Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces

We show the interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces by applying the boundedness of the Hardy–Littlewood maximal operator on Lp(⋅)Lp(⋅).As applications, we prove a new Landau–Komogorov type inequality for the second-order derivative and an embedding theorem and discuss the equivalent norms in the space W01,p(⋅)(Ω)∩W2,p(⋅)(Ω).