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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
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