Weak Musielak–Orlicz Hardy spaces and applications

Let φ :R n × [ 0 , ∞ ) → [ 0 , ∞ )satisfy that φ ( x , · ) , for any given x ∈ R n , is an Orlicz function and φ ( · , t ) is a MuckenhouptA ∞ ( R n )weight uniformly in t ∈ ( 0 , ∞ ) . In this article, the authors introduce the weak Musielak–Orlicz Hardy space W H φ ( R n )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 W H φ ( R n ) , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g ‐function or g λ * ‐function. All these characterizations for weighted weak Hardy spaces W H w p ( R n )(namely, φ ( x , t ) : = w ( x ) t pforallt ∈ [ 0 , ∞ )and x ∈ R nwith p ∈ ( 0 , 1 ] and w ∈ A ∞ ( R n ) ) are new and part of these characterizations even for weak Hardy spaces W H p ( R n )(namely, φ ( x , t ) : = t pforallt ∈ [ 0 , ∞ )and x ∈ R nwith p ∈ ( 0 , 1 ] ) are also new. As an application, the boundedness of Calderón–Zygmund operators fromH φ ( R n )to W H φ ( R n )in the critical case is presented.