Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Abstract

The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces H p (R n ) and their duals in the case when p < 1 and near to 1, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces H p (R n ) ×Λ α (R n ) and H p (R n ) × Λ α (R n ), where, for all p ∈ ( n n+1 , 1) and α := n( 1 p −1), H p (R n ) denotes the classical real Hardy space, anḋ Λ α and Λ α denote the homogeneous, respectively, the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are also proved. As an application, the authors establish some div-curl lemmas at the endpoint case.

Real-variable characterizations of Orlicz-slice Hardy spaces

“…In [8], Bonami and Feuto introduced the Hardy type spaces H Φ * (R n ) with respect to the amalgam space (L Φ , ℓ 1 )(R n ) = ℓ 1 (L Φ 1 )(R n ) with Φ(t) := t log(e+t) for any t ∈ [0, ∞), and applied these spaces to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ); see also [12]. Since ℓ 1 (L Φ 1 )(R n ) is a special case of the Orlicz-amalgam spaces introduced in Definition 2.9, from Proposition 2.12, we deduce that the space H Φ * (R n ) is also a special case of the Orlicz-slice Hardy spaces (HE q Φ ) t (R n ) considered in this article.…”

“…Moreover, very recently, Cao et al [12] applied h Φ * (R n ) to study the bilinear decomposition of the product of the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Recall that both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) were defined in [8] via the (local) radial maximal functions, while h Φ * (R n ) in [12] was defined via the local grand maximal function. Moreover, no other real-variable characterizations of both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) are known so far.…”

Some endpoint estimates for bilinear Coifman-Meyer multipliers

“…Lemma 2.10. Let w be an admissible weight and let Θ : (0, ∞) → (0, ∞) be a monotonic function satisfying (6). Then Θ(w(t)) is also an admissible weight.…”

“…Regarding the endpoint case s = 0, namely estimates for the product of two functions, one in BMO(R n ) and the other in the Hardy space H 1 (R n ), have been investigated by A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister in [2] and by A. Bonami, S. Grellier and L. D. Ky in [1]. Likewise, J. Cao, L. D. Ky and D. Yang [6] studied the counterpart problem where one of the terms lies in the local Hardy space h p (R n ), for 0 < p ≤ 1, and the other in the local bmo(R n ) space. In these studies, the product is decomposed in two terms, one belonging to L 1 (R n ), and the other in a suitable Musielak-Orlicz-Hardy space.…”

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

“…When p ∈ (0, 1], as an application of the obtained bilinear decomposition, in [4,6,8], some estimates of the div-curl products of elements in the Hardy space H p (R n ) and its dual space were also established. As for the local Hardy space, Cao et al [15] established an estimate of div-curl products of functions in the local Hardy space h 1 (R n ) and bmo (R n ), and no other estimates of div-curl products of elements in the local Hardy space h p (R n ) with p ∈ (0, 1) and its dual space are known so far.…”