Musielak–Orlicz Campanato spaces and applications

Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) be such that φ ( x , ⋅ ) is an Orlicz function and φ ( ⋅ , t ) is a Muckenhoupt A ∞ ( R n ) weight uniformly in t . In this article, the authors introduce the Musielak–Orlicz Campanato space L φ , q , s ( R n ) ; as an application, the authors prove that some of them is the dual space of the Musielak–Orlicz Hardy space H φ ( R n ) , which, in the case when q = 1 and s = 0 , was obtained by L.D. Ky [ arXiv:1105.0486]. The authors also establish a John–Nirenberg inequality for functions in L φ , 1 , s ( R n ) and, as an application, the authors also obtain several equivalent characterizations of L φ , q , s ( R n ) , which, in return, further induce the φ -Carleson measure characterization of L φ , 1 , s ( R n ) .