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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weighted Rellich type inequalities related to Baouendi-Grushin operators
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by Ismail Kombe and Abdullah Yener PDF
Proc. Amer. Math. Soc. 145 (2017), 4845-4857 Request permission

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
  • Email: ikombe@ticaret.edu.tr
  • Abdullah Yener
  • Affiliation: Department of Economics, Istanbul Commerce University, Sütlüce Mahallesi, İmrahor Caddesi, No: 90, Beyog̃lu 34445, İstanbul, Turkey
  • Email: ayener@ticaret.edu.tr
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4845-4857
  • MSC (2010): Primary 26D10, 35H10; Secondary 46E35
  • DOI: https://doi.org/10.1090/proc/13730
  • MathSciNet review: 3692000