No abstract

No abstract

Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

In this article, the authors first introduce the localized John-Nirenberg-Campanato space jn (p,q,s) α (X) and show that the localized Campanato space is the limit case of jn (p,q,s) α (X) as p → ∞. By means of local atoms and the weak- * topology, the authors then introduce the localized Hardy-kind space hk (p ′ ,q ′ ,s) α (X) which proves the predual space of jn (p,q,s) α (X). Moreover, the authors prove that hk (p ′ ,q ′ ,s) α (X) is invariant when 1 < q < p, where p ′ or q ′ denotes the conjugate number of p or q, respectively. All these results are new even for the localized John-Nirenberg space., where ⌊n( 1 p − 1)⌋ denotes the largest integer not greater than n( 1 p −1). Notice that C 0,1,0 (R n ) coincides with BMO (R n