Interpolation on Convex-set Valued Lebesgue Spaces and its Applications

In this paper, we obtain two interpolation theorems on convex-set valued Lebesgue spaces, which generalize the Marcinkiewicz interpolation theorem and Riesz-Thorin interpolation theorem on classical Lebesgue spaces, respectively. As applications, we obtain the boundedness of convex-set valued fractional averaging operators and fractional maximal operators, and prove the reverse factorization property of matrix weights.