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A Poincare’s inequality with non-uniformly degenerating gradient

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Abstract

In homogeneous space (\(\mathbb {R}^N, d, \mu \)) we explore Poincare’s type inequality

$$\begin{aligned} \Vert f-\overline{f}_{\Omega , v}\Vert _{q, v}\le C \Vert \nabla _\lambda f \Vert _{p}, \,\,\,\, q \ge p\ge 1 \end{aligned}$$

on estimation of weighted Lebesgue norm of a Lipschitz continuous function \(f: \Omega \rightarrow \mathbb {R}\) over the bounded convex domain \(\Omega \subset \mathbb {R}^N\) via such a norm of its non-uniformly degenerating gradient \(\nabla _\lambda f=\left\{ \lambda _1\frac{\partial f}{\partial x_1}, \lambda _2 \frac{\partial f}{\partial x_2}, \dots , \lambda _N \frac{\partial f}{\partial x_N} \right\} \) where \(\lambda _i/\lambda _j, \, i,j=1,2,\dots N\) may approach to zero and infinity when x varies in domain. For that, it is assumed that \(\lambda _i=\omega _i^{1/p}\) and \(\omega _i\in A_p \, (i=1,2,\dots N)\)-Muckenhoupt’s class and some compatibility condition is satisfied for the pares \((\omega _i,\, v), \, i=1,2,\dots N.\) Using this general result, useful inequalities is asserted for study the equations having in its part Grushin’s type operator.

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Acknowledgements

The author is grateful to the referees for their interest in the results and careful reading of the article.

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Correspondence to Farman Mamedov.

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Communicated by Adrian Constantin.

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Mamedov, F. A Poincare’s inequality with non-uniformly degenerating gradient. Monatsh Math 194, 151–165 (2021). https://doi.org/10.1007/s00605-020-01506-4

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