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We give a characterization of R-bounded families of operators on L 1 . We then use this result to study sectorial operators on L 1 . We show that if A is an R-sectorial operator on L 1 , then, for any e > 0, there is an invertible operator U: L 1 → L 1 with ∥U - I∥ < e such that for some strictly positive Borel function w, U(D(A)) contains the weighted L 1 -space L 1 (w).

Rademacher bounded families of operators on 𝐿₁