In homogeneous space ( $$\mathbb {R}^N, d, \mu $$ R N , d , μ ) we explore Poincare’s type inequality $$\begin{aligned} \Vert f-\overline{f}_{\Omega , v}\Vert _{q, v}\le C \Vert \nabla _\lambda f \Vert _{p}, \,\,\,\, q \ge p\ge 1 \end{aligned}$$ ‖ f - f ¯ Ω , v ‖ q , v ≤ C ‖ ∇ λ f ‖ p , q ≥ p ≥ 1 on estimation of weighted Lebesgue norm of a Lipschitz continuous function $$f: \Omega \rightarrow \mathbb {R}$$ f : Ω → R over the bounded convex domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N via such a norm of its non-uniformly degenerating gradient $$\nabla _\lambda f=\left\{ \lambda _1\frac{\partial f}{\partial x_1}, \lambda _2 \frac{\partial f}{\partial x_2}, \dots , \lambda _N \frac{\partial f}{\partial x_N} \right\} $$ ∇ λ f = λ 1 ∂ f ∂ x 1 , λ 2 ∂ f ∂ x 2 , ⋯ , λ N ∂ f ∂ x N where $$\lambda _i/\lambda _j, \, i,j=1,2,\dots N$$ λ i / λ j , i , j = 1 , 2 , ⋯ N may approach to zero and infinity when x varies in domain. For that, it is assumed that $$\lambda _i=\omega _i^{1/p}$$ λ i = ω i 1 / p and $$\omega _i\in A_p \, (i=1,2,\dots N)$$ ω i ∈ A p ( i = 1 , 2 , ⋯ N ) -Muckenhoupt’s class and some compatibility condition is satisfied for the pares $$(\omega _i,\, v), \, i=1,2,\dots N.$$ ( ω i , v ) , i = 1 , 2 , ⋯ N . Using this general result, useful inequalities is asserted for study the equations having in its part Grushin’s type operator.

中文翻译：

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具有非均匀退化梯度的 Poincare 不等式

当 x 在域中变化时，j = 1 , 2 , ⋯ N 可能趋近于零和无穷大。为此，假设 $$\lambda _i=\omega _i^{1/p}$$ λ i = ω i 1 / p 和 $$\omega _i\in A_p \, (i=1,2, \dots N)$$ ω i ∈ A p ( i = 1 , 2 , ⋯ N ) -Muckenhoupt 的类和一些兼容条件满足 $$(\omega _i,\, v), \, i=1 ,2,\dots N.$$ ( ω i , v ) , i = 1 , 2 , ⋯ N 。使用这个一般结果，可以断言有用的不等式来研究在其部分具有 Grushin 类型运算符的方程。i = 1 , 2 , ⋯ N 。使用这个一般结果，可以断言有用的不等式来研究在其部分具有 Grushin 类型运算符的方程。i = 1 , 2 , ⋯ N 。使用这个一般结果，可以断言有用的不等式来研究在其部分具有 Grushin 类型运算符的方程。