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Let φ:Rn×[0,∞)→[0,∞) satisfy that φ(x,·), for any given x∈Rn, is an Orlicz function and φ(·,t) is a Muckenhoupt A∞(double-struckRn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(double-struckRn) via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of WHφ(Rn), respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or gλ*‐function. All these characterizations for weighted weak Hardy spaces WHwp(double-struckRn) (namely, φ(x,t):=w(x)tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1] and w∈A∞(Rn)) are new and part of these characterizations even for weak Hardy spaces WHp(double-struckRn) (namely, φ(x,t):=tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1]) are also new. As an application, the boundedness of Calderón–Zygmund operators from Hφ(Rn) to WHφ(double-struckRn) in the critical case is presented.

In this paper, the authors establish new characterizations of the recently introduced Besov-type spacesḂ s,τ p,q (R n ) and Triebel-Lizorkin-type spacesḞ s,τ p,q (R n ) with p ∈ (0, ∞], s ∈ R, τ ∈ [0, ∞), and q ∈ (0, ∞], as well as their preduals, the Besov-Hausdorff spaces BḢ s,τ p,q (R n ) and Triebel-Lizorkin-Hausdorff spaces FḢ s,τ p,q (R n ), in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180 (2007), [237][238][239][240][241][242][243][244][245][246][247][248][249][250][251][252][253] and discretizations via the biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such asḞ s ∞,q (R n ) for s ∈ R, q ∈ (0, ∞] (including BMO(R n ) when s = 0 and q = 2), the Q space Q α (R n ), the Hardy-Hausdorff space HH −α (R n ) for α ∈ (0, min{n/2, 1}), the Morrey space M u p (R n ) for 1 < p ≤ u < ∞, and the Triebel-Lizorkin-Morrey spaceĖ s upq (R n ) for 0 < p ≤ u < ∞, s ∈ R and q ∈ (0, ∞], some of these results are new. investigated in [57,58,43,61,62]. These spaces establish the connection between the classical Besov-Triebel-Lizorkin spaces (see, for example, [47,22,49]), the Q spaces in [17], and the Hardy-Hausdorff spaces in [15], which has been posed as an open question in [15]. It was shown in [57,58,43] that these spaces unify and generalize many classical function spaces including classical Besov and Triebel-Lizorkin spaces, Triebel-Lizorkin-Morrey spaces (see, for example, [51,42,41,43]), Q spaces, and Hardy-Hausdorff spaces (see also [54,55]).These spaces are usually defined through building blocks constructed out of a dyadic decomposition of unity on the Fourier side. Several of the mentioned applications make it necessary to use more general convolution kernels for the definition of the spaces, especially the so-called local means of a function are of particular interest. By the pioneering work of Bui, Paluszyński and Taibleson [10,11], and later by Rychkov [37,38,39], we know that one rather uses kernels satisfying the Tauberian conditions (see (3.2) below), for characterizing classical Besov-Triebel-Lizorkin spaces. This also applies to the setting of the spaceṡ B s,τ p,q (R n ),Ḟ s,τ p,q (R n ), BḢ s,τ p,q (R n ) and FḢ s,τ p,q (R n ) considered here, which represents one of the main results of the paper; see Theorems 3.2, 3.6, 3.12 and 3.15 below. To establish these results we use a key estimate (see Lemma 3.5 below) which was obtained in [52] by a variant of a method from Rychkov [38,39] and originally from Strömberg and Torchinsky [46, Chapter 5]. In addition we make use of a certain involved decomposition of R n into proper sub-cubes in combination with some localized modifications of the approaches used for Besov and Triebel-Lizorkin spaces. For the technically more difficult Hausdorff type spaces BḢ s,τ p,q ...