Variable weak Hardy spaces and their applications

Abstract

Let p(·) : R n → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on R n , WH p(·) (R n ), via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WH p(·) (R n ), respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley g-function or g * λ -function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ-order Calderón-Zygmund operators from H p(·)The main purpose of this article is to introduce and to investigate the variable weak Hardy spaces on R n . It is well known that the classical weak Hardy spaces appear naturally in critical cases of the study on the boundedness of operators. Indeed, the classical weak Hardy space WH 1 (R n ) was originally introduced by Fefferman and Soria [18] when they tried to find out the biggest space from which the Hilbert transform is bounded to the weak Lebesgue space WL 1 (R n ). Via establishing the ∞-atomic characterization of WH 1 (R n ), they obtained the boundedness of some Calderón-Zygmund operators from WH 1 (R n ) to WL 1 (R n ). Moreover, it is also well known that, when studying the boundedness of some singular integral operators, H p (R n ) is a good substitute of the Lebesgue space L p (R n ) with p ∈ (0, 1]; while when studying the boundedness of operators in the critical case, the Hardy spaces H p (R n ) are usually further replaced by the weak Hardy space WH p (R n ). For example, if δ ∈ (0, 1] and T is a convolutional δ-type Calderón-Zygmund operator with T * (1) = 0, where T * denotes the adjoint operator of T , then T is bounded on H p (R n ) for all p ∈ (n/(n + δ), 1] (see [5]), but may not be bounded on H n/(n+δ) (R n ). For such an endpoint case, Liu [30] proved that T is bounded from H n/(n+δ) (R n ) to WH n/(n+δ) (R n ) via establishing the ∞-atomic characterization of the weak Hardy space WH p (R n ).Furthermore, when studying the real interpolation between the Hardy space H p (R n ) and the space L ∞ (R n ), Fefferman et al. [17] proved that the weak Hardy spaces WH p (R n ) also naturally appear as the intermediate spaces, which is another main motivation to develop a real-variable theory of WH p (R n ). Recently, He [23] and Grafakos and He [22] further investigated vector-valued weak Hardy spaces H p,∞ (R n , ℓ 2 ) with p ∈ (0, ∞). Very recently, Liang et al. [29] introduced a kind of generalized weak Hardy spaces of Musielak-Orlicz type WH ϕ (R n ), which covers both weak Hardy spaces WH p (R n ) and weighted 2010 Mathematics Subject Classification. Primary 42B30; Secondary 42B25, 42B20, 42B35, 46E30.

Variable Hardy–Lorentz spaces

“…Very recently, Yan et al. introduced the variable weak Hardy space via the radial grand maximal function in , where they used the notation to denote . By applying the method of atomic decompositions, they obtained various equivalent characterizations of .…”

“…However, compared with the classical case, our proof needs some extra technical estimates because of the complexity brought by the variable exponent . Also, one may find that our proof contains some techniques which are very different from those used in and . By mean of the atomic decomposition of , we describe as an interpolation space between variable Hardy spaces and via real method.…”

“…In Section 3, under the assumption that satisfies the log‐Hölder condition, we establish a Fefferman–Stein vector‐valued inequality on the variable Lorentz space for and . Note that the endpoint case of was obtained in by using an interpolation theorem of sublinear operators on the weak variable space . The establishment of our Fefferman–Stein inequality also rely on an interpolation theorem.…”

“…Particularly, we employ a different approach to get the interpolation result for sublinear operators on the space with and , which covers the interpolation theorem obtained by Yan et al. in [, Theorem 3.1]. Moreover, our method removes the additional restriction of in [, Theorem 3.1].…”

“…in [, Theorem 3.1]. Moreover, our method removes the additional restriction of in [, Theorem 3.1]. In the present paper, we mainly use this interpolation to get the Fefferman–Stein vector‐valued inequality; while the interpolation theorem itself may be of independent interest.…”