Product Hardy, BMO spaces and iterated commutators associated with Bessel Schrodinger operators

Abstract

In this paper we establish the product Hardy spaces associated with the Bessel Schrödinger operator introduced by Muckenhoupt and Stein, and provide equivalent characterizations in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions, and Littlewood-Paley theory, which are consistent with the classical product Hardy space theory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide another characterization via the Telyakovskií transform, which further implies that the product Hardy space associated with this Bessel Schrödinger operator is isomorphic to the subspace of suitable "odd functions" in the standard Chang-Fefferman product Hardy space. Based on the characterizations of these product Hardy spaces, we study the boundedness of the iterated commutator of the Bessel Riesz transforms and functions in the product BMO space associated with Bessel Schrödinger operator. We show that this iterated commutator is bounded above, but does not have a lower bound. √ S λ or T t := e −tS λ .We now define the product Hardy space via the Littlewood-Paley area functions as follows.We now define another version of the Littlewood-Paley area function. Let ∇ t 1 , y 1 := (∂ t 1 , ∂ y 1 ), ∇ t 2 , y 2 := (∂ t 2 , ∂ y 2 ). Then the Littlewood-Paley area function∇ t 1 , y 1 e −t 1 √ S λ ∇ t 2 , y 2 e −t 2 √ S λ (f )(y 1 , y 2 ) 2 dy 1 dy 2 dt 1 dt 2 1 2. (1.4)

Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

“…(ii) provide a new proof of equivalent characterizations via Littlewood-Paley square functions of the product Hardy spaces on spaces of homogeneous type in [18] whose proofs required the Hölder continuity and cancellation condition, (iii) provide the missing characterizations of product Hardy spaces via Littlewood-Paley square functions in the setting developed in [9] and [12], and (iv) recover the recent related known results in the setting of Bessel operators in [11] whose proofs relied on the Hölder regularity, and results for Bessel Schrödinger operators in [2] whose proofs used the Moser type inequality.…”

“…In [2], Betancor et al established the product Hardy space H p S λ (R + × R + ) associated with △ λ via the Littlewood-Paley area function and square functions. To prove the equivalence, they need to use the Poisson semigroup {e −t √ S λ }, the subordination formula and the Moser type inequality as a bridge.…”

End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

“…also see in [9]. After Wang [10] considered the g-function defined on BMO functions, more and more scholars pay attention to the end point estimate of the Littlewood-Paley operator [3,8,[11][12][13]. In this paper, we will also consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in BMO θ,L .…”

John--Nirenberg-$Q$ Spaces via Congruent Cubes

“…Another well-known space appeared in John and Nirenberg [23] is BMO (R n ), the space containing functions of bounded mean oscillation, which can be regarded as the limit space of JN con p,q (R n ) as p → ∞; see [19,Proposition 2.21] and also [31,Proposition 2.6]. The space BMO (R n ) has wide applications in harmonic analysis and partial differential equations; see, for instance, [2,11,12,14,15,16,17,25,28,34]. In particular, we refer the reader to [29] for a systematic survey on function spaces of John-Nirenberg type.…”