Let ψ be a generalized Orlicz function satisfying(A0),(A1),(A2),(aInc)and(aDec).We prove that the mapping f(→)f#:=supB 1/|B| ∫B |f(x)-fB|dx is continuous on Lψ(·)(Rn)by extrapolation.Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces,i.e.,‖▽f‖Lψ(·)(Ω)(＜)‖Df‖Lψ(·)(Ω).Using the Calderón-Zygmund theory on generalized Orlicz spaces,we obtain that the divergence equation divu= f has a solution u∈(W1,ψ0(·)(Ω))n such that ‖▽u‖Lψ(·)(Ω)(＜)‖f‖Lψ(Ω)·