Abstract:

 Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping

 f →f ^#:=sup_B 1/\int_|B||f(x)-f_B|dx is continuous on L^ϕ(·)(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∈(W0^1,ϕ(·)(Ω))ⁿ such that ||\triangledown u||_{L^{ϕ(·)}(Ω)} \lesssim ||f||_L^{ϕ}(Ω).

Key words:  Generalized Orlicz spaces, Korn’s inequality, Extrapolation.