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Science China Mathematics

Let X be a ball quasi-Banach function space on ℝn. In this article, we introduce the weak Hardy-type space W HX(ℝn), associated with X, via the radial maximal function. Assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, we then establish several real-variable characterizations of W HX (ℝn), respectively, in terms of various maximal functions, atoms and molecules. As an application, we obtain the boundedness of Calderón-Zygmund operators from the Hardy space HX (ℝn) to W HX (ℝn), which includes the critical case. All these results are of wide applications. Particularly, when $$X: = M_q^p({\mathbb{R}^n})$$ X : = M q p ( ℝ n ) (the Morrey space), $$X: = {L^{\vec p}}({\mathbb{R}^n})$$ X : = L p → ( ℝ n ) (the mixed-norm Lebesgue space) and $$X: = {(E_\Phi ^q)_t}({^n})$$ X : = ( E Φ q ) t ( ℝ n ) (the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasi-Banach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.

Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators