New Orlicz–Hardy spaces associated with divergence form elliptic operators

Abstract

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on ( 0 , ∞ ) of strictly critical lower type p ω ∈ ( 0 , 1 ] and ρ ( t ) = t − 1 / ω − 1 ( t − 1 ) for t ∈ ( 0 , ∞ ) . In this paper, the authors study the Orlicz–Hardy space H ω , L ( R n ) and its dual space BMO ρ , L * ( R n ) , where L * denotes the adjoint operator of L in L 2 ( R n ) . Several characterizations of H ω , L ( R n ) , including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John–Nirenberg inequality for the space BMO ρ , L ( R n ) are also given. As applications, the authors show that the Riesz transform ∇ L − 1 / 2 and the Littlewood–Paley g-function g L map H ω , L ( R n ) continuously into L ( ω ) . The authors further show that the Riesz transform ∇ L − 1 / 2 maps H ω , L ( R n ) into the classical Orlicz–Hardy space H ω ( R n ) for p ω ∈ ( n n + 1 , 1 ] and the corresponding fractional integral L − γ for certain γ > 0 maps H ω , L ( R n ) continuously into H ω ˜ , L ( R n ) , where ω ˜ is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω ( t ) = t p for all t ∈ ( 0 , ∞ ) and p ∈ ( 0 , 1 ) .