Short proofs for interpolation inequalities in Sobolev spaces with variable exponents
Volume 170 / 2022
Abstract
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.
