Bilinear decompositions and commutators of singular integral operators

Abstract

Abstract. Let b be a BM O-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuouslyIn this paper, we find the largest subspace H 1 b (R n ) such that all commutators of Calderón-Zygmund operators are continuous fromWe also study the commutators [b, T ] for T in a class K of sublinear operators containing almost all important operators in harmonic analysis. When T is linear, we prove that there exists a bilinear operators R = R T mapping continuouslywhere S is a bounded bilinear operator fromIn the particular case of T a Calderón-Zygmund operator satisfying T 1 = T * 1 = 0 and b in BM O log (R n )-the generalized BMO type space that has been introduced by Nakai and Yabuta to characterize multipliers of BMO(R n ) -we prove that the commutatorWhen T is sublinear, we prove that there exists a bounded subbilinear operator R = R T :The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong L 1 -estimates.

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

“…Here, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [11,12,13] for their definitions and properties). Moreover, the space H ϕ (R n ) also has already found many applications in analysis (see, for example, [3,4,17,22,23] and their references).…”

“…For example, for all (x, t) ∈ R n × [0, ∞), ϕ(x, t) := ω(x)Φ(t) satisfies Assumption (ϕ) if ω ∈ A ∞ (R n ) and Φ is an Orlicz function of lower type p for some p ∈ (0, 1] and upper type 1. A typical example of such an Orlicz function Φ is Φ(t) := t p , with p ∈ (0, 1], for all t ∈ [0, ∞); see, for example, [17,22,23] for more examples. Another typical example of functions satisfying Assumption (ϕ) is ϕ(x, t) := t α [ln(e+|x|)] β +[ln(e+t)] γ for all x ∈ R n and t ∈ [0, ∞) with any α ∈ (0, 1] and β, γ ∈ [0, ∞) (see [23] for further details).…”

“…Also, the Musielak-Orlicz-Hardy space H ϕ (R n ) has proved useful in the study of other analysis problems when we take various different Musielak-Orlicz functions ϕ (see, for example, [3,22,23]). …”

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

“…Moreover, we remark that the Musielak-Orlicz-Hardy space is a function space of Hardy-type which unifies the classical Hardy space, the weighted Hardy space, the Orlicz-Hardy space and the weighted Orlicz-Hardy space, in which the spatial and the time variables may not be separable (see [8], [15], [27], [22], [24], [28], [31] for more details on the developments of Hardy-type spaces and Musielak-Orlicz spaces). Furthermore, the Musielak-Orlicz-Hardy space appears naturally in many applications (see, for example, [1], [2], [21], [19]). This kind of Musielak-Orlicz-Hardy spaces associated with operators generalizes the (Orlicz-)Hardy space and the (weighted) Hardy space associated with operators, which has attracted great interests in recent years.…”

“…, associated with the Schrödinger operator L := −∆ + V on R n , appears naturally when studying the products of functions in H 1 (R n ) and BMO(R n ), the endpoint estimates for the div-curl lemma and the endpoint estimates for commutators of singular integrals related to the Schrödinger operator L (see [1], [2], [21], [19] for the details).…”