Hardy Spaces Associated with Generalized Degenerate Schrödinger Operators with Applications to Carleson Measure

• Published in: Bulletin of the Malaysian Mathematical Sciences Society 2023-07, Vol.46 (4), Article 132
• Main Authors: Liu, Xiong, He, Jianxun, Li, Jinxia
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Continuity (mathematics); Mathematics; Mathematics and Statistics; Norms; Operators (mathematics)
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-023-01527-w

Let L : = - 1 ω div ( A ∇ · ) + μ be the generalized degenerate Schrödinger operator in L ω 2 ( R n ) ( n ≥ 3 ) with suitable weight ω and nonnegative Radon measure μ . In this article, the authors first introduce the Hardy spaces H L , S h p ( R n , ω ) and H L p ( R n , ω ) , respectively, via the Lusin area function and the maximal function associated with L , where p ∈ ( 0 , 1 ] , and then, show that, when p ∈ ( n n + θ , 1 ] , H L , S h p ( R n , ω ) = H L p ( R n , ω ) with equivalent quasi-norms, where θ ∈ ( 0 , 1 ] is the critical index of Hölder continuity for the heat kernels { k t } t > 0 generated by L . As an application, the authors further obtain that the BMO type space BMO L ( R n , ω ) associated with L can be characterized by the Carleson measure. This result is also new even when L : = - 1 ω div ( A ∇ · ) + V , where V ≥ 0 belongs to the reverse Hölder class with respect to the measure ω ( x ) d x .