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.

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Abstract. In this article, the authors first introduce the Triebel-Lizorkintype space F

Let p(·) : R n → (0, ∞) be a measurable function satisfying some decay condition and some locally log-Hölder continuity. In this article, via first establishing characterizations of the variable exponent Hardy space H p(·) (R n ) in terms of the Littlewood-Paley g-function, the Lusin area function and the g * λ -function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley g-function, the intrinsic Lusin area function and the intrinsic g * λ -function. The p(·)-Carleson measure characterization for the dual space of H p(·) (R n ), the variable exponent Campanato space L 1,p(·),s (R n ), in terms of the intrinsic function is also presented.

Let θ ∈ (0, 1), s0, s1 ∈ ℝ, τ0, τ1 ∈ [0, ∞), p0, p1 ∈ (0, ∞), q0, q1 ∈ (0, ∞], s = s0(1 - θ) + s1θ, τ = τ0(1-θ) + τ1θ, [Formula: see text] and [Formula: see text]. In this paper, under the restriction [Formula: see text], the authors establish the complex interpolation, on Triebel–Lizorkin-type spaces, that [Formula: see text], where [Formula: see text] denotes the closure of the Schwartz functions in [Formula: see text]. Similar results on Besov-type spaces and Besov–Morrey spaces are also presented. As a corollary, the authors obtain the complex interpolation for Morrey spaces that, for all 1 < p0 ≤ u0 < ∞, 1 < p1 ≤ u1 < ∞ and 1 < p ≤ u < ∞ such that [Formula: see text], [Formula: see text] and p0u1 = p1u0, [Formula: see text], where [Formula: see text] denotes the closure of the Schwartz space in [Formula: see text]. It is known that, if p0u1 ≠ p1u0, these conclusions on Morrey spaces may not be true.