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Proceedings of the American Mathematical Society

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Weighted Rellich type inequalities related to Baouendi-Grushin operators

Authors: Ismail Kombe and Abdullah Yener
Journal: Proc. Amer. Math. Soc. 145 (2017), 4845-4857
MSC (2010): Primary 26D10, 35H10; Secondary 46E35
Published electronically: July 10, 2017
MathSciNet review: 3692000
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Abstract: We find a simple sufficient criterion on a pair of nonnegative weight functions $a\left ( x,y\right )$ and $b\left ( x,y\right )$ in $\mathbb {R} ^{m+k}$ so that the general weighted $L^{p}$ Rellich type inequality \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 $u\in C_{0}^{\infty }(\mathbb {R}^{m+k})$. Here $\Delta _{\gamma }=\Delta _{x}+|x|^{2\gamma }\Delta _{y}$ is the Baouendi-Grushin operator, $\gamma >0,$ $m,k\geq 1$ and $p>1$. 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 $\mathbb {\mathbb {R}}^{m+k}.$ We also obtain a sharp $L^{p}$ 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 $\Omega$ in $\mathbb {\mathbb {R}}^{m+k}$ via a nonlinear differential inequality.

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Additional Information

Ismail Kombe
Affiliation: Department of Electrical and Electronics Engineering, Istanbul Commerce University, Kucukyali E5 Kavsagi, Inonu Cad. No: 4, Kucukyali 34840, Istanbul, Turkey.
MR Author ID: 720054

Abdullah Yener
Affiliation: Department of Economics, Istanbul Commerce University, Sütlüce Mahallesi, İmrahor Caddesi, No: 90, Beyog̃lu 34445, İstanbul, Turkey

Keywords: Baouendi-Grushin vector fields, weighted Rellich inequality, two-weight Rellich inequality, remainder term
Received by editor(s): July 4, 2016
Received by editor(s) in revised form: July 5, 2016, and December 12, 2016
Published electronically: July 10, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society