Weight Poincaré inequality (Q2719411)

Let \(W_p^m(\mathbb{R}^N,w)\) be the weighted Sobolev space with function \(w\) which satisfies the Makenhaupt \(A_p\)-condition. The main result of the paper is the following assertion: NEWLINENEWLINENEWLINELet \( G\subset \mathbb{R}^N \) be a bounded domain of star-shape with respect to the ball \( B(x,\rho)\), \( \overline{B(x,\rho)}\subset G\), \( 1<p<\infty \) and the weight \( w\in A_p\). Then for any function \( u\in W_p^m(G,w) \) a polynomial NEWLINE\[NEWLINE P(x) = \sum\limits_{0\leq|\beta|\leq m-1} (u,\varphi_\beta)x^\beta NEWLINE\]NEWLINE is found, where \( \varphi_\beta\in C_0^\infty(B(x,\rho)) \) is such that NEWLINE\[NEWLINE \sum\limits_{k=0}^{m-1} \|\nabla^k(u-P)\mid L_p(G,w)\|\leq c\|\nabla^m u\mid L_p(G,w)\|, NEWLINE\]NEWLINE where \(c\) is a constant independent of \(u\).