Weighted Rellich type inequalities related to Baouendi-Grushin operators

Abstract

We find a simple sufficient criterion on a pair of nonnegative weight functions a ( x , y ) a\left ( x,y\right ) and b ( x , y ) b\left ( x,y\right ) in R m + k \mathbb {R} ^{m+k} so that the general weighted L p L^{p} Rellich type inequality ∫ R m + k a ( x , y ) | Δ γ u ( x , y ) | p d x d y ≥ ∫ R m + k b ( x , y ) | u ( x , y ) | p d x d y \begin{equation*} \int _{\mathbb {\mathbb {R}}^{m+k}}a\left ( x,y\right ) \left \vert \Delta _{\gamma }u\left ( x,y\right ) \right \vert ^{p}dxdy\geq \int _{\mathbb {\mathbb {R}}^{m+k}}b\left ( x,y\right ) \left \vert u\left ( x,y\right ) \right \vert ^{p}dxdy \end{equation*} holds for all u ∈ C 0 ∞ ( R m + k ) u\in C_{0}^{\infty }(\mathbb {R}^{m+k}) . Here Δ γ = Δ x + | x | 2 γ Δ y \Delta _{\gamma }=\Delta _{x}+|x|^{2\gamma }\Delta _{y} is the Baouendi-Grushin operator, γ > 0 , \gamma >0, m , k ≥ 1 m,k\geq 1 and p > 1 p>1 . It is important to point out here that our approach is constructive in the sense that it allows us to retrieve already established weighted sharp Rellich type inequalities as well as to get other new results with an explicit constant on R m + k . \mathbb {\mathbb {R}}^{m+k}. We also obtain a sharp L p L^{p} Rellich type inequality that connects first to second order derivatives and several new two-weight Rellich type inequalities with remainder terms on smooth bounded domains Ω \Omega in R m + k \mathbb {\mathbb {R}}^{m+k} via a nonlinear differential inequality.