A two-weight boundedness criterion and its applications.

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F , f) , on ℝ n in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley g -functions, Lusin area functions, Littlewood–Paley g λ ∗ -functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley g -functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ℝ n in the aforementioned function spaces. [ABSTRACT FROM AUTHOR]