Gagliardo–Nirenberg type inequality for variable exponent Lebesgue spaces

We prove analogies of the classical Gagliardo–Nirenberg inequalities‖∇kf‖s⩽C‖f‖q1−km‖∇mf‖pkm, when usual LpLp norms are replaced by variable Lp(⋅)Lp(⋅) norms.