Gagliardo–Nirenberg type inequality for variable exponent Lebesgue spaces

We prove analogies of the classical Gagliardo–Nirenberg inequalities ‖ ∇ k f ‖ s ⩽ C ‖ f ‖ q 1 − k m ‖ ∇ m f ‖ p k m , when usual L p norms are replaced by variable L p ( ⋅ ) norms.