Improved Hardy inequalities for Grushin operators

We show that the Hardy inequality for Grushin operators like Δx+(γ+1)2|x|2γΔyΔx+(γ+1)2|x|2γΔy on a bounded domain Ω containing the origin can be refined by adding remainder terms such as the improvement of Brezis–Vázquez, Vazquez–Zuazua and Filippas–Tertikas.