Improved Hardy inequalities in the Grushin plane

We show the Hardy inequality for Grushin operator like ∂x2+4x2∂y2 on a bounded domain Ω⊂R2Ω⊂R2 can be refined by adding remainder terms such as the improvement of Brezis–Vázquez, Vazquez–Zuazua and Filippas–Tertikas.