Maximal Function and Atomic Characterizations of Matrix-Weighted Hardy Spaces with Their Applications to Boundedness of Calder\'on--Zygmund Operators

Let $p\in(0,1]$ and $W$ be an $A_p$-matrix weight, which in scalar case is exactly a Muckenhoupt $A_1$ weight. In this article, we introduce matrix-weighted Hardy spaces $H^p_W$ via the matrix-weighted grand non-tangential maximal function and characterize them, respectively, in terms of various other maximal functions and atoms, both of which are closely related to matrix weights under consideration and their corresponding reducing operators. As applications, we first establish the finite atomic characterization of $H^p_W$, then using it we give a criterion on the boundedness of sublinear operators from $H^p_W$ to any $\gamma$-quasi-Banach space, and finally applying this criterion we further obtain the boundedness of Calder\'on--Zygmund operators on $H^p_W$. The main novelty of these results lies in that the aforementioned maximal functions related to reducing operators are new even in the scalar weight case and we characterize these matrix-weighted Hardy spaces by a fresh and natural variant of classical weighted atoms via first establishing a Calder\'on--Zygmund decomposition which is also new even in the scalar weight case.