Improved Hardy inequalities in the Grushin plane

We show the Hardy inequality for Grushin operator like ∂ x 2 + 4 x 2 ∂ y 2 on a bounded domain Ω ⊂ R 2 can be refined by adding remainder terms such as the improvement of Brezis–Vázquez, Vazquez–Zuazua and Filippas–Tertikas.