Weighted Mixed Weak-Type Inequalities for Littlewood–Paley Square Functions Related to Schrödinger Operators

Let L=−Δ+V be a Schrödinger operator, where Δ is the Laplacian defined on Rn and the nonnegative potential V satisfies the reverse Hölder inequality. In this paper, we mainly establish weighted mixed weak-type endpoint estimates for Littlewood–Paley square functions associated with the heat semigroup e−tL and their commutators. Such inequalities not only provide a refined characterization of the endpoint behavior of these operators in weighted spaces—extending the classical weak (1,1) bounds, but also have potential applications to the regularity theory of Schrödinger equations with non-smooth potentials and to the boundedness of spectral multipliers. Our approach is based on the Calderón–Zygmund decomposition adapted to Schrödinger setting, which allows for a direct proof without invoking the associated maximal operator. These results are new even in the unweighted setting, and their weighted formulations offer a robust foundation for further developments in Fourier analysis and elliptic equations.