General weighted Hardy type inequalities related to Baouendi-Grushin operators (original) (raw)

In this paper, we derive a sufficient condition on a pair of nonnegative weight functions ϑ and w in R m+k so that the general weighted Hardy type inequality with a remainder term R m+k ϑ ∇ γ φ p dxdy ≥ R m+k w |φ| p dxdy+c p R m+k ϑ ∇ γ φ f p f p dxdy holds for all φ ∈ C ∞ 0 (R m+k). Here m, k ≥ 1, p ≥ 2, c p > 0, 0 < f ∈ C ∞ (R m+k), γ > 0 and ∇ γ = ∂ ∂x 1 ,. .. , ∂ ∂x m , |x| γ ∂ ∂y 1 ,. .. , |x| γ ∂ ∂y k is the sub-elliptic gradient. It is worth emphasizing here that our unifying method may be readily used to recover most of the previously known sharp weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit constant. Furthermore, we also obtain new results on two-weight L p Hardy type inequalities with remainder terms on smooth bounded domains in R m+k via a non-linear partial differential inequality.