Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Abstract EN

Let φ : ℝ n × [ 0 , ∞ ) → [ 0 , ∞ ) be a Musielak-Orlicz function and A an expansive dilation.

In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ ( ℝ n ) , via the grand maximal function.

The authors then obtain some real-variable characterizations of H A φ ( ℝ n ) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p ( ℝ n ) with p ∈ ( 0,1 ] and are new even for its weighted variant.

Finally, the authors characterize these spaces by anisotropic atomic decompositions.

The authors also obtain the finite atomic decomposition characterization of H A φ ( ℝ n ) , and, as an application, the authors prove that, for a given admissible triplet ( φ , q , s ) , if T is a sublinear operator and maps all ( φ , q , s ) -atoms with q < ∞ (or all continuous ( φ , q , s ) -atoms with q = ∞ ) into uniformly bounded elements of some quasi-Banach spaces ℬ , then T uniquely extends to a bounded sublinear operator from H A φ ( ℝ n ) to ℬ .

These results are new even for anisotropic Orlicz-Hardy spaces on ℝ n .