Coercive inequalities on weighted Sobolev spaces. (English) Zbl 0827.46028

Let \(P_j= (P_{j1}, \dots, P_{jk})\) \((j=1, \dots, N)\) be scalar differential operators of order \(m\), acting on vector-valued functions \(f= (f_1, \dots, f_k)\): \[ P_j f=\sum_{i=1}^k P_{ji} f_i, \qquad P_{ji} g(x)= \sum_{|\alpha|\leq m} a_{\alpha, j,i} (x) Dg(x). \] Denote by \(P^m_j\) the principle part of \(P_j\), involving differentiations of highest order, and by \(P^0_j\) the part involving differentiations of order less than \(m\).
By \(W^{m,p} (\Omega)\) we denote the weighted Sobolev spaces: \[ W^{m,p} (\Omega):= \{f\in {\mathcal D}' (\Omega):\;D^\alpha f\in L^p_\rho (\Omega),\;|\alpha|\leq m\} \] with norm \(|f|_{W_\rho^{m,p} (\Omega)}:= \sum_{|\alpha|\leq m} |D^\alpha f|_{L^p_\rho (\Omega)}\), and \(\Omega\) be an open subset of \(\mathbb{R}^n\), \(\rho\geq 0\) a locally integrable function.
The main result of this paper is the following theorem:
Let \(\Omega\) be a bounded domain with cone property, \(\rho\in A_p\), \(1\leq p\leq \infty\) (where \(A_p\) is the class of Muckenhoupt type weights), and let \(\{P_j\}_{j= 1,\dots, N}\) be a family of differential operators of order \(m\) acting on vector-valued functions \(f= (f_1, \dots, f_k)\) such that
(i) the coefficients of \(P_j^m\) are continuous in \(\Omega\), and those of \(P^0_j\) are bounded in \(\Omega\),
(ii) the matrix \(\{P_{ij} (x, i\xi) \}^{j= 1,\dots, N}_{i= 1,\dots, k}\) has rank \(k\) for any \(\xi\neq (0, \dots, 0)\) with complex \(\xi_i\) and \(x\in \Omega\).
Then there exists a constant \(C\) such that for any \(f\in W_\rho^{m,p} (\Omega)\), \[ |\nabla^m f|_{L^p_\rho (\Omega)}\leq C \Biggl\{ |f|_{L^p_\rho (\Omega)}+ \sum_{j=1}^N |P_j f|_{L^p_\rho (\Omega)} \Biggr\}. \]


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B45 A priori estimates in context of PDEs
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