Short proofs for interpolation inequalities in Sobolev spaces with variable exponents

We present very short proofs for three versions of the Gagliardo–Nirenberg inequality in the setting of Sobolev spaces with variable exponents. These are formally expressed by $$ \|\nabla ^k f\|_{L^{r(\cdot )}(\mathbb {R}^d)} \le C(d,p,q,r,k,m) \, \|f\|_{L^{q(\cdot )}(\mathbb {R}^d)}^{1-\theta } \, \|\nabla ^m f\|_{L^{p(\cdot )}(\mathbb {R}^d)}^{\theta },$$ $$\|\nabla ^k f\|_{L^{p(\cdot )}(\mathbb {R}^d)} \le C(d,q,k,m) \, \|f\|_{L^{q(\cdot )}(\mathbb {R}^d)}^\theta \, \|\nabla ^m f\|_{{\rm BMO}(\mathbb {R}^d)}^{1-\theta },$$ $$\|\nabla ^k f\|_{L^{p(\cdot )}(\mathbb {R}^d)} \le C(d,q,k,m) \, \|f\|_{{\rm BMO}(\mathbb {R}^d)}^\theta \, \|\nabla ^m f\|_{L^{q(\cdot )}(\mathbb {R}^d)}^{1-\theta }.$$ The proofs employ Muramatu’s integral formula.