Upper bounds on homogeneous fractional Gagliardo–Nirenberg–Sobolev constants via parabolic estimates

Common proofs of the Gagliardo–Nirenberg–Sobolev (GNS) inequalities do not provide explicit bounds on the involved constants, unless a sharp constant is being determined. GNS inequalities naturally occur in error estimates for numerical approximations. In particular, bounds on GNS constants allow us to provide explicit a priori estimates. We provide an algorithm that determines explicit upper bounds on the non-endpoint homogeneous GNS inequalities in terms of explicit upper bounds for Young’s convolution inequality and parabolic estimates. Our method is based on the heat-kernel representation of the inverse Laplacian, from which we deduce interpolation estimates.