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Hardy inequalities related to Grushin type operators. (English) Zbl 1049.35077
Let $$\nabla_\gamma$$ be the gradient operator defined by $$\nabla_\gamma:= (\nabla_x,| x|^\gamma\nabla_y)$$ with $$\gamma> 0$$, $$n\in\mathbb{R}^d$$, $$y\in\mathbb{R}^k$$. Then $$\Delta_\gamma:= \nabla_\gamma\cdot\nabla_\gamma= \Delta_x+| x|^{2\gamma}\Delta_y$$ is a Grushin type operator. The author proves inequalities of the type $c \int_\Omega| u|^p w^p \,d\xi\leq \int_\Omega |\nabla_\gamma u|^p\, d\xi,\quad p> 1,\tag{$$*$$}$ where $$\Omega$$ is an open set of $$\mathbb{R}^{d+k}$$ and $$w$$ is one of the following functions: $$1/| x|$$, $$1/[[\xi]]$$ or $$| x|^\gamma/[[\xi]]^{1+\gamma}$$ ($$[[\cdot]]$$ denotes a suitable distance from the origin). Furthermore, he gives an estimate on the optimal constant in $$(*)$$ and in some cases he shows its sharp value.

##### MSC:
 35H10 Hypoelliptic equations 26D10 Inequalities involving derivatives and differential and integral operators 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
optimal constant
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