMO, and Riesz Commutators in the Bessel Setting

Abstract

Abstract. In this paper, we study the product BMO space, little bmo space and their connections with the corresponding commutators associated with Bessel operators studied by Weinstein, Huber, and by Muckenhoupt-Stein. We first prove that the product BMO space in the Bessel setting can be used to deduce the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little bmo space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators, where the main tool that we develop is the characterization of the predual of little bmo and its weak factorizations. We further show that in analogy with the classical setting, the little bmo space is a proper subspace of the product BMO space. These extend the previous related results studied by Cotlar-Sadosky and Ferguson-Sadosky on the bidisc to the Bessel setting, where the usual analyticity and Fourier transform do not apply.

Off-diagonal estimates for bi-commutators

“…For example, the two-weight commutator estimates, which include [17,18,19,22,31,32,33,34], have been one of the main lines of development. Commutators are also actively studied in other settings: see for example [10] for the flag setting, [11] for the Zygmund dilation setting and [12] for the Bessel setting.…”

𝐁𝐌𝐎 on shapes and sharp constants

“…There has also been some attention given to the space defined by a mean oscillation condition over rectangles with sides parallel to the axes, either in R n or on a domain in R n , appearing in the literature under various names. For instance, in [43], the space is called "anisotropic BMO" to highlight the contrast with cubes, while in papers such as [16,21,22,25], it goes by "little BMO" and is denoted by bmo. The notation bmo, however, had already been used for the "local BMO" space of Goldberg ([30]), a space that has been established as an independent topic of study (see, for instance, [8,20,63]).…”

Geometric maximal operators and BMO on product bases

“…Just as cubes can be replaced by rectangles in the definition of the maximal function, the same can be done with the definition of BMO(R n ) (and, likewise, with BLO(R n )). The resulting space, strong BMO, has appeared in the literature under different names (see [7,12,22]).…”