Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping
f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω) \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω).
