A Ne{c}as-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babu{s}ka-Aziz inequality

We prove a Ne{c}as-Lions inequality with symmetric gradients on two and three dimensional domains of diameter R that are star-shaped with respect to a ball of radius ρ; the constants in the inequality are explicit with respect to R and ρ. Crucial tools in deriving this inequality are a first order Babu{s}ka-Aziz inequality based on Bogovski{i}'s construction of a right-inverse of the divergence and Fourier transform techniques proposed by Dur\'an. As a byproduct, we derive arbitrary order estimates in arbitrary dimension for that operator.