Hardy inequalities

Here the following Hardy inequalities are studied \[ ∑ k = 0 m − 1 ∫ | ∇ k u | p d ∂ Ω ( x ) t d ∂ Ω ( x ) ( m − k ) p d x ≤ A Ω ∫ | ∇ m u | p d ∂ Ω