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

Interpolation on Convex-set Valued Lebesgue Spaces and its Applications