Weighted local Hardy spaces associated to elliptic operators with coefficients having minimal smoothness

Let L be a homogeneous higher-order divergence-form elliptic operator with complex, bounded, and measurable coefficients on R n , and let w be a local Muckenhoupt weight on R n . Given p ∈ ( 0 , ∞ ) , we develop a real-variable theory of weighted local Hardy spaces h L , w p associated with L . Specifically, h L , w p is defined through the Lusin area function corresponding to the semigroup generated by L . We then establish the molecular and various maximal function characterizations of h L , w p . Additionally, we obtain the boundedness of local Riesz transforms associated with L from h L , w p to the weighted Lebesgue space L w p . The class of local Muckenhoupt weights encompasses a much broader range of desirable weights than the classical Muckenhoupt class, including weights that grow or decay exponentially. The main results of this article extend the corresponding results established by Tang (2012) [58] by generalizing the Laplace operator to more general elliptic operators.